Representations of *-Algebras

by Unbounded Operators:

C*-Hulls, Local–Global Principle, and Induction

Ralf Meyer

Received: September 29, 2016 Revised: July 27, 2017

Communicated by Joachim Cuntz

Abstract. We define a C^{∗}-hull for a ^{∗}-algebra, given a notion of integrability for its representations on Hilbert modules. We establish a local–global principle which, in many cases, characterises integrable representations on Hilbert modules through the integrable representations on Hilbert spaces. The induction theorem constructs a C^{∗}-hull for a certain class of integrable representations of a graded ^{∗}-algebra, given a C^{∗}-hull for its unit fibre.

2010 Mathematics Subject Classification: Primary 47L60; Secondary 46L55 Keywords and Phrases: unbounded operator; regular Hilbert module operator; integrable representation; induction of representations; graded ^{∗}-algebra; Fell bundle; C^{∗}-algebra generated by unbounded operators; C^{∗}-envelope; C^{∗}-hull; host algebra; Weyl algebra; canonical commutation relations; Local–Global Principle; Rieffel deformation

1 Introduction

Savchuk and Schmüdgen [27] have introduced a method to define and classify the integrable representations of certain^{∗}-algebras by an inductive construction. The original goal of this article was to clarify this method and thus make it apply to more situations. This has led me to reconsider some foundational aspects of the theory of representations of ^{∗}-algebras by unbounded operators. This is best explained by formulating an induction theorem that is inspired by [27].

Let G be a discrete group with unit element e ∈ G. Let A = ⨁ _{g∈G} A_{g} be a G-graded unital ^{∗}-algebra. That is, A_{g} · A_{h} ⊆ A_{gh}, A_{g}^{∗} = A_{g^{−1}} , and 1 ∈ A_{e}. In particular, the unit fibre A_{e} is a unital ^{∗}-algebra. Many interesting examples of this situation are studied in [7, 27]. A Fell bundle over G is a family of subspaces (B_{g})_{g∈G} of a C^{∗}-algebra B (which is not part of the data) such that B_{g} · B_{h} ⊆ B_{gh} and B_{g}^{∗} = B_{g^{−1}} . The universal choice for B is the section C^{∗}-algebra of the Fell bundle. Briefly, our main result says the following. Let B_{e} be a C^{∗}-algebra such that “integrable” “representations” of A_{e} are “equivalent” to “representations” of B_{e}. Under some technical conditions, we construct a Fell bundle (B_{g}^{+} )_{g∈G} over G such that “integrable” “representations” of A are “equivalent” to “representations” of its section C^{∗}-algebra. Here the words in quotation marks must be interpreted carefully to make this true. A representation of a ^{∗}-algebra A on a Hilbert D-module E is an algebra homomorphism π from A to the algebra of D-module endomorphisms of a dense D-submodule 𝔈 ⊆ E with ⟨ξ, π(a)η⟩ = ⟨π(a^{∗})ξ, η⟩ for all ξ, η ∈ 𝔈, a ∈ A. The representation induces a graph topology on 𝔈. We restrict to closed representations most of the time, that is, we require 𝔈 to be complete in the graph topology. The difference from usual practice is that we consider representations on Hilbert modules over C^{∗}-algebras. A representation of a C^{∗}-algebra B on a Hilbert module E is a nondegenerate ^{∗}-homomorphism B → 𝔹(E), where 𝔹(E) denotes the C^{∗}-algebra of adjointable operators on E. The notion of “integrability” for representations is a choice. The class of all Hilbert space representations of a ^{∗}-algebra may be quite wild. Hence it is customary to limit the study to some class of “nice” or “integrable” representations. For instance, for the universal enveloping algebra of the Lie algebra of a Lie group G, we may call those representations “integrable” that come from a unitary representation of G. This example suggests the name “integrable” representations. We fix a notion of integrability for representations of A_{e} ⊆ A on all Hilbert modules over all C^{∗}-algebras. A representation of A is called integrable if its restriction to A_{e} is integrable. The induction theorem describes the integrable representations of A in terms of integrable representations of A_{e}. For instance, if A_{e} is finitely generated and commutative, then we may call a representation π on a Hilbert module integrable if the closure \overline{π(a)} is a regular, self-adjoint operator for each a ∈ A_{e} with a = a^{∗}. All examples in [7, 27] are of this type. An “equivalence” between the integrable representations of a unital^{∗}-algebra A and the representations of a C^{∗}-algebra B is a family of bijections – one for each Hilbert module E over each C^{∗}-algebra D – between the sets of integrable representations of A and of representations of B on E; these bijections must be compatible with isometric intertwiners and interior tensor products. These properties require some more definitions. First, an isometric intertwiner between two representations is a Hilbert module isometry – not necessarily adjointable – between the underlying Hilbert modules that restricts to a left module map between the domains of the representations. For an equivalence between integrable representations of A and representations of B we require an isometry to intertwine two representations of B if and only if it intertwines the corresponding integrable representations of A. Secondly, a C^{∗}-correspondence from D_{1} to D_{2} is a Hilbert D_{2}-module F with a representation of D_{1}. Given such a correspondence and a Hilbert D_{1}-module E, the interior tensor product E ⊗_{D_{1}} F is a Hilbert D_{2}-module. A representation of A or B on E induces a representation on E ⊗_{D_{1}} F. We require our bijections between integrable representations of A and representations of B to be compatible with this interior tensor product construction on representations. We call B a C^{∗}-hull for the integrable representations of A if the integrable representations of A are equivalent to the representations of B as explained above, that is, through a family of bijections compatible with isometric intertwiners and interior tensor products. The Induction Theorem builds a C^{∗}-hull for the integrable representations of A using a C^{∗}-hull for the integrable representations of A_{e} and assuming a further mild technical condition, which we explain below. Many results of the general theory remain true if we only require the equivalence of representations to be compatible with interior tensor products and unitary ^{∗}-intertwiners, that is, isomorphisms of representations; we speak of a weak C^{∗}-hull in this case. The Induction Theorem, however, fails for weak C^{∗}-hulls. We show this by a counterexample. Some results only need the class of integrable representations to have some properties that are clearly necessary for the existence of a C^{∗}-hull or weak C^{∗}-hull, but they do not need the (weak) C^{∗}-hull itself. This is formalised in our notions of admissible and weakly admissible classes of representations. For example, let A be commutative. Let Â be the space of characters of A with the topology of pointwise convergence. If Â is locally compact and A is countably generated, then C_{0}(Â) is a C^{∗}-hull for the integrable representations of A as defined above, that is, those representations where each \overline{π(a)} for a ∈ A with a = a^{∗} is regular and self-adjoint. If, say, A = ℂ[x] with x = x^{∗}, then the C^{∗}-hull is C_{0}(ℝ). Here the equivalence of representations maps an integrable representation π of ℂ[x] to the functional calculus homomorphism for the regular, self-adjoint operator \overline{π(x)}. If Â is not locally compact, then the integrable representations of A defined above still form an admissible class, but they have no C^{∗}-hull. If, say, A is the algebra of polynomials in countably many variables, then Â = ℝ^{∞} , which is not locally compact. The problem of associating C^{∗}-algebras to this ^{∗}-algebra has recently been studied by Grundling and Neeb [12]. From our point of view, this amounts to choosing a smaller class of “integrable” representations that does admit a C^{∗}-hull. We have now explained the terms in quotation marks in our Induction Theorem and how we approach the representation theory of ^{∗}-algebras. Most previous work focused either on representations on Hilbert spaces or on single unbounded operators on Hilbert modules. Hilbert module representations occur both in the assumptions and in the conclusions of the Induction Theorem, and hence we cannot prove it without them. In addition, taking into account Hilbert module representations makes our C^{∗}-hulls unique. Besides the Induction Theorem, the other main strand of this article are Local– Global Principles, which aim at reducing the study of integrability for representations on general Hilbert modules to representations on Hilbert space. We may use a state ω on the coefficient C^{∗}-algebra D of a Hilbert module E to complete E to a Hilbert space. Thus a representation of A on E induces Hilbert space representations for all states on D. The Local–Global Principle says that a representation of A on E is integrable if and only if these induced Hilbert space representations are integrable for all states; the Strong Local–Global Principle says the same with all states replaced by all pure states. We took these names from [15]. Earlier results of Pierrot [21] show that the Strong Local–Global Principle holds for any class of integrable representations that is defined by certain types of conditions, such as the regularity and self-adjointness of \overline{π(a)} for certain a ∈ A with a = a^{∗}. For instance, this covers the integrable representations of commutative ^{∗}-algebras and universal enveloping algebras. In all examples that we treat, the regularity of \overline{π(a)} for certain a ∈ A is part of the definition of an integrable representation. Other elements of A may, however, act by irregular operators in some integrable representations. Thus affiliation and regularity are important to study the integrable representations in concrete examples, but cannot play a foundational role for the general representation theory of ^{∗}-algebras. If B is generated in the sense of Woronowicz [32] by some self-adjoint, affiliated multipliers that belong to A, then it is a C^{∗}-hull and the Strong Local–Global Principle holds (see Theorem 5.19). A counterexample shows that this theorem breaks down if the generating affiliated multipliers are not self-adjoint: both the Local–Global Principle and compatibility with isometric intertwiners fail in the counterexample. So regularity without self-adjointness seems to be too weak for many purposes. The combination of regularity and self-adjointness is an easier notion than regularity alone. A closed operator T is regular and self-adjoint if and only if T − λ is surjective for all λ ∈ ℂ \ ℝ, if and only if the Cayley transform of T is unitary, if and only if T has a functional calculus homomorphism on C_{0}(ℝ). Now we describe the Fell bundle in the Induction Theorem and, along the way, the further condition besides compatibility with isometric intertwiners that it needs. Our input data is a graded ^{∗}-algebra A = ⨁ _{g∈G} A_{g} and a C^{∗}-hull B_{e} for A_{e}. A representation of A is integrable if its restriction to A_{e} is integrable. We seek a C^{∗}-hull for the integrable representations of A. As in [27], we induce representations from A_{e} to A, and this requires a positivity condition. We call representations of A_{e} that may be induced to A inducible. We describe a quotient C^{∗}-algebra B_{e}^{+} of B_{e} that is a C^{∗}-hull for the inducible, integrable representations of A. It is the unit fibre of our Fell bundle. If a representation π of A is integrable, then its restriction to A_{e} is integrable and inducible. Thus it corresponds to a representation π̄_{e}^{+} of B_{e}^{+} . The identity correspondence on B_{e}^{+} corresponds to a particular (“universal”) inducible, integrable representation of A_{e} on B_{e}^{+} . Its domain is a dense right ideal 𝔅_{e}^{+} ⊆ B_{e}^{+} . The operators π(a)π̄_{e}^{+} (b) on E for a ∈ A_{g}, b ∈ 𝔅_{e}^{+} are adjointable. Their closed linear span is the fibre B_{g}^{+} of our Fell bundle at g provided π_{e}^{+} is faithful. The most difficult point is to prove B_{e}^{+} · B_{g}^{+} = B_{g}^{+} for all g ∈ G. This easily implies B_{g}^{+} · B_{h}^{+} ⊆ B_{gh}^{+} and (B_{g}^{+} )^{∗} = B_{g^{−1}}^{+} , so that the subspaces B_{g}^{+} ⊆ 𝔹(E) form a Fell bundle. To prove B_{e}^{+} ·B_{g}^{+} = B_{g}^{+} , we need compatibility with isometric intertwiners and that induction maps inducible, integrable representations of A_{e} to integrable representations of A. Two counterexamples show that both assumptions are necessary for the Induction Theorem. Fell bundles are noncommutative partial dynamical systems. More precisely, a Fell bundle (B_{g}^{+} )_{g∈G} over G is equivalent to an action of G on B_{e}^{+} by partial Morita–Rieffel equivalences; this is made precise in [4]. In the examples in [7, 27], the group G is almost always ℤ; the C^{∗}-algebras B_{e} and hence B_{e}^{+} are commutative; and the resulting Fell bundle comes from a partial action of G on the spectrum of B_{e}^{+} . In these examples, the section C^{∗}-algebra is a partial crossed product. This may also be viewed as the groupoid C^{∗}-algebra of the transformation groupoid for the partial action of G on the spectrum of B_{e}^{+} . We show that the C^{∗}-hull B for the integrable representations of A is a twisted groupoid C^{∗}-algebra of this transformation groupoid whenever B_{e} is commutative. We give some criteria when the twist is absent, and examples where the twist occurs. One way to insert such twists is by Rieffel deformation, using a 2-cocycle on the group G. We show that Rieffel deformation is compatible with the construction of C^{∗}-hulls. We describe commutative and noncommutative C^{∗}-hulls for the polynomial algebra ℂ[x] in §4 and §6; the noncommutative C^{∗}-hulls for ℂ[x] make very good counterexamples. We classify and study commutative C^{∗}-hulls in §8. Many results about them generalise easily to locally bounded representations. Roughly speaking, these are representations where the vectors on which the representation acts by bounded operators form a core. The only ^{∗}-algebras for which we treat locally bounded representations in some detail are the commutative ones. Through the Induction Theorem, the representation theory of commutative ∗-algebras is important even for noncommutative algebras because they may admit a grading by some group with commutative unit fibre. Many examples of this are treated in detail in [7, 27]. We discuss untwisted and twisted Weyl algebras in finitely and infinitely many generators in §13. The twists involved are Rieffel deformations. Since these examples have commutative unit fibres, the resulting C^{∗}-hulls are twisted groupoid C^{∗}-algebras. As it turns out, all twists of the relevant groupoids are trivial, so that the twists do not change the representation theory of the Weyl algebras up to equivalence. I am grateful to Yuriy Savchuk for several discussions, which led me to pursue this project and eliminated mistakes from early versions of this article. And I am grateful to the referee as well for several useful suggestions.

2 Representations by unbounded operators on Hilbert modules

Let A be a unital ^{∗}-algebra, D a C^{∗}-algebra, and E a Hilbert D-module. Our convention is that inner products on Hilbert spaces and Hilbert modules are linear in the second and conjugate-linear in the first variable.

Definition 2.1. A representation of A on E is a pair (𝔈, π), where 𝔈 ⊆ E is a dense D-submodule and π: A → End_{D}(𝔈) is a unital algebra homomorphism to the algebra of D-module endomorphisms of 𝔈, such that

⟨π(a)ξ, η⟩_{D} = ⟨ξ, π(a^{∗})η⟩_{D} for all a ∈ A, ξ, η ∈ 𝔈.

We call 𝔈 the domain of the representation. We may drop π from our notation by saying that 𝔈 is an A, D-bimodule with the right module structure inherited from E, or we may drop 𝔈 because it is the common domain of the partial linear maps π(a) on E for all a ∈ A. We equip 𝔈 with the graph topology, which is generated by the graph norms

∥ξ∥_{a} := ∥(ξ, π(a)ξ)∥ := ∥⟨ξ, ξ⟩ + ⟨π(a)ξ, π(a)ξ⟩∥^{^{1}/2} = ∥⟨ξ, π(1 + a^{∗}a)ξ⟩∥^{^{1}/2}

for a ∈ A. The representation is closed if 𝔈 is complete in this topology. A core for (𝔈, π) is an A, D-subbimodule of 𝔈 that is dense in 𝔈 in the graph topology.

Definition 2.1 for D = ℂ is the usual definition of a representation of a^{∗}-algebra on a Hilbert space by unbounded operators. This situation has been studied extensively (see, for instance, [28]). For E = D with the canonical Hilbert D-module structure, we get representations of A by densely defined unbounded multipliers. The domain of such a representation is a dense right ideal 𝔇 ⊆ D. This situation is a special case of the “compatible pairs” defined by Schmüdgen [29]. Given two norms p, q, we write p ⪯ q if there is a scalar c > 0 with p ≤ cq.

Lemma 2.2. The set of graph norms partially ordered by ⪯ is directed: for all a_{1}, …, a_{n} ∈ A there are b ∈ A and c ∈ ℝ_{>0} so that ∥ξ∥_{a_{i}} ≤ c∥ξ∥_{b} for any representation (𝔈, π), any ξ ∈ 𝔈, and i = 1, …, n.

Proof. Let b = ∑ _{j=1}^{n} a_{j}^{∗}a_{j}. The following computation implies ∥ξ∥_{a_{i}} ≤ ^{5}/4∥ξ∥_{b}:

0 ≤ ⟨ξ, π(1 + a_{i}^{∗}a_{i})ξ⟩

≤ ⟨ξ, π(1 + a_{i}^{∗}a_{i})ξ⟩ + ∑_{i≠j} ⟨π(a_{j})ξ, π(a_{j})ξ⟩ + ⟨π(b −^{1}/2)ξ, π(b −^{1}/2)ξ⟩

= ⟨ξ, π(1 + b + (b −^{1}/2)^{2})ξ⟩ = ⟨ξ, π(^{5}/4 + b^{2})ξ⟩ ≤ ^{5}/4⟨ξ, π(1 + b^{∗}b)ξ⟩.

Definition 2.3 ([20], [17, Chapter 9]). A densely defined operator t on a Hilbert module E is semiregular if its adjoint is also densely defined. It is regular if it is closed, semiregular and 1 + t^{∗}t has dense range. An affiliated multiplier of a C^{∗}-algebra D is a regular operator on D viewed as a Hilbert D-module.

The closability assumption in [20, Definition 2.1.(ii)] is redundant by [15, Lemma 2.1]. Regularity was introduced by Baaj and Julg [1], affiliation by Woronowicz [31]. Remark 2.4. Let (𝔈, π) be a representation of A on E and let a ∈ A. The operator π(a) is automatically semiregular because π(a)^{∗} is defined on 𝔈. The closure \overline{π(a)} of π(a) need not be regular. The regularity of \overline{π(a)} for some a ∈ A is often assumed in the definition of integrable representations. For noncommutative A, we should expect that \overline{π(a)} is irregular for some a ∈ A even if π is integrable. For instance, a remark after Corollaire 1.27 in [21] says that this happens for certain symmetric elements in the universal enveloping algebra U(𝔤) for a simply connected Lie group G: they act by irregular operators in certain representations that integrate to unitary representations of G. The usual norm on E is the graph norm for 0 ∈ A. Hence the inclusion map 𝔈 ↪ E is continuous for the graph topology on 𝔈 and extends continuously to the completion \overline{𝔈} of 𝔈 in the graph topology.

Proposition 2.5. The canonical map \overline{𝔈} → E is injective, and its image is

\overline{𝔈} = ⋂_{a∈A} dom \overline{π(a)}.  (2.6)

Thus (𝔈, π) is closed if and only if 𝔈 = ⋂ _{a∈A} dom \overline{π(a)}. Each π(a) extends uniquely to a continuous operator \overline{π(a)} on \overline{𝔈}. This defines a closed representation (\overline{𝔈}, \overline{π}) of A, called the closure of (𝔈, π).

Proof. The operator π(a) for a ∈ A is semiregular and hence closable by [15, Lemma 2.1]. Equivalently, the canonical map from the completion of 𝔈 in the graph norm for a to E is injective. Its image is dom \overline{π(a)}, the domain of the closure of π(a). The graph norms for a ∈ A form a directed set that defines the graph topology on 𝔈. So the completion of 𝔈 in the graph topology is the projective limit of the graph norm completions for a ∈ A. Since each of these graph norm completions embeds into E, the projective limit in question is just an intersection in E, giving (2.6). For Hilbert space representations, this is [28, Proposition 2.2.12]. The operators π(a) ∈ End_{D}(𝔈) for a ∈ A are continuous in the graph topology. Thus they extend uniquely to continuous linear operators \overline{π}(a) ∈ End_{D}(\overline{𝔈}). These are again D-linear and the map \overline{π} is linear and multiplicative because extending operators to a completion is additive and functorial. The set of (ξ, η) ∈ \overline{𝔈} × \overline{𝔈} with ⟨ξ, \overline{π}(a)η⟩ = ⟨\overline{π}(a^{∗})ξ, η⟩ for all a ∈ A is closed in the graph topology and contains 𝔈×𝔈, which is dense in \overline{𝔈}×\overline{𝔈}. Hence this equation holds for all ξ, η ∈ \overline{𝔈}. So (\overline{𝔈}, \overline{π}) is a representation of A on E. The graph topology on \overline{𝔈} for \overline{π} extends the graph topology on 𝔈 for π and hence is complete. So (\overline{𝔈}, \overline{π}) is a closed representation.

We shall need a generalisation of (2.6) that replaces A by a sufficiently large subset.

Definition 2.7. A subset S ⊆ A is called a strong generating set if it generates A as an algebra and the graph norms for a ∈ S generate the graph topology in any representation. That is, for any representation on a Hilbert module, any vector ξ in its domain and any a ∈ A, there are c ≥ 1 in ℝ and b_{1}, …, b_{n} ∈ S with ∥ξ∥_{a} ≤ c ∑ _{i=1}^{n} ∥ξ∥_{b_{i}}.

An estimate ∥ξ∥_{a} ≤ c ∑ _{i=1}^{n} ∥ξ∥_{b_{i}} is usually shown by finding d_{1}, …, d_{m} ∈ A with a^{∗}a + ∑ _{j=1}^{m} d_{j}^{∗}d_{j} = c · ∑ _{i=1}^{n} b_{i}^{∗}b_{i}, compare the proof of Lemma 2.2. Example 2.8. Let A_{h} := {a ∈ A | a = a^{∗}} be the set of symmetric elements. Call an element of A positive if it is a sum of elements of the form a^{∗}a. The positive elements and, a fortiori, the symmetric elements form strong generating sets for A. Any element is of the form a_{1} + ia_{2} with a_{1}, a_{2} ∈ A_{h}, and

a = ( \frac{a + 1}{2} ) ^{2} − ( \frac{a − 1}{2} ) ^{2}

for a ∈ A_{h}. Thus the positive elements generate A as an algebra. The graph norms for positive elements generate the graph topology by the proof of Lemma 2.2.

Proposition 2.9. Let S ⊆ A be a strong generating set. Two closed representations (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) of A on the same Hilbert module E are equal if and only if \overline{π_{1}(a)} = \overline{π_{2}(a)} for all a ∈ S.

Proof. One direction is trivial. To prove the non-trivial direction, assume \overline{π_{1}(a)} = \overline{π_{2}(a)} for all a ∈ S. Let (𝔈, π) = (𝔈_{i}, π_{i}) for i = 1, 2. The completion of 𝔈 for the graph norm of a is dom \overline{π(a)}, compare the proof of Proposition 2.5. Hence the completion of 𝔈 in the sum of graph norms ∑ _{k=1}^{n} ∥ξ∥_{b_{k}} for b_{1}, …, b_{n} ∈ S is ⋂ _{k=1}^{n} dom \overline{π(b_{k})}. These sums of graph norms for b_{1}, …, b_{n} ∈ S form a directed set that generates the graph topology on 𝔈. Hence

\overline{𝔈} = ⋂_{a∈S} dom \overline{π(a)},  (2.10)

compare the proof of (2.6). So 𝔈_{1} = 𝔈_{2}. Moreover, π_{1}(a) = \overline{π_{1}(a)}|_{𝔈_{1}} = \overline{π_{2}(a)}|_{𝔈_{2}} = π_{2}(a) for all a ∈ S. Since S generates A as an algebra and π_{i}(A)𝔈_{i} ⊆ 𝔈_{i}, this implies π_{1} = π_{2}.

Proposition 2.9 may fail for generating sets that are not strong, see Example 4.2.

Corollary 2.11. Let S be a strong generating set of A and let (𝔈, π) be a closed representation of A with dom \overline{π(a)} = E for each a ∈ S. Then 𝔈 = E and π is a ^{∗}-homomorphism to the C^{∗}-algebra 𝔹(E) of adjointable operators on E.

Proof. Equation (2.10) gives 𝔈 = E. Since π(a^{∗}) ⊆ π(a)^{∗} and π(a^{∗}) is defined everywhere, it is adjoint to π(a). So π(a) ∈ 𝔹(E) and π is a ^{∗}-homomorphism to 𝔹(E).

Lemma 2.12. Let A be a unital C^{∗}-algebra. Any closed representation of A on E has domain 𝔈 = E and is a unital ^{∗}-homomorphism to 𝔹(E).

Proof. Let a ∈ A. There are a positive scalar C > 0 and b ∈ A with a^{∗}a+b^{∗}b = C; say, take C = ∥a∥^{2} and b = \sqrt{C − a^{∗}a} . Then

⟨π(a)ξ, π(a)ξ⟩ ≤ ⟨π(a)ξ, π(a)ξ⟩ + ⟨π(b)ξ, π(b)ξ⟩ = ⟨ξ, π(a^{∗}a + b^{∗}b)ξ⟩ = C⟨ξ, ξ⟩

for all ξ ∈ 𝔈. Thus the graph topology on 𝔈 is equivalent to the norm topology on E. Hence E = 𝔈 for any closed representation.

An isometry I: E_{1} ↪ E_{2} between two Hilbert D-modules E_{1} and E_{2} is a right D-module map with ⟨Iξ_{1}, Iξ_{2}⟩ = ⟨ξ_{1}, ξ_{2}⟩ for all ξ_{1}, ξ_{2} ∈ E_{1}.

Definition 2.13. Let (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) be representations on Hilbert D-modules E_{1} and E_{2}, respectively. An isometric intertwiner between them is an isometry I: E_{1} ↪ E_{2} with I(𝔈_{1}) ⊆ 𝔈_{2} and I ∘ π_{1}(a)(ξ) = π_{2}(a) ∘ I(ξ) for all a ∈ A, ξ ∈ 𝔈_{1}. Equivalently, I ∘ π_{1}(a) ⊆ π_{2}(a) ∘ I for all a ∈ A, that is, the graph of π_{2}(a) ∘ I contains the graph of I ∘ π_{1}(a). We neither ask I to be adjointable nor I(𝔈_{1}) = 𝔈_{2}. Let Rep(A, D) be the category with closed representations of A on Hilbert D-modules as objects, isometric intertwiners as arrows, and the usual composition. The unit arrow on (𝔈, π) is the identity operator on E.

Lemma 2.14. Let (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) be representations on Hilbert D-modules E_{1} and E_{2}, respectively, and let I: E_{1} ↪ E_{2} be an isometric intertwiner. Then I is also an intertwiner between the closures of (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}).

Proof. Since I intertwines the representations, it is continuous for the graph topologies on 𝔈_{1} and 𝔈_{2}. Hence I maps the domain of the closure \overline{π_{1}} into the domain of \overline{π_{2}}. This extension is still an intertwiner because it is an intertwiner on a dense subspace.

Proposition 2.15. Let (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) be closed representations of A on Hilbert D-modules E_{1} and E_{2}, respectively. Let S ⊆ A be a strong generating set. An isometry I: E_{1} ↪ E_{2} is an intertwiner from (𝔈_{1}, π_{1}) to (𝔈_{2}, π_{2}) if and only if I ∘ \overline{π_{1}(a)} ⊆ \overline{π_{2}(a)} ∘ I for all a ∈ S.

Proof. First let I satisfy I ∘ \overline{π_{1}(a)} ⊆ \overline{π_{2}(a)} ∘ I for all a ∈ S. Then I maps the domain of \overline{π_{1}(a)} into the domain of \overline{π_{2}(a)} for each a ∈ S. Now (2.10) implies I(𝔈_{1}) ⊆ 𝔈_{2}. Since π_{i}(a) = \overline{π_{i}(a)}|_{𝔈_{i}}, we get I(π_{1}(a)(ξ)) = π_{2}(a)(I(ξ)) for all a ∈ S, ξ ∈ 𝔈_{1}. Since S generates A as an algebra and π_{i}(A)𝔈_{i} ⊆ 𝔈_{i}, this implies I ∘ π_{1}(a) = π_{2}(a) ∘ I for all a ∈ A, that is, I is an intertwiner. Conversely, assume that I is an intertwiner from (𝔈_{1}, π_{1}) to (𝔈_{2}, π_{2}). Equivalently, I ∘ π_{1}(a) ⊆ π_{2}(a) ∘ I for all a ∈ A. We have I ∘ \overline{π_{1}(a)} = \overline{I ∘ π_{1}(a)} because I is an isometry, and \overline{π_{2}(a) ∘ I} ⊆ \overline{π_{2}(a)} ∘ I. Thus I ∘ \overline{π_{1}(a)} ⊆ \overline{π_{2}(a)} ∘ I for all a ∈ A.

Now we relate the categories Rep(A, D) for different C^{∗}-algebras D.

Definition 2.16. Let D_{1} and D_{2} be two C^{∗}-algebras. A C^{∗}-correspondence from D_{1} to D_{2} is a Hilbert D_{2}-module with a representation of D_{1} by adjointable operators (representations of C^{∗}-algebras are tacitly assumed nondegenerate). An isometric intertwiner between two correspondences from D_{1} to D_{2} is an isometric map on the underlying Hilbert D_{2}-modules that intertwines the left D_{1}-actions. Let Rep(D_{1}, D_{2}) denote the category of correspondences from D_{1} to D_{2} with isometric intertwiners as arrows and the usual composition.

By Lemma 2.12, our two definitions of Rep(A, D) for unital ^{∗}-algebras and C^{∗}-algebras coincide if A is a unital C^{∗}-algebra. So our notation is not ambiguous. There is no need to define representations of a non-unital ^{∗}-algebra A because we may adjoin a unit formally. A representation of A extends uniquely to a representation of the unitisation Ã. Thus the nondegenerate representations of A are contained in Rep(Ã). To get rid of degenerate representations, we may require nondegeneracy on A when defining the integrable representations of Ã, compare Example 5.13. Let E be a Hilbert D_{1}-module and F a correspondence from D_{1} to D_{2}. The interior tensor product E ⊗_{D_{1}} F is the (Hausdorff) completion of the algebraic tensor product E ⊙ F to a Hilbert D_{2}-module, using the inner product

⟨ξ_{1} ⊗ η_{1}, ξ_{2} ⊗ η_{2}⟩ = ⟨η_{1}, ⟨ξ_{1}, ξ_{2}⟩_{D_{1}} · η_{2}⟩_{D_{2}} ,  (2.17)

see the discussion around [17, Proposition 4.5] for more details. We may use the balanced tensor product E ⊙_{D_{1}} F instead of E ⊙ F because the inner product (2.17) descends to this quotient. If we want to emphasise the left action φ: D_{1} → 𝔹(F) in the C^{∗}-correspondence F, we write E ⊗_{φ} F for E ⊗_{D_{1}} F. In addition, let (𝔈, π) be a closed representation of A on E. We are going to build a closed representation (𝔈 ⊗_{D_{1}} F, π ⊗_{D_{1}} 1) of A on E ⊗_{D_{1}} F. First let X ⊆ E ⊗_{D_{1}} F be the image of 𝔈 ⊙ F or 𝔈 ⊙_{D_{1}} F under the canonical map to E ⊗_{D_{1}} F.

Lemma 2.18. For a ∈ A, there is a unique linear operator π(a) ⊗ 1: X → X with (π(a)⊗1)(ξ ⊗η) = π(a)(ξ)⊗η for all ξ ∈ 𝔈, η ∈ F. The map a ↦ π(a)⊗1 is a representation of A with domain X.

Proof. Write ω, ζ ∈ X as images of elements of 𝔈 ⊙ F:

ω = ∑_{i=1}^{n} ξ_{i} ⊗ η_{i}, ζ = ∑_{j=1}^{m} α_{j} ⊗ β_{j}

with ξ_{i}, α_{j} ∈ 𝔈, η_{i}, β_{j} ∈ F for 1 ≤ i ≤ n, 1 ≤ j ≤ m. Then

⟨ ζ, ∑_{i=1}^{n} π(a)ξ_{i} ⊗ η_{i}⟩ = ⟨ ∑_{j=1}^{m} π(a^{∗})α_{j} ⊗ β_{j} , ω⟩ .  (2.19)

An element ω′ ∈ E ⊗_{D_{1}} F is determined uniquely by its inner products ⟨ζ, ω′⟩ = 0 for all ζ ∈ X because X is dense in E⊗_{D_{1}} F. The right hand side in (2.19) does not depend on how we decomposed ω. Hence (π(a) ⊗ 1)ω := ∑ _{i=1}^{n} π(a)ξ_{i} ⊗ η_{i} well-defines an operator π(a) ⊗ 1: X → X. This is a right D_{2}-module map, and a ↦ π(a) ⊗ 1 is linear and multiplicative because π is. Equation (2.19) says that ⟨ζ, (π(a) ⊗ 1)ω⟩ = ⟨(π(a^{∗}) ⊗ 1)ζ, ω⟩ for all ω, ζ ∈ X. Thus π ⊗ 1 is a representation.

Definition 2.20. Let (𝔈 ⊗_{D_{1}} F, π ⊗_{D_{1}} 1) be the closure of the representation on E ⊗_{D_{1}} F defined in Lemma 2.18.

Lemma 2.21. Let I: E_{1} ↪ E_{2} be an isometric intertwiner between two representations (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}), and let J: F_{1} ↪ F_{2} be an isometric intertwiner of C^{∗}-correspondences. Then I ⊗_{D_{1}} J: E_{1} ⊗_{D_{1}} F_{1} ↪ E_{2} ⊗_{D_{1}} F_{2} is an isometric intertwiner between (𝔈_{1} ⊗_{D_{1}} F_{1}, π_{1} ⊗ 1) and (𝔈_{2} ⊗_{D_{1}} F_{2}, π_{2} ⊗ 1).

Proof. The isometry I ⊗_{D_{1}} J maps the image X_{1} of 𝔈_{1} ⊙ F_{1} to the image X_{2} of 𝔈_{2} ⊙ F_{2} and intertwines the operators π_{1}(a) ⊗ 1 on X_{1} and π_{2}(a) ⊗ 1 on X_{2} for all a ∈ A. That is, it intertwines the representations defined in Lemma 2.18. It also intertwines their closures by Lemma 2.14.

The lemma gives a bifunctor

⊗_{D_{1}} : Rep(A, D_{1}) × Rep(D_{1}, D_{2}) → Rep(A, D_{2}).  (2.22)

The corresponding bifunctor

⊗_{D_{1}} : Rep(B, D_{1}) × Rep(D_{1}, D_{2}) → Rep(B, D_{2})

for a C^{∗}-algebra B is the usual composition of C^{∗}-correspondences. This composition is associative up to canonical unitaries

E ⊗_{D_{1}} (F ⊗_{D_{2}} G) ⟶^{∼} (E ⊗_{D_{1}} F) ⊗_{D_{2}} G,  ξ ⊗ (η ⊗ ζ) ↦ (ξ ⊗ η) ⊗ ζ, (2.23)

for all triples of composable C^{∗}-correspondences.

Lemma 2.24. If E carries a representation (𝔈, π) of a ^{∗}-algebra A, then the unitary in (2.23) is an intertwiner (𝔈, π)⊗_{D_{1}} (F⊗_{D_{2}} G) ⟶^{∼} ((𝔈, π)⊗_{D_{1}} F)⊗_{D_{2}} G.

Proof. The bilinear map from 𝔈 × F to 𝔈 ⊗_{D_{1}} F is separately continuous with respect to the graph topologies on 𝔈 and 𝔈 ⊗_{D_{1}} F and the norm topology on F. Since the image of F ⊙ G in the Hilbert module F ⊗_{D_{2}} G is dense in the norm topology, the image of 𝔈 ⊙ F ⊙ G in E ⊗_{D_{1}} (F ⊗_{D_{2}} G) is a core for the representation (𝔈, π) ⊗_{D_{1}} (F ⊗_{D_{2}} G). Since the image of 𝔈 ⊙ F in 𝔈 ⊗_{D_{1}} F is dense in the graph topology, the image of 𝔈⊙F ⊙G in (E ⊗_{D_{1}} F)⊗_{D_{2}} G is a core for the representation ((𝔈, π) ⊗_{D_{1}} F) ⊗_{D_{2}} G. The unitary in (2.23) intertwines between these cores. Hence it also intertwines between the resulting closed representations by Lemma 2.14.

Definition 2.25. Let (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) be two representations of A on Hilbert D-modules E_{1} and E_{2}. An adjointable operator x: E_{1} → E_{2} is an intertwiner if x(𝔈_{1}) ⊆ 𝔈_{2} and xπ_{1}(a)ξ = π_{2}(a)xξ for all a ∈ A, ξ ∈ 𝔈_{1}. It is a ∗-intertwiner if both x and x^{∗} are intertwiners.

Any adjointable intertwiner between two representations of a C^{∗}-algebra B is a ∗-intertwiner. In contrast, for a general ^{∗}-algebra, even the adjoint of a unitary intertwiner u fails to be an intertwiner if u(𝔈_{1}) ⊊ 𝔈_{2}.

Example 2.26. Let t be a positive symmetric operator on a Hilbert space H. Assume that ⋂ _{n∈ℕ} domt^{n} is dense in H, so that t generates a representation π of the polynomial algebra ℂ[x] on H. The Friedrichs extension of t is a positive self-adjoint operator t′ on H. It generates another representation π′ of ℂ[x] on H. The identity map on H is a unitary intertwiner π ↪ π′. It is not a ∗-intertwiner unless t = t′.

The following proposition characterises when an adjointable isometry I: E_{1} ↪ E between two representations on Hilbert D-modules is a ^{∗}-intertwiner. Since E ≅ E_{1} ⊕ E_{1}^{⊥} if I is adjointable, we may as well assume that I is the inclusion of a direct summand.

Proposition 2.27. Let E_{1} and E_{2} be Hilbert modules over a C^{∗}-algebra D and let (𝔈_{1}, π_{1}) and (𝔈, π) be representations of A on E_{1} and E_{1} ⊕ E_{2}, respectively. The following are equivalent:

(1) the canonical inclusion I: E_{1} ↪ E_{1} ⊕ E_{2} is a ^{∗}-intertwiner from π_{1} to π;

(2) the canonical inclusion I: E_{1} ↪ E_{1} ⊕ E_{2} is an intertwiner from π_{1} to π and 𝔈 = 𝔈_{1} + (𝔈 ∩ E_{2});

(3) there is a representation (𝔈_{2}, π_{2}) on E_{2} such that π = π_{1} ⊕ π_{2}.

Proof. We view E_{1} and E_{2} as subspaces of E_{1}⊕E_{2}, so we may drop the isometry I from our notation. The implication (3)⇒(1) is trivial. We are going to prove (1)⇒(2)⇒(3). First assume that I is a ^{∗}-intertwiner. Then I is an intertwiner. In particular, 𝔈_{1} ⊆ 𝔈. Write ξ ∈ 𝔈 as ξ = ξ_{1} + ξ_{2} with ξ_{1} ∈ E_{1}, ξ_{2} ∈ E_{2}. Since I^{∗} is an intertwiner, ξ_{1} = I^{∗}(ξ) ∈ 𝔈_{1}. Hence ξ_{2} = ξ − ξ_{1} ∈ 𝔈 ∩ E_{2}. Thus (1) implies (2). If (2) holds, then 𝔈_{1} ⊆ 𝔈 is π-invariant and π|_{𝔈_{1}} = π_{1} because I is an intertwiner. We claim that 𝔈_{2} := 𝔈 ∩ E_{2} is π-invariant as well. Let ξ ∈ 𝔈_{2} and η ∈ 𝔈_{1}. Then ⟨η, π(a)ξ⟩ = ⟨π(a^{∗})η, ξ⟩ = ⟨π_{1}(a^{∗})η, ξ⟩ = 0 because π_{1}(a^{∗})η ∈ E_{1} is orthogonal to E_{2}. Since 𝔈_{1} is dense in E_{1}, this implies π(a)ξ ∈ E_{1}^{⊥} = E_{2}, and this implies our claim. The condition (2) implies 𝔈 = 𝔈_{1} ⊕ 𝔈_{2} as a vector space with 𝔈_{2} = E_{2} ∩ 𝔈 because E_{1} ∩ E_{2} = {0}. Then 𝔈_{2} is dense in E_{2} because 𝔈 is dense in E_{1} ⊕ E_{2}. Thus (𝔈_{2}, π|_{𝔈_{2}}) is a representation of A on E_{2}. And (𝔈, π) is the direct sum of (𝔈_{1}, π_{1}) and (𝔈_{2}, π|_{𝔈_{2}}) because 𝔈 = 𝔈_{1} ⊕ 𝔈_{2} and π_{1} = π|_{𝔈_{1}}. Thus (2) implies (3).

3 Integrable representations and C^{∗}-hulls

From now on, we tacitly assume representations to be closed. Proposition 2.5 shows that this is no serious loss of generality. Let A be a unital^{∗}-algebra. We assume that a class of “integrable” (closed) representations of A on Hilbert modules is chosen. Let Rep_{int}(A, D) ⊆ Rep(A, D) be the full subcategory with integrable representations on Hilbert D-modules as objects. Being full means that the set of arrows between two integrable representations of A is still the set of all isometric intertwiners. We sometimes write Rep_{int}(A) and Rep(A) for the collection of all the categories Rep_{int}(A, D) and Rep(A, D) for all C^{∗}-algebras D. A C^{∗}-hull is a C^{∗}-algebra B with natural isomorphisms Rep(B, D) ≅ Rep_{int}(A, D) for all C^{∗}-algebras D. More precisely:

Definition 3.1. A C^{∗}-hull for the integrable representations of A is a C^{∗}-algebra B with a family of bijections Φ = Φ^{E} from the set of representations of B on E to the set of integrable representations of A on E for all Hilbert modules E over all C^{∗}-algebras D with the following properties:

• compatibility with isometric intertwiners: an isometry E_{1} ↪ E_{2} (not necessarily adjointable) is an intertwiner between two representations ϱ_{1} and ϱ_{2} of B if and only if it is an intertwiner between Φ(ϱ_{1}) and Φ(ϱ_{2});

• compatibility with interior tensor products: if F is a correspondence from D_{1} to D_{2}, E is a Hilbert D_{1}-module, and ϱ is a representation of B on E, then Φ(ϱ ⊗_{D_{1}} 1_{F}) = Φ(ϱ) ⊗_{D_{1}} 1_{F} as representations of A on E ⊗_{D_{1}} F.

The compatibility with isometric intertwiners means that the bijections Φ for all E with fixed D form an isomorphism of categories Rep(B, D) ≅ Rep_{int}(A, D) which, in addition, does not change the underlying Hilbert D-modules. The compatibility with interior tensor products expresses that these isomorphisms of categories for different D are natural with respect to C^{∗}-correspondences.

Definition 3.2. A weak C^{∗}-hull for the integrable representations of A is a C^{∗}-algebra B with a family of bijections Φ between representations of B and integrable representations of A on Hilbert modules that is compatible with unitary ^{∗}-intertwiners and interior tensor products.

Much of the general theory also works for weak C^{∗}-hulls. But the Induction Theorem 9.26 fails for weak C^{∗}-hulls, as shown by a counterexample in §9.6.

Proposition 3.3. Let a class of integrable representations of A have a weak C^{∗}-hull B. Let (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) be integrable representations of A on Hilbert D-modules E_{1} and E_{2}, and let ϱ_{i} be the corresponding representations of B on E_{i} for i = 1, 2. An adjointable operator x: E_{1} → E_{2} is a ^{∗}-intertwiner from (𝔈_{1}, π_{1}) to (𝔈_{2}, π_{2}) if and only if it is an intertwiner from ϱ_{1} to ϱ_{2}.

Proof. Working with the direct sum representations on E_{1} ⊕ E_{2} and the adjointable operator ( \begin{matrix}0 & x \\ 0 & 0 \end{matrix} ), we may assume without loss of generality that E_{1} = E_{2} = E, 𝔈_{1} = 𝔈_{2} = 𝔈, π_{1} = π_{2} = π, and ϱ_{1} = ϱ_{2} = ϱ. The adjointable intertwiners for the representation ϱ of B form a C^{∗}-algebra B′: the commutant of B in 𝔹(E). We claim that the ^{∗}-intertwiners for the representation π of A also form a C^{∗}-algebra A′. Intertwiners and hence ^{∗}-intertwiners form an algebra. Thus A′ is a ^{∗}-algebra. We show that it is closed. Let (x_{i})_{i∈ℕ} be a sequence of adjointable intertwiners for (𝔈, π) that converges in norm to x ∈ 𝔹(E). Let ξ ∈ 𝔈. Then x_{i}(ξ) ∈ 𝔈 because each x_{i} is an intertwiner. Since π(a)(x_{i}ξ) = x_{i}π(a)ξ is norm-convergent for each a ∈ A, the sequence x_{i}(ξ) is a Cauchy sequence for the graph topology on 𝔈. Since representations are tacitly assumed to be closed, this Cauchy sequence converges in 𝔈, so that x(𝔈) ⊆ 𝔈. Moreover, x(π(a)ξ) = π(a)x(ξ) for all a ∈ A, ξ ∈ 𝔈, so x is again an intertwiner. Thus the algebra of intertwiners is norm-closed. This implies that A′ is a C^{∗}-algebra. Since the family of bijections Rep_{int}(A) ≅ Rep(B) is compatible with unitary ∗-intertwiners, a unitary operator on E is a^{∗}-intertwiner for A if and only if it is an intertwiner for B. That is, the unital C^{∗}-subalgebras A′, B′ ⊆ 𝔹(E) contain the same unitaries. A unital C^{∗}-algebra is the linear span of its unitaries because any self-adjoint element t of norm at most 1 may be written as

t = ^{1}/2(t + i\sqrt{1 − t^{2}} ) +^{1}/2(t − i\sqrt{1 − t^{2}} )

and t ± i\sqrt{1 − t^{2}} are unitary. Thus A′ = B′. This is what we had to prove.

Corollary 3.4. Let Rep_{int}(A) have a weak C^{∗}-hull B. Direct sums and summands of integrable representations remain integrable, and the family of bijections Rep_{int}(A) ≅ Rep(B) preserves direct sums.

Proof. Let π_{1}, π_{2} be representations of A on Hilbert D-modules E_{1}, E_{2}. Let S_{i}: E_{i} ↪ E_{1} ⊕ E_{2} for i = 1, 2 be the inclusion maps. First we assume that π_{1}, π_{2} are integrable. Let ϱ_{i} be the representation of B on E_{i} corresponding to π_{i}, and let π be the integrable representation of A on E_{1} ⊕ E_{2} corresponding to the representation ϱ_{1} ⊕ ϱ_{2} of B. The isometries S_{i} are intertwiners from ϱ_{i} to ϱ_{1} ⊕ ϱ_{2}. By Proposition 3.3, they are ^{∗}-intertwiners from π_{i} to π. Hence π = π_{1} ⊕ π_{2} by Proposition 2.27. Thus π_{1} ⊕ π_{2} is integrable and the family of bijections Rep_{int}(A) ≅ Rep(B) preserves direct sums. The same argument works for infinite direct sums. Now we assume instead that π_{1} ⊕ π_{2} is integrable. Let ϱ be the representation of B corresponding to π_{1} ⊕ π_{2}. The orthogonal projection onto E_{1} is a ∗-intertwiner on the representation π_{1} ⊕ π_{2} by Proposition 2.27, and hence also on ϱ by Proposition 3.3. Thus ϱ = ϱ_{1} ⊕ ϱ_{2} for some representations ϱ_{i} of B on E_{i}. Let π′_{i} be the integrable representation of A corresponding to ϱ_{i}. The isometry S_{i} is a ^{∗}-intertwiner from ϱ_{i} to ϱ_{1} ⊕ϱ_{2} and hence from π′_{i} to π_{1} ⊕π_{2} by Proposition 3.3. This implies π′_{i} = π_{i}, so that π_{i} is integrable for i = 1, 2.

Definition 3.5. Let B be a weak C^{∗}-hull for A. The universal integrable representation of A is the integrable representation (𝔅, μ) of A on B that corresponds to the identity representation of B on itself.

Proposition 3.6. Let B with a family of bijections Φ between representations of B and integrable representations of A on Hilbert modules be a weak C^{∗}-hull for the integrable representations of A. Let (𝔅, μ) be the universal integrable representation of A. Then Φ(E) ≅ (𝔅, μ) ⊗_{B} E for any C^{∗}-correspondence E from B to D. (The proof makes this isomorphism more precise.)

Proof. Let ϱ: B → 𝔹(E) be a representation of B on a Hilbert module E. Then u: B ⊗_{ϱ} E ⟶^{∼} E, b ⊗ ξ ↦ ϱ(b)ξ, is a unitary ^{∗}-intertwiner between the interior tensor product of the identity representation of B with E and the representation ϱ on E. As Φ is compatible with interior tensor products and unitary ^{∗}-intertwiners, u is a unitary ^{∗}-intertwiner between (𝔅, μ) ⊗_{B} E and Φ(ϱ). Therefore, the image u(𝔅⊙E) = ϱ(𝔅)E is a core for Φ(ϱ), and a ∈ A acts on this core by a ↦ u(μ(a) ⊗ 1)u^{∗} or, explicitly, a · (ϱ(b)ξ) = ϱ(μ(a)b)ξ for all a ∈ A, b ∈ 𝔅, ξ ∈ E.

Put in a nutshell, the whole isomorphism between integrable representations of A and representations of B is encoded in the single representation (𝔅, μ) of A on B. This is similar to Schmüdgen’s approach in [29]. In the following, we disregard the canonical unitary u in the proof of Proposition 3.6 and write Φ(ϱ) = (𝔅, μ) ⊗_{B} E. A (weak) C^{∗}-hull B does not solve the problem of describing the integrable representations of A. It only reduces it to the study of the representations of the C^{∗}-algebra B. This reduction is useful because it gets rid of unbounded operators. If B is of type I, then any Hilbert space representation of B is a direct integral of irreducible representations, and irreducible representations may, in principle, be classified. Thus integrable Hilbert space representations of A are direct integrals of irreducible integrable representations, and the latter may, in principle, be classified. But if B is not of type I, then the integrable Hilbert space representations of A are exactly as complicated as the Hilbert space representations of B, and giving the C^{∗}-algebra B may well be the best one can say about them.

Proposition 3.7. A class of integrable representations has at most one weak C^{∗}-hull.

Proof. Let B_{1} and B_{2} be weak C^{∗}-hulls for the same class of integrable representations of A. The identity map on B_{1}, viewed as a representation of B_{1} on itself, corresponds first to an integrable representation of A on B_{1} and further to a representation of B_{2} on B_{1}. This is a “morphism” from B_{2} to B_{1}, that is, a nondegenerate ^{∗}-homomorphism B_{2} → M(B_{1}). Similarly, we get a morphism from B_{1} to B_{2}. These morphisms B_{1} ↔ B_{2} are inverse to each other with respect to the composition of morphisms because the maps they induce on representations of B_{1} and B_{2} on B_{1} and B_{2} are inverse to each other. An isomorphism in the category of morphisms is an isomorphism of C^{∗}-algebras in the usual sense by [6, Proposition 2.10].

Now take any representation (𝔅, μ) of A on B. When is this the universal integrable representation of a (weak) C^{∗}-hull? Let D be a C^{∗}-algebra and E a Hilbert D-module. For a representation ϱ: B → 𝔹(E), let Φ(ϱ) = (𝔅, μ) ⊗_{ϱ} E be the induced representation of A on E as in the proof of Proposition 3.6. A representation of A is called B-integrable if it is in the image of Φ.

Proposition 3.8. The C^{∗}-algebra B is a weak C^{∗}-hull for the B-integrable representations of A if and only if

(1) if two representations ϱ_{1}, ϱ_{2} : B ⇉ 𝔹(H) on the same Hilbert space H satisfy μ ⊗_{B} ϱ_{1} = μ ⊗_{B} ϱ_{2} as closed representations of A, then ϱ_{1} = ϱ_{2}.

It is a C^{∗}-hull if and only if (1) and the following equivalent conditions hold:

(2) Let (ℌ, π) be a representation of A on a Hilbert space H and let (ℌ_{0}, π_{0}) be a subrepresentation on a closed subspace H_{0} ⊆ H; that is, ℌ_{0} ⊆ ℌ and π_{0}(a) = π(a)|_{ℌ_{0}} for all a ∈ A. If both π_{0} and π are B-integrable, then ℌ = ℌ_{0} ⊕ (ℌ ∩ H_{0}^{⊥} ) as vector spaces.

(3) Isometric intertwiners between B-integrable Hilbert space representations of A are ^{∗}-intertwiners.

(4) B-integrable subrepresentations of B-integrable Hilbert space representations of A are direct summands.

The conditions (1)–(4) together are equivalent to

(5) let ϱ: B → 𝔹(H) be a Hilbert space representation and let (ℌ, π) be the associated representation of A on H. If (ℌ_{0}, π|_{ℌ_{0}}) is a B-integrable subrepresentation of (ℌ, π) on a closed subspace H_{0} ⊆ H, then the projection onto H_{0} commutes with ϱ(B).

Proof. The map Φ is compatible with interior tensor products by Lemma 2.24. The condition (1) says that Φ is injective on Hilbert space representations. We claim that this implies injectivity also for representations on a Hilbert module E over a C^{∗}-algebra D. Let ϱ_{1}, ϱ_{2} be representations of B on E with μ ⊗_{B} ϱ_{1} = μ ⊗_{B} ϱ_{2}. Let D → 𝔹(H) be a faithful representation. Then the representations ϱ_{1} ⊗_{D} 1 and ϱ_{2} ⊗_{D} 1 on the Hilbert space E ⊗_{D} H satisfy μ ⊗_{B} ϱ_{1} ⊗_{D} 1 = μ ⊗_{B} ϱ_{2} ⊗_{D} 1 by Lemma 2.24. Then condition (1) implies ϱ_{1} ⊗_{D} 1 = ϱ_{2} ⊗_{D} 1. Since the representation 𝔹(E) → 𝔹(E ⊗_{D} H) is faithful, this implies ϱ_{1} = ϱ_{2}. So Φ is injective also for representations on E. The image of Φ consists exactly of the B-integrable representations of A by definition. A unitary operator u ^{∗}-intertwines two representations (𝔈_{1}, π_{1}) and (𝔈_{2}, π_{2}) of A if and only if π_{2} = uπ_{1}u^{∗}, where uπ_{1}u^{∗} denotes the representation with domain u(𝔈_{1}) and (uπ_{1}u^{∗})(a) = uπ_{1}(a)u^{∗}. Similarly, u intertwines two representations ϱ_{1} and ϱ_{2} of B if and only if ϱ_{2} = uϱ_{1}u^{∗}. Hence (1) implies that a unitary that ^{∗}-intertwines two B-integrable representations of A also intertwines the corresponding representations of B. The converse is clear. So B is a weak C^{∗}-hull for the B-integrable representations if and only if (1) holds.

The equivalence between (2), (3) and (4) follows from Proposition 2.27 by writing H = H_{0} ⊕ H_{0}^{⊥} . Assume that B is a C^{∗}-hull. An isometric intertwiner for A is also one for B. Then it is a ^{∗}-intertwiner for A and its range projection is an intertwiner for B by Proposition 3.3. Thus both (3) and (5) follow if B is a C^{∗}-hull. Conversely, assume (1) and (3). We are going to prove that B is a C^{∗}-hull for the B-integrable representations of A. We have already seen that B is a weak C^{∗}-hull. We must check compatibility with isometric intertwiners. Let D be a C^{∗}-algebra and let E_{1}, E_{2} be Hilbert D-modules with representations ϱ_{1}, ϱ_{2} of B. The corresponding representations (𝔈_{i}, π_{i}) of A for i = 1, 2 are the closures of the representations on ϱ_{i}(𝔅)E_{i} given by π_{i}(a)(ϱ_{i}(b)ξ) := ϱ_{i}(μ(a)b)(ξ) for a ∈ A, b ∈ 𝔅, ξ ∈ E_{i}. Hence an isometric intertwiner for B is also one for A. Conversely, let I: E_{1} ↪ E_{2} be a Hilbert module isometry with I(𝔈_{1}) ⊆ 𝔈_{2} and π_{2}(a)(Iξ) = I(π_{1}(a)ξ) for all a ∈ A, ξ ∈ 𝔈_{1}. We must prove ϱ_{2}(b)I = Iϱ_{1}(b) for all b ∈ B. Let φ: D ↪ 𝔹(K) be a faithful representation on a Hilbert space K. Equip H_{i} := E_{i} ⊗_{φ} K with the induced representations ϱ̃_{i} of B and π̃_{i} of A for i = 1, 2. Since the family of bijections Φ: Rep(B) ⟶^{∼} Rep_{int}(A) is compatible with interior tensor products, it maps ϱ̃_{i} to π̃_{i}. The operator I induces an isometric intertwiner Ĩ from π̃_{1} to π̃_{2} by Lemma 2.21. Since π̃_{1} and π̃_{2} are B-integrable, we are in the situation of (3). So Ĩ is a ^{∗}-intertwiner from π̃_{1} to π̃_{2}. Thus Ĩ is an intertwiner from ϱ̃_{1} to ϱ̃_{2} by Proposition 3.3. That is, Ĩϱ̃_{1}(b) = ϱ̃_{2}(b)Ĩ for all b ∈ B. Equivalently, (Iϱ_{1}(b)ξ) ⊗ η = (ϱ_{2}(b)Iξ) ⊗ η in E_{2} ⊗_{φ} H for all b ∈ B, ξ ∈ E, η ∈ H. Since the representation φ is faithful, this implies Iϱ_{1}(b)ξ = ϱ_{2}(b)Iξ for all b, ξ, so that Iϱ_{1}(b) = ϱ_{2}(b)I for all b, that is, I intertwines ϱ_{1} and ϱ_{2}. Thus Φ is compatible with isometric intertwiners. Since (5) holds for C^{∗}-hulls, we have proved along the way that (1) and (3) imply (5). It remains to show, conversely, that (5) implies (3) and (1). In the situation of (3), the projection P onto H_{0} commutes with B by (5). Thus the representation of B on H is a direct sum of representations on H_{0} and H_{0}^{⊥} . This is inherited by the induced representation of A and its domain. So (5) implies (3). In the situation of (1), form the direct sum representation ϱ_{1} ⊕ ϱ_{2} on H ⊕ H and let H_{0} := {(ξ, ξ) | ξ ∈ H}. The representation of A corresponding to ϱ_{1} ⊕ ϱ_{2} is μ ⊗_{ϱ_{1}} H ⊕ μ ⊗_{ϱ_{2}} H. Since μ ⊗_{ϱ_{1}} H = μ ⊗_{ϱ_{2}} H by assumption, the domain of μ ⊗_{ϱ_{1}} H ⊕ μ ⊗_{ϱ_{2}} H is ℌ ⊕ ℌ for some dense subspace ℌ ⊆ H, and ℌ_{0} := {(ξ, ξ) | ξ ∈ ℌ} is a dense subspace in H_{0} that is invariant for the representation μ ⊗_{ϱ_{1}} H ⊕ μ ⊗_{ϱ_{2}} H. The restricted representation on this subspace is B-integrable because it is unitarily equivalent to μ⊗_{ϱ_{1}} H = μ⊗_{ϱ_{2}} H. Therefore, the projection onto H_{0} commutes with the representation of B by (5). Thus ϱ_{1} = ϱ_{2}. So (5) implies (1).

The equivalent conditions (2)–(4) may be easier to check than (5) because they do not involve the C^{∗}-hull.

Corollary 3.9. Let A be a ^{∗}-algebra and let B_{i} be C^{∗}-algebras with representations (𝔅_{i}, μ_{i}) of A for i = 1, 2. Assume that for each Hilbert space H, the maps Φ_{i}: Rep(B_{i}, H) → Rep(A, H), ϱ_{i} ↦ (𝔅_{i}, μ_{i})⊗_{ϱ_{i}} H, are injective and have the same image. Then there is a unique isomorphism B_{1} ≅ B_{2} intertwining the representations (𝔅_{i}, μ_{i}) of A for i = 1, 2.

Hence a C^{∗}-envelope as defined in [7] is unique if it exists.

Proof. Both B_{1} and B_{2} are weak C^{∗}-hulls for the same class of representations of A by Proposition 3.8. Proposition 3.7 gives the isomorphism B_{1} ≅ B_{2}.

Remark 3.10. The Hilbert space representations of a C^{∗}-algebra only determine its bidual W^{∗}-algebra, not the C^{∗}-algebra itself. Hence it is remarkable that the conditions in Proposition 3.8 and Corollary 3.9 only need Hilbert space representations. For Corollary 3.9, this works because the bijection between the representations is of a particular form, induced by representations of A.

The condition (1) in Proposition 3.8 is required in several other theories that associate a C^{∗}-algebra to a^{∗}-algebra, such as the host algebras of Grundling [10, 11], the C^{∗}-envelopes of Dowerk and Savchuk [7], or the notion of a C^{∗}-algebra generated by affiliated multipliers by Woronowicz [32], see [32, Theorem 3.3] or the proof of Theorem 5.19 below.

Definition 3.11. Let A be a ^{∗}-algebra. A class of “integrable” representations of A on Hilbert modules over C^{∗}-algebras is admissible if it satisfies the conditions (1)–(4) below, and weakly admissible if it satisfies (1)–(3).

(1) If there is a unitary ^{∗}-intertwiner from an integrable representation to another representation, then the latter is integrable.

(2) If D and D′ are C^{∗}-algebras, F is a correspondence from D to D′, and (𝔈, π) is an integrable representation of A on a Hilbert D-module E, then the representation (𝔈, π) ⊗_{D} F on E ⊗_{D} F is integrable.

(3) Direct sums and summands of integrable representations are integrable.

(4) Any integrable subrepresentation of an integrable representation of A on a Hilbert space is a direct summand.

Lemma 3.12. Any class of integrable representations with a (weak) C^{∗}-hull is (weakly) admissible.

Proof. If there is a C^{∗}-hull, Proposition 3.8 implies (4) in Definition 3.11. If there is a weak C^{∗}-hull, then (1) and (2) in Definition 3.11 follow from the compatibility with unitary ^{∗}-intertwiners and interior tensor products in the definition of a C^{∗}-hull, and (3) follows from Corollary 3.4.

Proposition 3.13. Let A be a unital ^{∗}-algebra and let E be a Hilbert module over a C^{∗}-algebra D. There is a natural bijection between the sets of representations of A on E and 𝕂(E). It preserves integrability if the class of integrable representations of A is weakly admissible or, in particular, if it has a weak C^{∗}-hull.

Proof. We may view E as an imprimitivity bimodule between 𝕂(E) and the ideal I in D that is spanned by the inner products ⟨ξ, η⟩_{D} for ξ, η ∈ E. Let E^{∗} be the inverse imprimitivity bimodule, which is a Hilbert module over 𝕂(E) with 𝕂(E^{∗}) ≅ I. Then 𝕂(E) ≅ E ⊗_{D} E^{∗} and E^{∗} ⊗_{𝕂(E)} E = I. If (π, 𝔈) is a representation of A on E, then (π, 𝔈) ⊗_{D} E^{∗} is a representation of A on E ⊗_{D} E^{∗} = 𝕂(E). This maps Rep(A, E) to Rep(A, 𝕂(E)). If (ϱ, 𝔎) is a representation of A on 𝕂(E), then (ϱ, 𝔎) ⊗_{𝕂(E)} E is a representation of A on 𝕂(E) ⊗_{𝕂(E)} E ≅ E. This maps Rep(A, 𝕂(E)) to Rep(A, E). We claim that these two maps are inverse to each other. Both preserve integrability by (2) in Definition 3.11. The map Rep(A, E) → Rep(A, 𝕂(E)) → Rep(A, E) sends a representation (π, 𝔈) of A on E to the representation (π, 𝔈) ⊗_{D} (E^{∗} ⊗_{𝕂(E)} E) = (π, 𝔈) ⊗_{D} I of A on E ≅ E ⊗_{D} I by Lemma 2.24. This is the restriction of π to 𝔈 · I ⊆ 𝔈. Since E is also a Hilbert module over I, it is nondegenerate as a right I-module. Therefore, if (u_{i}) is an approximate unit in I, then limξu_{i} = ξ for all ξ ∈ E. Then also limπ(a)ξu_{i} = π(a)ξ for all ξ ∈ 𝔈, a ∈ A, so limξu_{i} = ξ in the graph topology for all ξ ∈ 𝔈. Thus 𝔈·I = 𝔈, and we get the identity map on Rep(A, E). A similar, easier argument shows that we also get the identity map on Rep(A, 𝕂(E)).

4 Polynomials in one variable I

Let A = ℂ[x] with x = x^{∗}. A (not necessarily closed) representation of A on a Hilbert D-module E is determined by a dense D-submodule 𝔈 ⊆ E and a single symmetric operator π(x): 𝔈 → 𝔈, that is, π(x) ⊆ π(x)^{∗}. Then π(x^{n}) = π(x)^{n}.

Lemma 4.1. The graph topology on 𝔈 is generated by the increasing sequence of norms ∥ξ∥_{n} := ∥⟨ξ, (1 + π(x^{2n}))ξ⟩∥ for n ∈ ℕ.

Proof. We must show that for any a ∈ ℂ[x] there are C > 0 and n ∈ ℕ with ∥ξ∥_{a} ≤ C∥ξ∥_{n}. We choose n so that a has degree at most n. Then there is C > 0 so that C(1 + t^{2n}) > 1 + |a(t)|^{2} for all t ∈ ℝ. Thus the polynomial b := C(1 + x^{2n}) − (1 + a^{∗}a) is positive on ℝ. So the zeros of b are complex and come in pairs λ_{j} ± iμ_{j} for j = 1, …, n with λ_{j} , μ_{j} ∈ ℝ by the Fundamental Theorem of Algebra. Then b = ∏ _{j=1}^{n} ((x − λ_{j})^{2} + μ_{j}^{2}) = ∑ _{k=1}^{2^{n}} b_{k}^{2}, where each b_{k} is a product of either x − λ_{j} or μ_{j} for j = 1, …, k, so b_{k} = b_{k}^{∗}. Thus ∥ξ∥_{a} ≤ C∥ξ∥_{n}.

Thus the monomials {x^{n} | n ∈ ℕ} form a strong generating set for ℂ[x]. A representation of ℂ[x] is determined by the closed operators \overline{π(x^{n})} for n ∈ ℕ by Proposition 2.9. In contrast, it is not yet determined by the single closed operator \overline{π(x)} because \{x\} is not a strong generating set:

Example 4.2. We construct a closed representation of ℂ[x] on a Hilbert space with \overline{π(x^{2})} ⊊ (\overline{π(x)})^{2}. Let H := L^{2}(𝕋), viewed as the space of ℤ-periodic functions on ℝ. Let ℌ_{0} := C^{∞} (𝕋) and let π_{0} : ℂ[x] → End(ℌ_{0}) be the polynomial functional calculus for the operator i\frac{d}{dt}. The graph topology generated by this representation of ℂ[x] is the usual Fréchet topology on C^{∞} (𝕋). So the representation of ℂ[x] on C^{∞} (𝕋) is closed. Now for some λ ∈ 𝕋, let

ℌ := {f ∈ C^{∞} (𝕋) | f^{(n)} (λ) = 0 for all n ≥ 1}.

This is a closed, ℂ[x]-invariant subspace in ℌ_{0}. Let π be the restriction of π_{0} to ℌ_{0}. This is also a closed representation of ℂ[x]. Its domain ℌ is dense in ℌ_{0} in the graph norm of x, but not in the graph norm of x^{2}. So \overline{π(x)} = \overline{π_{0}(x)} and \overline{π(x^{2})} ⊊ \overline{π_{0}(x^{2})} = (\overline{π(x)})^{2}.

All notions of integrability for representations of ℂ[x] that we shall consider imply \overline{π(x^{n})} = \overline{π(x)}^{n}. Under this assumption, an integrable representation of ℂ[x] is determined by the single closed operator \overline{π(x)}. Let B := C_{0}(ℝ). Let X be the identity function on ℝ, viewed as an unbounded multiplier of B. We define a closed representation (𝔅, μ) of A on C_{0}(ℝ) by

𝔅 := {f ∈ B | ∀n: X^{n} · f ∈ B} and μ(x^{n})f := X^{n} · f for f ∈ 𝔅, n ∈ ℕ. (4.3)

Theorem 4.4. Let (𝔈, π) be a representation of A = ℂ[x] on a Hilbert module E over a C^{∗}-algebra D. The following are equivalent:

(1) π = μ ⊗_{ϱ} 1_{E} for a representation ϱ: B → 𝔹(E);

(2) \overline{π(a)} is regular and self-adjoint for each a ∈ A_{h} := \{a ∈ A | a = a^{∗}\};

(3) \overline{π(x^{n})} is regular and self-adjoint for each n ∈ ℕ;

(4) \overline{π(x)} is regular and self-adjoint and \overline{π(x^{n})} = \overline{π(x)}^{n} for all n ∈ ℕ;

(5) \overline{π(x)} is regular and self-adjoint and 𝔈 = ⋂ _{n=1}^{∞} dom \overline{π(x)}^{n}.

Call representations with these equivalent properties integrable. The C^{∗}- algebra C_{0}(ℝ) is a C^{∗}-hull for the integrable representations of A with (𝔅, μ) as the universal integrable representation.

Proof. If a ∈ A_{h}, then \overline{μ(a)} is a self-adjoint, affiliated multiplier of B. Hence μ(a) ⊗_{D} 1 is a regular, self-adjoint operator on B ⊗_{ϱ} E ≅ E for any representation ϱ of B on E by [17, Proposition 9.10]. Thus (1) implies (2). The implication (2)⇒(3) is trivial. The operator \overline{π(x^{n})} is always contained in the n-fold power \overline{π(x)}^{n}. The latter is symmetric, and a proper suboperator of a symmetric operator cannot be self-adjoint. Thus (3) implies (4). The set {x^{n} | n ∈ ℕ} is a strong generating set for ℂ[x] by Lemma 4.1. Equation (2.10) gives 𝔈 = ⋂ _{n=1}^{∞} dom \overline{π(x^{n})} for any (closed) representation. Thus (4) implies (5). Assume (5) and abbreviate t = \overline{π(x)}. The functional calculus for t is a nondegenerate ^{∗}-homomorphism ϱ: C_{0}(ℝ) → 𝔹(E) (see [17, Theorem 10.9]). Let π′ be the representation μ ⊗_{ϱ} 1 of A on E associated to ϱ. We claim that π = π′. The functional calculus extends to affiliated multipliers and maps the identity function on ℝ to the regular, self-adjoint operator t. This means that \overline{π′(x)} = t. Then \overline{π′(x^{n})} ⊆ t^{n}. This implies \overline{π′(x^{n})} = t^{n} because \overline{π′(x^{n})} is self-adjoint and t^{n} is symmetric. Since the set {x^{n} | n ∈ ℕ} is a strong generating set for ℂ[x], the domain of π′ is ⋂ dom \overline{π′(x^{n})} = 𝔈 by condition (5) and (2.10). On this domain, π(x) and π′(x) act by the same operator because they have the same closure. Thus π = π′ and (5) implies (1). So all five conditions in the theorem are equivalent. To show that B is a C^{∗}-hull for the class of representations described in (1), we check (5) in Proposition 3.8. An integrable representation of A on a Hilbert space H corresponds to a self-adjoint operator t on H by (5). An integrable subrepresentation is a closed subspace H_{0} of H with a self-adjoint operator t_{0} on H_{0} whose graph is contained in that of t. Since t_{0} is self-adjoint, the subspaces (t_{0} ± i)(dom(t_{0})) = (t ± i)(dom(t_{0})) are equal to H_{0}. The Cayley transform u of t maps (t + i)(dom(t_{0})) onto (t − i)(dom(t_{0})). Thus it maps H_{0} onto itself. Since u − 1 generates the image of B = C_{0}(ℝ) under the functional calculus, the projection onto H_{0} is B-invariant.

Example 4.5. Regularity and self-adjointness are independent properties of a symmetric operator. Examples of regular symmetric operators that are not selfadjoint are easy to find, see §6. We are going to construct a representation π of ℂ[x] on a Hilbert module for which \overline{π(a)} is self-adjoint for each a ∈ ℂ[x] with a = a^{∗}, but \overline{π(x)} is not regular. We follow the example after Théorème 1.3 in [21], which Pierrot attributes to Hilsum. Let H be the Hilbert space L^{2}([0, 1]) and let T_{1} and T_{2} be the operators i\frac{d}{dx} on H with the following domains. For T_{1}, we take 1-periodic smooth functions; for T_{2}, we take the restrictions to [0, 1] of smooth functions on ℝ satisfying f(x + 1) = −f(x). Both T_{1} and T_{2} are essentially self-adjoint. Let D := C([−1, 1]) and E := C([−1, 1], H). Let 𝔈 ⊆ E be the dense subspace of all functions f: [−1, 1] × [0, 1] → ℂ such that \frac{∂^{n}}{∂^{n} x} f(t, x) is continuous for each n ∈ ℕ,

\frac{∂^{n}}{∂^{n}x}f(t, 1) = sign(t) · \frac{∂^{n}}{∂^{n}x}f(t, 0)  (4.6)

for all t ∈ [−1, 1], x ∈ ℝ, t ≠ 0, and

\frac{∂^{n}}{∂^{n}x}f(0, 0) = \frac{∂^{n}}{∂^{n}x}f(0, 1) = 0.  (4.7)

Equivalently, f(t, ␣) belongs to the domain of \overline{T_{1}}^{n} = \overline{T_{1}^{n}} for all n ∈ ℕ, t ≤ 0 and to the domain of \overline{T_{2}}^{n} = \overline{T_{2}^{n}} for all n ∈ ℕ, t ≥ 0; indeed, this forces \frac{∂^{n}}{∂^{n} x} f to be continuous on [−1, 1] × [0, 1] and to satisfy the boundary conditions (4.6). These imply (4.7) by continuity. Let x^{n} ∈ ℂ[x] act on 𝔈 by (i\frac{d}{dx} )^{n}. This defines a closed ^{∗}-representation of ℂ[x] on E with 𝔈 = ⋂ _{n∈ℕ} dom \overline{π(x)}^{n}. The closure \overline{π(x)} is the irregular self-adjoint operator described in [21]. Let a ∈ ℂ[x] with a = a^{∗}. Then \overline{ϱ(a)} is (regular and) self-adjoint for any integrable representation ϱ of ℂ[x] by Theorem 4.4. Therefore, the restriction of \overline{π(a)} to a single fibre of E at some t ∈ [−1, 1] \ {0} is a self-adjoint operator on L^{2}([0, 1]) because \overline{T_{1}} and \overline{T_{2}} are self-adjoint and 𝔈 = ⋂ _{n∈ℕ} dom \overline{π(x)}^{n}. The restriction of \overline{π(a)}^{∗} at t = 0 is contained in the self-adjoint operators \overline{a(T_{1})} and \overline{a(T_{2})} by continuity. We claim that \overline{a(T_{1})}∩ \overline{a(T_{2})} = \overline{π(a)}|_{t=0} . This claim implies that \overline{π(a)}^{∗} is contained in \overline{π(a)}, that is, \overline{π(a)} is self-adjoint. Let a ∈ ℂ[x] have degree n. Then the graph norms for a and x^{n} are equivalent in any representation by the proof of Lemma 4.1. Hence \overline{a(T_{i})} and \overline{T_{i}^{n}} have the same domain. The domain of \overline{T_{i}^{n}} consists of functions [0, 1] → ℂ whose nth derivative lies in L^{2} and whose derivatives of order strictly less than n satisfy the boundary condition for T_{i}. Hence the domain of \overline{T_{1}^{n}}∩ \overline{T_{2}^{n}} consists of those functions [0, 1] → ℂ whose nth derivative lies in L^{2} and whose derivatives of order strictly less than n vanish at the boundary points 0 and 1. This is exactly the domain of the closure of (T_{1} ∩ T_{2})^{n} = π(x^{n})|_{t=0} . On this domain the operators \overline{a(T_{1})}∩ \overline{a(T_{2})} and \overline{π(a)}|_{t=0} both act by the differential operator a(i\frac{d}{dx}).

The algebra A = ℂ[x] has many Hilbert space representations coming from closed symmetric operators that are not self-adjoint. There is, however, no larger admissible class of integrable representations:

Proposition 4.8. Assume that an admissible class of integrable representations of A = ℂ[x] contains all representations coming from self-adjoint Hilbert space operators. Then any integrable representation of A on a Hilbert module comes from a regular, self-adjoint operator.

Proof. We first prove that there can be no more integrable Hilbert space representations than those coming from self-adjoint operators. Let (ℌ, π) be an integrable representation on a Hilbert space H. We may extend the closed symmetric operator t := \overline{π(x)} on H to a self-adjoint operator t_{2} on a larger Hilbert space H_{2}. This gives a representation π_{2} of A on H_{2} as in Theorem 4.4, which is integrable by assumption. The inclusion map H ↪ H_{2} is an isometric intertwiner from π to π_{2}. Hence π is a direct summand of π_{2} by (4) in Definition 3.11. Thus \overline{π(x^{n})} is self-adjoint for each n ∈ ℕ, and π is the representation induced by t. Now let (𝔈, π) be an integrable representation of A on a Hilbert D-module E. For any Hilbert space representation φ: D → 𝔹(H), the induced representation of A on the Hilbert space E ⊗_{φ} H is also integrable by (2) in Definition 3.11. Thus \overline{π(x^{n}) ⊗_{φ} 1_{H}} is self-adjoint for any Hilbert space representation φ: D → 𝔹(H). A closed, densely defined, symmetric operator T on a Hilbert D-module E is self-adjoint and regular if and only if, for any state ω on D, the closure of T ⊗_{D} 1 on the Hilbert spaces E ⊗_{D} H_{ω} is self-adjoint; here H_{ω} means the GNSrepresentation for ω. This is called the Local–Global Principle by Kaad and Lesch ([15, Theorem 1.1]); the result was first proved by Pierrot ([21, Théorème 1.18]). We will take up Local–Global Principles more systematically in §5.

Thus \overline{π(x^{n})} is regular and self-adjoint for each n ∈ ℕ. So π is obtained from the regular self-adjoint operator \overline{π(x)} as in Theorem 4.4.

Example 4.9. There are many admissible classes of representations of ℂ[x] that are smaller than the class in Theorem 4.4. There are even many such classes that contain the same Hilbert space representations. For instance, let B := C_{0}((−∞, 0))⊕C_{0}([0, ∞)) with the representation of polynomials by pointwise multiplication. This is a C^{∗}-hull for a class of representations of ℂ[x] by Theorem 8.2 below. Since the standard topologies on ℝ and (−∞, 0) ⊔ [0, ∞) have the same Borel sets, both C^{∗}-hulls C_{0}((−∞, 0)) ⊕ C_{0}([0, ∞)) and C_{0}(ℝ) give the same integrable Hilbert space representations because of the Borel functional calculus. But there are regular, self-adjoint operators on Hilbert modules that do not give a B-integrable representation. The obvious example is the multiplier X of C_{0}(ℝ) that generates the universal integrable representation of ℂ[x]. Can there be an admissible class of representations of ℂ[x] that contains some representation on a Hilbert space that does not come from a self-adjoint operator? We cannot rule this out completely. But such a class would have to be rather strange. By Proposition 4.8, it cannot contain all self-adjoint operators. By Example 2.26, it cannot contain all representations coming from positive symmetric operators because then there would be isometric intertwiners among integrable representations that are not ^{∗}-intertwiners. The following example rules out symmetric operators with one deficiency index 0: Example 4.10. Let t be a closed symmetric operator on a Hilbert space H of deficiency indices (0, n) for some n ∈ [1, ∞]. Then dom^{∞} (t) := ⋂ _{n=1}^{∞} dom(t^{n}) is a core for each power t^{k} by [28, Proposition 1.6.1]. Thus there is a closed representation π of ℂ[x] with domain dom^{∞} (t) and \overline{π(x^{k})} = t^{k} for all k ∈ ℕ. By assumption, the operator t + i is surjective, but t − i is not. That is, the Cayley transform c := (t − i)(t + i)^{−1} is a non-unitary isometry. The operator t may be reconstructed from c as in [17, Equation (10.11)]. Here c^{∗} is surjective, so this simplifies to dom(t) = (1 − c)c^{∗}H = (1 − c)H, and t(1 − c)ξ = i(1 + c)ξ for all ξ ∈ H. Thus c(domt) ⊆ domt and ct ⊆ tc because

ct((1 − c)ξ) = ic(1 + c)ξ = i(1 + c)(cξ) = t(1 − c)(cξ) = (tc)((1 − c)ξ).

Then ct^{n} ⊆ t^{n}c for all n ∈ ℕ. Thus c is an isometric intertwiner from π to itself by Proposition 2.15. If c^{∗} were an intertwiner as well, then c^{∗}(domt) ⊆ domt and c^{∗}(t ± i)ξ = (t ± i)c^{∗}ξ for all ξ ∈ dom(t). So

c^{∗}c(t + i)ξ = c^{∗}(t − i)ξ = (t − i)c^{∗}ξ = c(t + i)c^{∗}ξ = cc^{∗}(t + i)ξ.

This is impossible because c^{∗}c ≠ cc^{∗} and t + i is surjective. So the isometry c is an intertwiner, but not a ^{∗}-intertwiner. This is forbidden for admissible classes of integrable representations. If t has deficiency indices (n, 0) instead, then −t has deficiency indices (0, n) and its Cayley transform is an isometric intertwiner that is not a ^{∗}-intertwiner by the argument above.

5 Local–Global principles

Definition 5.1. Let A be a ^{∗}-algebra with a weakly admissible class of integrable representations (Definition 3.11). The Local–Global Principle says that a representation π of A on a Hilbert D-module E is integrable if (and only if) the representations π⊗_{ϱ}1 are integrable for all Hilbert space representations ϱ: D → 𝔹(H). The Strong Local–Global Principle says that a representation π of A on a Hilbert D-module E is integrable if (and only if) the representations π⊗_{ϱ}1 are integrable for all irreducible Hilbert space representations ϱ: D → 𝔹(H).

Roughly speaking, the Local–Global Principle says that the class of integrable representations on Hilbert modules is determined by the class of integrable representations on Hilbert spaces. Examples where the Local–Global Principle fails are constructed in §6 and §8. We do not know an example with the Local–Global Principle for which the Strong Local–Global Principle fails. An irreducible representation ϱ: D → 𝔹(H) is unitarily equivalent to the GNSrepresentation for a pure state ψ on D. The tensor product E⊗_{ϱ}H is canonically isomorphic to the completion E_{ψ} of E to a Hilbert space for the scalar-valued inner product ⟨x, y⟩_{ℂ} := ψ(⟨x, y⟩_{D}). The induced representation π ⊗_{D} 1 of A on H_{ψ} is the closure of the representation π with domain 𝔈 ⊆ E ⊆ E_{ψ} . Any representation ϱ: D → 𝔹(H) is a direct sum of cyclic representations, and these are GNS-representations of states. Since any weakly admissible class of integrable representations is closed under direct sums, the Local–Global Principle holds if and only if integrability of π⊗_{ϱ}1 for all GNS-representations ϱ of states on D implies integrability of π. Example 5.2. Define integrable representations of the polynomial algebra ℂ[x] as in Theorem 4.4. Thus they correspond to regular, self-adjoint operators on Hilbert modules. The main result in [15] says that the integrable representations of ℂ[x] satisfy the Local–Global Principle. This is where our notation comes from. We already used this to prove Proposition 4.8. The Strong Local– Global Principle for integrable representations of ℂ[x] is only conjectured in [15]. This conjecture had already been proved by Pierrot in [21, Théorème 1.18] before [15] was written. It is based on the following Hahn–Banach type theorem for Hilbert submodules:

Theorem 5.3 ([21, Proposition 1.16]). Let D be a C^{∗}-algebra and let E be a Hilbert D-module. Let F ⊊ E be a proper, closed Hilbert submodule. There is an irreducible Hilbert space representation ϱ: D → 𝔹(H) with F ⊗_{ϱ}H ⊊ E ⊗_{ϱ}H.

Corollary 5.4 ([21, Corollaire 1.17]). Let E be a Hilbert module over a C^{∗}-algebra D. Let F_{1}, F_{2} ⊊ E be two closed Hilbert submodules. If F_{1} ≠ F_{2}, then there is an irreducible Hilbert space representation ϱ: D → 𝔹(H) with F_{1} ⊗_{ϱ} H ≠ F_{2} ⊗_{ϱ} H as closed subspaces in E ⊗_{ϱ} H.

Corollary 5.5 ([21, Théorème 1.18]). Let T be a closed, semiregular operator on a Hilbert D-module E. The operator T is regular if and only if, for each irreducible representation ϱ: D → 𝔹(H) on a Hilbert space H, the closures of T ⊗_{ϱ} 1 and T^{∗} ⊗_{ϱ} 1 on E ⊗_{ϱ} H are adjoints of each other. Hence T is regular and self-adjoint if and only if \overline{T ⊗_{ϱ} 1} is a self-adjoint operator on E ⊗_{ϱ} H for each irreducible Hilbert space representation ϱ: D → 𝔹(H).

We now apply the above results of Pierrot. First we deduce a criterion for representations to be equal. Then we prove that certain definitions of integrability automatically satisfy the Strong Local–Global Principle.

Theorem 5.6. Let A be a ^{∗}-algebra and let π_{i} for i = 1, 2 be (closed) representations of A on a Hilbert module E over a C^{∗}-algebra D. The following are equivalent:

(1) π_{1} = π_{2};

(2) π_{1} ⊗_{ϱ} H = π_{2} ⊗_{ϱ} H for each irreducible Hilbert space representation ϱ of D;

(3) \overline{π_{1}(a)} = \overline{π_{2}(a)} for each a ∈ A.

Proof. The equivalence (3) ⟺ (1) is Proposition 2.9, and (1) clearly implies (2). Thus we only have to prove that not (3) implies not (2). Assume that there is a ∈ A with \overline{π_{1}(a)} ≠ \overline{π_{2}(a)}. The graphs Γ_{1} and Γ_{2} of \overline{π_{1}(a)} and \overline{π_{2}(a)} are different Hilbert submodules of E ⊕ E. Corollary 5.4 gives an irreducible representation ϱ of D with Γ_{1} ⊗_{ϱ} H ≠ Γ_{2} ⊗_{ϱ} H. This says that \overline{π_{1}(a) ⊗_{ϱ} 1_{H}} ≠ \overline{π_{2}(a) ⊗_{ϱ} 1_{H}} because Γ_{i} ⊗_{ϱ} H is the graph of \overline{π_{i}(a) ⊗_{ϱ} 1_{H}} .

How do we specify which representations π of a ^{∗}-algebra A are integrable? There are two basically different ways. The “universal way” specifies the universal integrable representation. That is, it starts with a representation (𝔅, μ) on a C^{∗}-algebra B that satisfies (1) in Proposition 3.8 and takes the class of B-integrable representations. The “operator way” imposes conditions on the operators π(a), such as regularity and self-adjointness of \overline{π(a)} or strong commutation relations. In good cases, the same class of integrable representations may be specified in both ways. For instance, Theorem 4.4 shows that several classes of representations of ℂ[x] are equal. The first is defined by the universal representation on C_{0}(ℝ). The second asks \overline{π(a)} to be regular and self-adjoint for all a ∈ A_{h}. We are going to make the “operator way” more precise so that all classes of representations defined in this way satisfy the Strong Local–Global Principle. This is a powerful method to prove Local–Global Principles.

Definition 5.7. Let A be a ^{∗}-algebra and Rep′(A) some weakly admissible class of representations of A on Hilbert modules over C^{∗}-algebras. A natural construction of Hilbert submodules (of rank n ∈ ℕ_{≥1} ) associates to each representation π on a Hilbert module E that belongs to Rep′(A) a Hilbert submodule F(π) ⊆ E^{n}, such that (1) if u: E_{1} ⟶^{∼} E_{2} is a unitary ^{∗}-intertwiner between two representations π_{1}

and π_{2} in Rep′(A), then u^{⊕n} : E_{1}^{n} → E_{2}^{n} maps F(π_{1}) onto F(π_{2});

(2) let D_{1} and D_{2} be C^{∗}-algebras and let G be a D_{1}, D_{2}-correspondence; let π be a representation in Rep′(A) on a Hilbert D_{1}-module E; then the canonical isomorphism E^{n} ⊗_{D_{1}} G ⟶^{∼} (E ⊗_{D_{1}} G)^{n} maps F(π) ⊗_{D_{1}} G onto F(π ⊗_{D_{1}} G);

(3) if π_{i} for i in a set I are representations in Rep′(A) on Hilbert D-modules E_{i} over the same C^{∗}-algebra D, then the canonical isomorphism (⨁ E_{i})^{n} ⟶^{∼} ⨁ E_{i}^{n} maps F(⨁ π_{i}) onto ⨁ F(π_{i}).

In brief, F(π) ⊆ E^{n} is compatible with unitary ^{∗}-interwiners, interior tensor products, and direct sums. A smaller class of representations Rep′′(A) ⊆ Rep′(A) is defined by a submodule condition if there are two natural constructions of Hilbert submodules F_{i}(π), i = 1, 2, of the same rank n, such that a representation π in Rep′(A) belongs to Rep′′(A) if and only if F_{1}(π) = F_{2}(π). A class of representations Rep_{int}(A) ⊆ Rep(A) is defined by submodule conditions if it is defined by transfinite recursion by repeating the step in the previous paragraph. More precisely, there are a well-ordered set I with a greatest element M and least element 0 and subclasses Rep^{i}(A) ⊆ Rep(A) for i ∈ I such that

(1) Rep^{0}(A) = Rep(A) and Rep^{M} (A) = Rep_{int}(A);

(2) Rep^{i+1} (A) ⊆ Rep^{i}(A) is defined by a submodule condition for each i ∈ I;

(3) Rep^{i}(A) = ⋂ _{i′<i} Rep^{i′}(A) if i ≠ 0 and i ≠ i′+ 1 for all i′ ∈ I.

The following lemma makes this definition meaningful, the following theorem makes it interesting.

Lemma 5.8. If Rep′(A) ⊆ Rep(A) is weakly admissible and Rep′′(A) ⊆ Rep′(A) is defined by a submodule condition, then Rep′′(A) is also weakly admissible. If (Rep^{i}(A))_{i∈I} is a set of weakly admissible subclasses, then ⋂ _{i∈I} Rep^{i}(A) is weakly admissible. Any class of representations defined by submodule conditions is weakly admissible.

Theorem 5.9. If Rep_{int}(A) ⊆ Rep(A) is defined by submodule conditions, then it satisfies the Strong Local–Global Principle.

Before we prove these two results, we give examples of classes of representations defined by one or more submodule conditions, and a few counterexamples. These show that a class of integrable representations defined in the operator way is often but not always defined by submodule conditions.

Example 5.10. The regularity condition for a ∈ A_{h} requires \overline{π(a)} to be regular and self-adjoint. Equivalently, the closures of (π(a) ± i)(𝔈) for both signs are dense in E; this is equivalent to \overline{π(a)} having a unitary Cayley transform. Sending π to the image of \overline{π(a) + i} or \overline{π(a) − i} is a natural construction of a Hilbert submodule. Hence the condition that \overline{π(a)} is regular and self-adjoint is equivalent to the combination of two submodule conditions of rank 1. Alternatively, we may proceed as in the definition of regularity for non-self-adjoint operators. Let Γ(T) denote the closure of the graph of an operator T. A closed operator T is regular if and only if the direct sum of Γ(T) and U_{0}(Γ(T^{∗})) is E ⊕ E, where U_{0}(ξ_{1}, ξ_{2}) := (ξ_{2}, −ξ_{1}). If a ∈ A_{h}, then regularity and selfadjointness of \overline{π(a)} together are equivalent to the equality of

F_{1}(π) := Γ(π(a)) ⊕ U_{0}(Γ(π(a^{∗}))) and F_{2}(π) := E ⊕ E.

We claim that F_{1} and F_{2} are natural constructions of Hilbert submodules of rank 2. This is trivial for F_{2}. That F_{1} is compatible with unitary intertwiners and direct sums is an easy exercise. The construction F_{1} is compatible with interior tensor products because the graph of \overline{(π ⊗_{D_{1}} 1_{G})(a)} is Γ(π(a)) ⊗_{D_{1}} G. For instance, (2) in Theorem 4.4 defines integrable representations of ℂ[x] by regularity conditions. We generalise this in Theorem 5.17 below.

Example 5.11. The class of representations where \overline{π(a)} is regular for some a ∈ A is always weakly admissible by [17, Proposition 9.10]. The first example in §6 shows a class of representations defined by such a condition that does not satisfy the Local–Global Principle, in contrast to Theorem 5.9. Hence asking for \overline{π(a)} to be regular for some a ∈ A cannot be a submodule condition. The problem is that the inclusion Γ(π(a)^{∗}) ⊗_{D} 1_{G} ⊆ Γ((π(a) ⊗_{D} 1_{G})^{∗}) for a correspondence G may be strict.

Example 5.12. Let a_{1}, a_{2} ∈ A_{h} and suppose that t_{1} := \overline{π(a_{1})} and t_{2} := \overline{π(a_{2})} are self-adjoint, regular operators for all representations in Rep′(A); we may achieve this by submodule conditions as in Example 5.10 in previous steps of a recursive definition. We say that t_{1} and t_{2} strongly commute if their Cayley transforms u_{1} and u_{2} commute. Equivalently, u_{1} commutes with t_{2}, that is, u_{1}t_{2}u_{1}^{∗} = t_{2}. The graphs of t_{2} and u_{1}t_{2}u_{1}^{∗} are natural constructions of Hilbert submodules of rank 2. Therefore, strong commutation of \overline{π(a_{1})} and \overline{π(a_{2})} is a submodule condition.

Example 5.13. Let I ◁ A be an ideal. A nondegeneracy condition for I asks the closed linear span of π(I)𝔈 to be all of E; here 𝔈 is the domain of π. This means that F_{1}(π), the closed linear span of π(a)ξ for a ∈ I, ξ ∈ 𝔈, is equal to F_{2}(π) = E. These are natural constructions of Hilbert submodules of rank 1. So a nondegeneracy condition is a submodule condition. For instance, let I be a non-unital ^{∗}-algebra and let A = Ĩ be its unitisation. Any representation of I extends uniquely to a unital representation of A. The class of nondegenerate representations of I inside the class of all representations of A is defined by a submodule condition.

More generally, let V_{1}, V_{2} ⊆ A be vector subspaces and ask the closed linear spans of π(a)ξ for a ∈ V_{j}, ξ ∈ 𝔈 to be equal for j = 1, 2. This is a submodule condition as well. For instance, the condition \overline{π(a + i)𝔈} = E for a ∈ A_{h} is of this form. It holds if and only if the Cayley transform of \overline{π(a)} is an isometry (possibly without adjoint). Often we need a mild generalisation of the above construction, see Example 5.14 below. Suppose that we have constructed a representation φ(π) of a unital ∗-algebra A′ on E for any representation π in Rep′(A), such that π ↦ φ(π) is compatible with unitary ^{∗}-intertwiners, direct sums, and interior tensor products; the last property means that φ(π⊗_{D_{1}} 1_{G}) = φ(π)⊗_{D_{1}} 1_{G} as representations on E ⊗_{D_{1}} G. Then we may ask the nondegeneracy condition for an ideal in A′instead. In particular, A′may be a weak C^{∗}-hull for some class of representations containing Rep′(A).

Example 5.14. Let a_{1}, …, a_{n} ∈ A_{h} be commuting, symmetric elements and suppose that \overline{π(a_{j})} for j = 1, …, n are strongly commuting, self-adjoint, regular operators for all representations in Rep′(A); we may achieve all this by previous submodule conditions as in Examples 5.10 and 5.12. A closed spectral condition asks the joint spectrum of \overline{π(a_{1})}, …, \overline{π(a_{n})} to be contained in a closed subset X ⊆ ℝ^{n}. We claim that this is a submodule condition. Under our assumptions, the functional calculus Φ: C_{0}(ℝ^{n}) → 𝔹(E) exists. Our spectral condition means that Φ(C_{0}(ℝ^{n} \ X))E = 0. The construction of Φ is clearly compatible with unitary ^{∗}-intertwiners and direct sums. It is also compatible with interior tensor products, that is, the functional calculus for \overline{π ⊗ 1(a_{1})} , …, \overline{π ⊗ 1(a_{n})} maps f ↦ Φ(f) ⊗ 1. Hence Φ(C_{0}(ℝ^{n} \ X))E is a naturally constructed Hilbert submodule of E. So our spectral condition for closed X ⊆ ℝ^{n} is a submodule condition. More generally, let X ⊆ ℝ^{n} be locally closed, that is, X is relatively open in its closure \overline{X}. Suppose that the spectral condition for \overline{X} holds for all representations in Rep′(A), say, by previous recursion steps. Then the functional calculus homomorphism for \overline{π(a_{1})}, …, \overline{π(a_{n})} exists and descends to C_{0}(\overline{X}). The spectral condition for X asks the restriction of this homomorphism to the ideal C_{0}(X) ◁ C_{0}(\overline{X}) to be nondegenerate. This is a submodule condition by Example 5.13.

Example 5.15 (see [31, §3]). Let A_{μ} for some μ ∈ ℝ\{0} be the unital ^{∗}-algebra generated by two elements v, n with the relations v^{∗}v = vv^{∗} = 1, n^{∗}n = nn^{∗}, v^{∗}nv = μn. This is the algebra of polynomial functions on the quantum group E_{μ}(2). The relations allow to write any element as a linear combination of v^{k} ·g(n, n^{∗}) for k ∈ ℤ and a polynomial g. It follows that the graph topology of a representation of A_{μ} is generated by the graph norms of (n^{∗}n)^{k} for k ∈ ℕ. Thus a representation is closed if and only if its domain is ⋂ _{k=0}^{∞} \overline{π((n^{∗}n)^{k})}, compare the proof of (2.10). The C^{∗}-algebra of E_{μ}(2) is a C^{∗}-hull for a certain class of integrable representations of A_{μ} that is defined by submodule conditions. First, we require \overline{π(n)} to be a regular, normal operator; equivalently, \overline{π(n + n^{∗})} and −i\overline{π(n − n^{∗})} are regular and self-adjoint, and they strongly commute; these are submodule conditions by Examples 5.10 and 5.12. Secondly, we require the spectrum of \overline{π(n)} (or the joint spectrum of its real and imaginary part) to be contained in X_{μ} := {z ∈ ℂ | |z| ∈ μ^{ℤ}} ∪ {0}; this is a submodule condition by Example 5.14. Finally, we require \overline{π(n^{∗}n)^{k}} to be regular and self-adjoint for all k ≥ 1. These are submodule conditions by Example 5.10. We claim that an integrable representation on E is equivalent to a pair (V, N) consisting of a unitary operator V and a regular, normal operator N on E with spectrum contained in X_{μ}, subject to the relation V^{∗}NV = μN. First, any such pair (V, N) gives an integrable representation of A_{μ} with domain ⋂ _{k=0}^{∞} dom(N^{k}). Conversely, if π is an integrable representation, then let N := \overline{π(n)}, V := \overline{π(v)}. These have the properties required above. Since \overline{π((n^{∗}n)^{k})} is self-adjoint and contained in the symmetric operator (N^{∗}N)^{k}, we must have \overline{π((n^{∗}n)^{k})} = (N^{∗}N)^{k}. So the domain of the representation of A_{μ} is ⋂ _{k=0}^{∞} dom(N^{k}). The regular, normal operator N with spectrum in X_{μ} defines a functional calculus ϱ on C_{0}(X_{μ}). The commutation relation v^{∗}nv = μn is equivalent to V^{∗}ϱ(f)V = ϱ(α(f)) for the automorphism α(f)(x) := f(μx) on C_{0}(X_{μ}). As a consequence, the crossed product C^{∗}-algebra C_{0}(X_{μ}) ⋊_{α} ℤ is a C^{∗}-hull for our class of integrable representations. By the way, this also follows from our Induction Theorem. For this, we give A_{μ} the unique ℤ-grading where v has degree 1 and n has degree 0. Then (A_{μ})_{0} = ℂ[n, n^{∗}], and we call a representation of ℂ[n, n^{∗}] integrable if n is regular and normal with spectrum contained in X_{μ}. The C^{∗}-hull for this class of integrable representations of ℂ[n, n^{∗}] is C_{0}(X_{μ}). In this case, all representations of ℂ[n, n^{∗}] are inducible to A_{μ}, and the induced C^{∗}-hull for A_{μ} is C_{0}(X_{μ}) ⋊_{α} ℤ.

Interesting classes of representations defined by submodule conditions occur in Theorems 5.21 and 8.6. The examples in [7,27] are also defined by submodule conditions, compare Proposition 9.4.

Example 5.16. If the algebra A carries a topology, then we may restrict attention to representations of A that are continuous in some sense. For instance, if G is a topological group and A = ℂ[G] is the group ring of the underlying discrete group, then representations of A are unitary representations of G, possibly discontinuous. Among them, we may restrict to the continuous representations (compare the definition of a host algebra for G in [11]). If G is an infinite-dimensional Lie group, we may restrict further to representations of ℂ[G] that are smooth in the sense that the smooth vectors are dense. I do not expect continuity or smoothness to be a submodule condition, and I do not know when the classes of continuous or smooth representations satisfy the Local–Global Principle or its strong variant. Semiboundedness conditions ask for certain (regular) self-adjoint operators to be bounded above, see [19]. If we specify the upper bound on the spectrum, this is a spectral condition as in Example 5.14. When we let the upper bound go to ∞, however, then direct sums no longer preserve semiboundedness. Therefore, semiboundedness conditions seem close enough to submodule conditions to be tractable, but the details require further thought.

Proof of Lemma 5.8. Let F_{1} and F_{2} be natural constructions of Hilbert submodules of rank n that define Rep′′(A) inside Rep′(A), and let Rep′(A) be weakly admissible. Let π_{i} for i = 1, 2 be representations on Hilbert D-modules E_{i} for a C^{∗}-algebra D that belong to Rep′(A). Let u: E_{1} ⟶^{∼} E_{2} be a unitary ^{∗}-intertwiner from π_{1} to π_{2}. If F_{1}(π_{1}) = F_{2}(π_{1}), then F_{1}(π_{2}) = u^{⊕n} (F_{1}(π_{1})) = u^{⊕n} (F_{2}(π_{1})) = F_{2}(π_{2}). Thus π_{2} belongs to Rep′′(A) if π_{1} does. This verifies (1) in Definition 3.11 using (1) in Definition 5.7. Similarly, (2) and (3) in Definition 5.7 show that Rep′′(A) inherits (2) and (3) in Definition 3.11 from Rep′(A). Thus Rep′′(A) is again weakly admissible. It is trivial that weak admissibility is hereditary for intersections. By transfinite induction, it follows that any class of representations defined by submodule conditions is weakly admissible.

Proof of Theorem 5.9. Let F_{1} and F_{2} be natural constructions of Hilbert submodules of rank n that define Rep′′(A) inside Rep′(A), and assume that Rep′(A) satisfies the Strong Local–Global Principle. Let π be a representation on a Hilbert D-module E that does not belong to Rep′′(A). We must find an irreducible representation ϱ of D on a Hilbert space H such that π ⊗_{ϱ} H does not belong to Rep′′(A). If the representation does not even belong to Rep′(A), this is possible because Rep′(A) satisfies the Strong Local–Global Principle by assumption. So we may assume that π belongs to Rep′(A) but not to Rep′′(A). Thus F_{1}(π) and F_{2}(π) are well defined and different Hilbert submodules of E^{n}. Corollary 5.4 gives an irreducible representation ϱ of D on a Hilbert space H such that F_{1}(π) ⊗_{ϱ} H ≠ F_{2}(π) ⊗_{ϱ} H as closed subspaces of E^{n} ⊗_{ϱ} H. Identify these with subspaces of (E ⊗_{ϱ} H)^{n}. The condition (2) in Definition 5.7 gives

F_{1}(π ⊗_{ϱ} H) = F_{1}(π) ⊗_{ϱ} H ≠ F_{2}(π) ⊗_{ϱ} H = F_{2}(π ⊗_{ϱ} H).

That is, π ⊗_{ϱ} H does not belong to Rep′′(A). Thus Rep′′(A) inherits the Strong Local–Global Principle from Rep′(A). The Strong Local–Global Principle is easily seen to be hereditary for intersections. Hence any class of representations defined by submodule conditions satisfies the Strong Local–Global Principle by transfinite induction.

Theorem 5.17. Let A be a ^{∗}-algebra and let S ⊆ A_{h}. Let Rep^{S}(A) be the class of all representations where the elements of S act by regular, self-adjoint operators. This class is defined by submodule conditions and hence satisfies the Strong Local–Global Principle. It is admissible if S is a strong generating set for A.

Proof. Asking \overline{π(a)} to be regular and self-adjoint for a single a ∈ S is a submodule condition by Example 5.10. In order to ask this simultaneously for a set S, let ≺ be a well-ordering on S, and add an element M with a ≺ M for all a ∈ S. Let Rep^{a}(A) ⊆ Rep(A) for a ∈ S ∪ {M} be the class of all representations π where \overline{π(b)} is regular and self-adjoint for all b ∈ S with b ≺ a. These subclasses form a recursive definition of Rep^{S}(A) by submodule conditions as in Definition 5.7. Thus Rep^{S}(A) is defined by submodule conditions. Then it is weakly admissible and satisfies the Strong Local–Global Principle by Lemma 5.8 and Theorem 5.9. From now on, we assume that S is a strong generating set. For Rep^{S}(A) to be admissible, we must prove that any isometric intertwiner I: (𝔈_{0}, π_{0}) ↪ (𝔈, π) between two Hilbert space representations in Rep^{S}(A) is a ^{∗}-intertwiner. If a ∈ S, then \overline{π(a)} and \overline{π_{0}(a)} are regular, self-adjoint operators. Hence they generate integrable representations of ℂ[x] as in Theorem 4.4. The isometry I intertwines these representations of ℂ[x]. Hence it is a ^{∗}-intertwiner by Theorem 4.4. In particular, I^{∗} maps dom \overline{π(a)} to dom \overline{π_{0}(a)} for each a ∈ S. Since S is a strong generating set, (2.10) gives 𝔈_{0} = ⋂ _{a∈S} dom \overline{π_{0}(a)} and similarly for π. So I^{∗}(𝔈) ⊆ 𝔈_{0}. Then 𝔈 = 𝔈_{0} + (𝔈 ∩ 𝔈_{0}^{⊥} ) and I is a ^{∗}-intertwiner by Proposition 2.27.

Corollary 5.18. Let S ⊆ A_{h} be a strong generating set for a ^{∗}-algebra A and let B with universal representation μ be a weak C^{∗}-hull. If the closed multipliers \overline{μ(a)} for a ∈ S are self-adjoint and affiliated with B, then B is a C^{∗}-hull.

Proof. All B-integrable representations belong to Rep^{S}(A) because the latter is weakly admissible and contains the universal B-integrable representation. Since Rep^{S}(A) is admissible by Theorem 5.17, any smaller class of integrable representations inherits the equivalent conditions (2)–(4) in Proposition 3.8, which characterise C^{∗}-hulls among weak C^{∗}-hulls.

Theorem 5.19. Let A be a ^{∗}-algebra, B a C^{∗}-algebra, (𝔅, μ) a representation of A on B, and T_{1}, …, T_{n} ∈ A. Assume that \overline{μ(T_{1})}, …, \overline{μ(T_{n})} are self-adjoint and affiliated with B and generate B in the sense of Woronowicz, see [32, Definition 3.1]. Then B is a C^{∗}-hull for the B-integrable representations of A defined by (𝔅, μ), and these satisfy the Strong Local–Global Principle.

Proof. To show that B is a C^{∗}-hull, we check the condition (5) in Proposition 3.8. Let ϱ: B → 𝔹(H) be a representation of B on a Hilbert space H and let (𝔈, π) be the corresponding B-integrable representation of A. Let (𝔈_{0}, π|_{𝔈_{0}}) be a B-integrable representation on a closed subspace E_{0} ⊆ E and let P ∈ 𝔹(E) be the projection onto E_{0}. We must show that ϱ(B) is contained in the commutant of P. Equivalently, ϱ is a morphism in the notation of [32] to the algebra K = 𝕂(E_{0}) ⊕ 𝕂(E_{0}^{⊥} ) of all compact operators on E that commute with P. Let 1 ≤ i ≤ n. Since T_{i} is self-adjoint and regular as an adjointable operator on the Hilbert B-module B, it generates an integrable representation of the polynomial algebra ℂ[x] on B as in Theorem 4.4. These integrable representations form an admissible class. Therefore, a B-integrable representation of A gives an integrable representation of ℂ[x] when we compose with the canonical map j_{i}: ℂ[x] → A, x ↦ T_{i}, and take the closure. And since π and π|_{𝔈_{0}} are both B-integrable, \overline{π|_{𝔈_{0}} ∘ j_{i}} is a direct summand in \overline{π ∘ j_{i}} . Equivalently, the unbounded operator \overline{π(T_{i})} is affiliated with K. The extension of ϱ to affiliated multipliers maps \overline{μ(T_{i})} to \overline{π(T_{i})}, which is affiliated with K. Hence ϱ is a morphism to K because these affiliated multipliers generate B. Thus B is a C^{∗}-hull for the B-integrable representations by Proposition 3.8. Now we check the Strong Local–Global Principle. Let (𝔈, π) be a representation of A on a Hilbert D-module E. Assume that the representation (𝔈, π) ⊗_{ω} H_{ω} is integrable for each irreducible representation ω of D on a Hilbert space H_{ω} in the sense that it comes from a representation of B. We must show that the representation (𝔈, π) is integrable. The condition that \overline{π(T_{i})}be self-adjoint and regular is a submodule condition by Example 5.10. Hence the class of representations with this property satisfies the Strong Local–Global Principle by Theorem 5.9. Therefore, \overline{π(T_{i})} is a regular, self-adjoint operator on E for i = 1, …, n. Let ω be the direct sum of all irreducible representations of D; this is a faithful representation of D on some Hilbert space H. The induced representation j of 𝕂(E) on K := E ⊗_{D} H is faithful as well. By assumption, the representation π ⊗_{D} 1 of A on K is integrable, so it comes from a representation σ of B. The extension of σ to affiliated multipliers maps \overline{μ(T_{i})} η B to \overline{(π ⊗_{D} 1)(T_{i})} . Since \overline{π(T_{i})} is a regular operator on E, it is an affiliated multiplier of 𝕂(E), see [20] or Proposition 3.13. Thus \overline{(π ⊗_{D} 1)(T_{i})} is affiliated with the image of 𝕂(E) on K by [17, Proposition 9.10]. Thus σ(\overline{μ(T_{i})}) η 𝕂(E) for i = 1, …, n. Since the affiliated multipliers \overline{μ(T_{i})} generate B in the sense of Woronowicz, σ factors through a morphism τ: B → 𝕂(E). This is the same as a representation of B on E. Let π′ be the representation of A on E associated to τ. If ϱ is an irreducible Hilbert space representation of D, then π ⊗_{ϱ} H = π′⊗_{ϱ} H by construction of τ. Hence Theorem 5.6 gives π = π′. Since π′ is integrable by construction, so is π.

The first counterexample in §6 exhibits a symmetric affiliated multiplier that generates a C^{∗}-algebra, such that the Local–Global Principle fails and B is not a C^{∗}-hull. Without self-adjointness, we only get the following much weaker statement:

Lemma 5.20. Let A be a ^{∗}-algebra, B a C^{∗}-algebra, (𝔅, μ) a representation of A on B, and T_{1}, …, T_{n} ∈ A. Assume that \overline{μ(T_{1})}, …, \overline{μ(T_{n})} are affiliated with B and generate B in the sense of Woronowicz. Then B is a weak C^{∗}-hull for the B-integrable representations of A.

Proof. To show that B is a weak C^{∗}-hull, we check (1) in Proposition 3.8. Let ϱ_{1}, ϱ_{2} be representations of B on a Hilbert space H with (𝔅, μ) ⊗_{ϱ_{1}} H = (𝔅, μ) ⊗_{ϱ_{2}} H. We claim that ϱ_{1} ⊕ ϱ_{2} : B → 𝔹(H^{2}) = 𝕄_{2}(𝔹(H)) maps B into the multiplier algebra of the diagonally embedded copy K of 𝕂(H). This is equivalent to ϱ_{1} = ϱ_{2}. Since (𝔅, μ)⊗_{ϱ_{1}} H = (𝔅, μ)⊗_{ϱ_{2}} H, the extension of ϱ_{1}⊕ϱ_{2} to affiliated multipliers maps \overline{μ(T_{i})} η B to an operator of the form (X_{i}, X_{i}) for i = 1, …, n; these are affiliated with K. Since these affiliated multipliers generate B, ϱ_{1} ⊕ ϱ_{2} is a morphism from B to K. Thus B is a weak C^{∗}-hull for A.

5.1 Universal enveloping algebras

We illustrate our theory by an example. Let 𝔤 be a finite-dimensional Lie algebra over ℝ and let A = U(𝔤) be its universal enveloping algebra with the usual involution, where elements of 𝔤 are skew-symmetric. A representation of A on E, possibly not closed, is equivalent to a dense submodule 𝔈 ⊆ E with a Lie algebra representation π: 𝔤 → End_{D}(𝔈) satisfying ⟨ξ, π(X)(η)⟩ = −⟨π(X)(ξ), η⟩ for all X ∈ 𝔤, ξ, η ∈ 𝔈. Let G be a simply connected Lie group with Lie algebra 𝔤 and let B := C^{∗}(G). A representation of C^{∗}(G) on a Hilbert module E is equivalent to a strongly continuous, unitary representation of G on E. Given such a representation, let E^{∞} ⊆ E be its subspace of smooth vectors. This is the domain of a closed representation of U(𝔤). We call a representation of U(𝔤) integrable if it comes from a unitary representation of G in this way. In particular, G acts continuously on C^{∗}(G) by left multiplication with unitary multipliers. Let 𝔅 = C^{∗}(G)^{∞} be the right ideal of smooth elements for this G-action, equipped with the canonical U(𝔤)-module structure μ. By the universal property of C^{∗}(G), the pair (𝔅, μ) is the universal integrable representation. That is, a representation of U(𝔤) is integrable if and only if it is of the form (𝔅, μ) ⊗_{ϱ} E for a representation ϱ of C^{∗}(G). Let X_{1}, …, X_{d} form a basis of 𝔤. The Laplacian is L := − ∑ _{i=1}^{d} X_{i}^{2} ∈ U(𝔤).

Theorem 5.21 ([21, Théorème 2.12]). A representation (π, 𝔈) of U(𝔤) is integrable if and only if \overline{π(L^{n})} is regular and self-adjoint for all n ∈ ℕ.

Proof. Since Pierrot does not require representations to be closed, his statement is slightly different from ours. Pierrot shows that there is a continuous representation ϱ of G with differential X ↦ \overline{π(X)} if and only if T := \overline{π(L)} is self-adjoint and regular. His proof shows that all elements of ⋂ _{n=1}^{∞} domT^{n} are smooth vectors for ϱ. Conversely, all smooth vectors must belong to this intersection. A representation of U(𝔤) is determined by its domain and the closed operators \overline{π(X)} for X ∈ 𝔤. So a closed representation (𝔈, π) of U(𝔤) is integrable if and only if T is self-adjoint and regular and 𝔈 = ⋂ _{n=1}^{∞} domT^{n}. Moreover, the proof shows that the graph topology for a representation with regular self-adjoint T is determined by the graph norms of L^{n} for all n ∈ ℕ. If \overline{π(L^{n})} is self-adjoint, then it must be equal to T^{n} because \overline{π(L^{n})} ⊆ T^{n} and T^{n} is symmetric. Therefore, if \overline{π(L)} is regular and self-adjoint, then the domain of π is ⋂ _{n=1}^{∞} domT^{n} if and only if \overline{π(L^{n})} is regular and self-adjoint also for all n ≥ 2.

Theorem 5.22. The class of integrable representations of U(𝔤) has C^{∗}(G) as a C^{∗}-hull and is defined by submodule conditions. So it satisfies the Strong Local–Global Principle.

Proof. By Theorem 5.21, a representation is integrable if and only if all elements of the set {L^{n} | n ∈ ℕ} act by a regular and self-adjoint operator. Hence the assertion follows from Theorem 5.17. Alternatively, the closed multipliers of C^{∗}(G) associated to iX_{1}, …, iX_{d} are regular and affiliated with C^{∗}(G) and generate C^{∗}(G) by [32, Example 3 in §3]. Hence C^{∗}(G) is a C^{∗}-hull and the Strong Local–Global Principle holds by Theorem 5.19.

The results of Vassout [30] get close to proving an analogue of Theorem 5.22 for an s-simply connected Lie groupoid G with compact base. This analogue would replace 𝔤 by the space of smooth sections of the Lie algebroid A(G), and U(𝔤) by the ^{∗}-algebra of G-equivariant differential operators on G, a subalgebra of the ∗-algebra of G-pseudodifferential operators. Any symmetric, elliptic element of U(𝔤) should be a possible replacement for the Laplacian in Theorem 5.22.

6 Polynomials in one variable II

We discuss two classes of “integrable” representations of the^{∗}-algebra ℂ[x] with x = x^{∗} which are weakly admissible, but not admissible, and which violate the Local–Global Principle. Both examples have a weak C^{∗}-hull, on which all powers of the generator x act by an affiliated multiplier. In the first example, these affiliated multipliers generate the weak C^{∗}-hull, but not in the second. Neither Theorem 5.17 nor Theorem 5.19 apply because the generating affiliated multipliers are not self-adjoint. The first example shows that a C^{∗}-algebra generated by affiliated multipliers in the sense of Woronowicz need not be a C^{∗}-hull, though it is always a weak C^{∗}-hull by Lemma 5.20. The second example shows that a weak C^{∗}-hull need not be generated by affiliated multipliers. Let S ∈ 𝔹(ℓ^{2}ℕ) be the unilateral shift. Let Q be the closed symmetric operator on ℓ^{2}ℕ with Cayley transform S. Thus Q has deficiency index (0, 1). The domain of Q is (1 − S)ℓ^{2}ℕ, and Q(1 − S)ξ := i(1 + S)ξ for all ξ ∈ ℓ^{2}ℕ (see also Example 4.10). We may identify ℓ^{2}ℕ with the Hardy space H^{2}. Then Q becomes the Toeplitz operator with the unbounded symbol i(1 + z)(1 − z)^{−1} . Let T be the Toeplitz C^{∗}-algebra, that is, the C^{∗}-subalgebra of 𝔹(ℓ^{2}ℕ) generated by S. Every element in T is of the form T_{φ} + K, where T_{φ} is the Toeplitz operator with symbol φ ∈ C(S^{1}) and K is a compact operator. Let T_{0} ◁ T be the kernel of the unique ^{∗}-homomorphism T → ℂ that maps S to 1.

Proposition 6.1. There is a symmetric, affiliated multiplier Q of T_{0} with domain (1−S)·T_{0} and Q·(1−S)·t := i(1+S)·t for all t ∈ T_{0}. It generates T_{0} in the sense of Woronowicz.

Proof. We claim that the right ideal (1−S)S^{∗}T_{0} ⊆ T_{0} is dense. This would fail for T because the continuous ^{∗}-homomorphism T → ℂ, S ↦ 1, annihilates this right ideal. First, (1 − S)S^{∗}𝕂(ℓ^{2}ℕ) is dense in 𝕂(ℓ^{2}ℕ) because (1 − S)S^{∗} has dense range on ℓ^{2}ℕ. So the closure of (1 − S)S^{∗}T_{0} contains 𝕂(ℓ^{2}ℕ). Secondly, (1−S)S^{∗}T_{0}/𝕂(ℓ^{2}ℕ) is dense in T_{0}/𝕂(ℓ^{2}ℕ) ≅ C_{0}(S^{1} \{1}) because the function (1 − z)\overline{z} on S^{1} vanishes only at 1. An affiliated multiplier of T_{0} is the same as a regular operator on T_{0}, viewed as a Hilbert module over itself. Since (1 − S)S^{∗}T_{0} is dense in T_{0}, there is a regular, symmetric operator Q′ on T_{0} that has S as its Cayley transform, see [17, Chapter 10]. The operator Q′ has the domain (1 − S)S^{∗}T_{0} and acts by Q′ · (1 − S)S^{∗}t := i(1 + S)S^{∗}t. Rewriting any t ∈ T_{0} as t = S^{∗}St, we may replace S^{∗}t by t here. Thus Q′ = Q. Since Q + i maps (1 − S)t to i(1 + S)t + i(1 − S)t = 2it, it is surjective, and (Q + i)^{−1} = \frac{1}{2i} (1 − S) belongs to T_{0}. Hence (Q + i)^{∗} = Q^{∗} − i is the inverse of \frac{1}{−2i} (1−S^{∗}). So Q^{∗} has domain (1−S^{∗})T_{0} and maps (1−S^{∗})t ↦ i(1−S^{∗})t−2it = −i(1+S^{∗})t. As expected, Q^{∗} contains Q: we may write (1−S)t = S^{∗}St−St = (1 − S^{∗})(−St), and Q^{∗} maps this to −i(1 + S^{∗})(−St) = i(S + 1)t. Next we show that Q^{∗}Q+1 is the inverse of \frac{1}{4}(1−S)(1−S^{∗}) ∈ T_{0}. We compute

Q^{∗}Q(1 − S)(1 − S^{∗})t = iQ^{∗}(1 + S)(1 − S^{∗})t = iQ^{∗}(1 + S − S^{∗} − SS^{∗})t = iQ^{∗}(1−S^{∗})(2+S−SS^{∗})t = (1+S^{∗})(2+S−SS^{∗})t = (4−(1−S)(1−S^{∗}))t.

This implies (Q^{∗}Q + 1)(1 − S)(1 − S^{∗})t = 4t. Since this is already surjective and Q^{∗}Q + 1 is injective, the domain of Q^{∗}Q + 1 is exactly (1 − S)(1 − S^{∗})T_{0}, and Q^{∗}Q + 1 is the inverse of \frac{1}{4}(1 − S)(1 − S^{∗}) ∈ T_{0} as asserted. Let ϱ_{1} and ϱ_{2} be two Hilbert space representations of T_{0} whose extension to affiliated multipliers maps Q to the same unbounded operator. Then they also map the Cayley transform S of Q to the same partial isometry. So ϱ_{1}(S) = ϱ_{2}(S), which gives ϱ_{1} = ϱ_{2}. Thus Q separates the representations of T_{0}. Since (Q^{∗}Q + 1)^{−1} ∈ T_{0} as well, [32, Theorem 3.3] shows that the affiliated multiplier Q generates T_{0}.

The domain of Q^{n} is the right ideal (1 − S)^{n} · T_{0}, which is dense in T_{0} for the same reason as (1−S)·T_{0}. Even more, the right ideal (1−S)^{n+1} ·T_{0} is dense in (1−S)^{n} ·T_{0} in the graph norm of Q^{n}. Thus the intersection 𝔗 of this decreasing chain of dense right ideals (1 − S)^{n}T_{0} is still dense in T_{0} by [28, Lemma 1.1.2]. This intersection is the domain of a closed representation μ of ℂ[x] on T_{0} with \overline{μ(x^{n})} = Q^{n}. We call a representation of ℂ[x] on a Hilbert module E Toeplitz integrable if it is of the form (𝔗, μ) ⊗_{ϱ} E for some representation ϱ: T_{0} → 𝔹(E).

Proposition 6.2. The class of Toeplitz integrable representations of ℂ[x] is weakly admissible with the weak C^{∗}-hull T_{0}. It is not admissible, so T_{0} is not a C^{∗}-hull. The Toeplitz integrable representations violate the Local–Global Principle. A representation (𝔈, π) of ℂ[x] on a Hilbert module E over a C^{∗}-algebra D is Toeplitz integrable if and only if it has the following properties:

(1) \overline{π(x + i)^{n}𝔈} = E for all n ∈ ℕ_{≥1} ;

(2) \overline{π(x)} is regular.

Toeplitz integrable representations on E are in bijection with regular, symmetric operators T on E for which T + i is surjective.

Proof. We checked condition (1) in Proposition 3.8 in the proof of Proposition 6.1. Thus T_{0} is a weak C^{∗}-hull for the Toeplitz integrable representations, and this class is weakly admissible. Any self-adjoint operator on a Hilbert space generates a Toeplitz integrable representation of ℂ[x] because T_{0}/𝕂(ℓ^{2}ℕ) ≅ C_{0}(ℝ); so does Q itself. Thus both Example 4.10 and Proposition 4.8 show that the class of Toeplitz integrable representations is not admissible. So T_{0} is not a C^{∗}-hull. We claim that the representation (𝔗, μ) of ℂ[x] on T_{0} has the properties (1) and (2) in the proposition. First, (μ(x) + i)^{n} acts by (2i)^{n}(1 − S)^{−n} on its dense domain 𝔗 := ⋂ _{k=1}^{∞} (1 − S)^{k}T_{0}. Since (1 − S)^{k+1} T_{0} is norm dense in (1 − S)^{k}T_{0}, the closure of (μ(x) + i)^{n} is equal to (2i)^{n}(1 − S)^{−n} with its natural domain (1 − S)^{n}T_{0}, and this operator is surjective. Secondly, μ(x) = Q is regular. The property (1) is a sequence of submodule conditions, see Example 5.13. Hence it is inherited by interior tensor products by Lemma 5.8. So is the property (2) by [17, Proposition 9.10]. Hence both (1) and (2) are necessary for a representation (𝔈, π) to be Toeplitz integrable. Conversely, let (𝔈, π) be a representation of ℂ[x] on E that satisfies (1) and (2). Then the closed, symmetric operator T := \overline{π(x)} on E is regular by (2). So its Cayley transform s is an adjointable partial isometry such that (1 − s)s^{∗} has dense range (see [17, Chapter 10]). Even more, s is an isometry because \overline{(T + i)𝔈} = E. Thus s generates a unital representation ϱ of T . The restriction of ϱ to T_{0} is nondegenerate because (1 − s)s^{∗} has dense range. Let π′:= μ ⊗_{ϱ} 1 be the representation of ℂ[x] associated to ϱ. Then

\overline{π′((x + i)^{n})} = (2i)^{n}(1 − s)^{−n} ⊇ \overline{π((x + i)^{n})} .

Assumption (1) implies that 𝔈 is dense in the domain of (2i)^{n}(1 − s)^{−n} in the graph norm of (2i)^{n}(1 − s)^{−n} . Hence even \overline{π′((x + i)^{n})} = (2i)^{n}(1 − s)^{−n} = \overline{π((x + i)^{n})} . Since the domains of π(x)^{n} form a decreasing sequence, induction on n now shows that \overline{π′(x^{n})} = \overline{π(x^{n})}. The set \{x^{n}\} is a strong generating set for ℂ[x] by Lemma 4.1. Thus π = π′ by Proposition 2.9. This finishes the proof that Toeplitz integrable representations of ℂ[x] are characterised by the properties (1) and (2) and that they are in bijection with regular, symmetric operators T for which T + i is surjective. For a counterexample to the Local–Global Principle, let ℕ̄ = ℕ ∪ {∞} be the one-point compactification of ℕ and D = C(ℕ̄). Let E ⊆ C(ℕ̄, ℓ^{2}ℕ) consist of all continuous functions f: ℕ̄ → ℓ^{2}ℕ with f(∞)⊥δ_{0}. The unilateral shift S on C(ℕ̄, ℓ^{2}ℕ) restricts to a non-adjointable isometry s on this subspace. Let T be the inverse Cayley transform of s. This is a closed, symmetric operator on E that is irregular because its Cayley transform is not adjointable. If ϱ: D → 𝔹(H) is a Hilbert space representation, then the induced representation of ℂ[x] is associated to the closed operator T ⊗_{ϱ} 1. The operator (T ⊗_{ϱ} 1) + i remains surjective, and T ⊗_{ϱ} 1 is regular because it acts on a Hilbert space. So T ⊗_{ϱ} 1 generates a Toeplitz integrable representation for all representations ϱ of D. Since T itself does not generate a Toeplitz integrable representation, the Local– Global Principle is violated.

Condition (1) in Proposition 6.2 is a submodule condition. If regularity without self-adjointness were a submodule condition as well, then the Toeplitz integrable representations of ℂ[x] would be defined by submodule conditions; so the failure of the Local–Global Principle for them would contradict Theorem 5.19. The identical inclusion T_{0} ↪ M(𝕂(ℓ^{2}ℕ)) is a representation of the weak C^{∗}-hull T_{0} on 𝕂(ℓ^{2}ℕ) and thus corresponds to a Toeplitz integrable representation of ℂ[x] on 𝕂(ℓ^{2}ℕ). This is simply the restriction of (𝔗, μ) to the Hilbert T_{0}-submodule 𝕂(ℓ^{2}ℕ) ⊂ T_{0}, with domain 𝔗 ∩ 𝕂(ℓ^{2}ℕ) and the same action μ of ℂ[x]. Call a representation purely Toeplitz integrable if it is of the form (𝔗 ∩ 𝕂(ℓ^{2}ℕ), μ) ⊗_{ϱ} E for some representation ϱ: 𝕂(ℓ^{2}ℕ) → 𝔹(E).

Proposition 6.3. The purely Toeplitz integrable representations of ℂ[x] form a weakly admissible class that is not admissible, and 𝕂(ℓ^{2}ℕ) is a weak C^{∗}-hull for it, but not a C^{∗}-hull. This class violates the Local–Global Principle. The closed multiplier Q = \overline{μ(x)} of T_{0} is affiliated with 𝕂(ℓ^{2}ℕ) but does not generate 𝕂(ℓ^{2}ℕ). A representation (𝔈, π) of ℂ[x] on a Hilbert module E over a C^{∗}-algebra D is purely Toeplitz integrable if and only if it has the following property in addition to those in Proposition 6.2:

(3) the closure of ⋃ _{n=1}^{∞} (π(x − i)^{n}𝔈)^{⊥} is E.

Proof. Since 𝕂(ℓ^{2}ℕ) has fewer representations than T_{0}, the condition (1) in Proposition 3.8 for 𝕂(ℓ^{2}ℕ) follows from the corresponding property for T_{0}, which we have already checked in the proof of Proposition 6.1. Hence 𝕂(ℓ^{2}ℕ) is a weak C^{∗}-hull for the purely Toeplitz representations of ℂ[x]. Since Q gives a purely Toeplitz representation of ℂ[x] on ℓ^{2}(ℕ), the class of purely Toeplitz integrable representations is not admissible by Example 4.10. Therefore, its weak C^{∗}-hull is not a C^{∗}-hull. The same counterexample as in the proof of Proposition 6.2 shows that the Local–Global Principle fails for the purely Toeplitz representations. Any closed operator on ℓ^{2}ℕ is affiliated with 𝕂(ℓ^{2}ℕ). In particular, so is Q. In the identical representation of 𝕂(ℓ^{2}ℕ) on the Hilbert space ℓ^{2}ℕ, the image of Q is affiliated with T_{0} by Proposition 6.1. But the representation of 𝕂(ℓ^{2}ℕ) is not by a morphism to T_{0} because the inclusion map 𝕂(ℓ^{2}ℕ) ↪ T_{0} is degenerate. Hence Q does not generate 𝕂(ℓ^{2}ℕ) in the sense of Woronowicz. The element P_{n} := 1 − S^{n}(S^{∗})^{n} ∈ 𝕂(ℓ^{2}ℕ) ⊆ T_{0} is the orthogonal projection onto the span of δ_{0}, …, δ_{n−1} . A representation of T_{0} maps P_{n} to an orthogonal projection whose image is the orthogonal complement of the image of S^{n}. This is also the orthogonal complement of the image of π(x − i)^{n}. These orthogonal complements form an increasing chain of complementable submodules, and π is purely Toeplitz if and only if their union is all of E. This proves our characterisation of purely Toeplitz representations.

7 Bounded and locally bounded representations

Let A be a ^{∗}-algebra. A bounded representation of A on a Hilbert module E is a ^{∗}-homomorphism π: A → 𝔹(E). Corollary 2.11 says that a closed representation is bounded once \overline{π(a)} is globally defined for a in a strong generating set of A. Finite-dimensional representations are always bounded. In particular, characters are bounded. Thus commutative ^{∗}-algebras have many bounded representations. Many other ^{∗}-algebras, such as the Weyl algebra, have no bounded representations. In this section, we are going to study C^{∗}-hulls related to bounded representations. These are only relevant if A has many bounded representations. Any bounded representation π of A is bounded in some C^{∗}-seminorm q on A, that is, ∥π(a)∥ ≤ q(a) for all a ∈ A. Then π extends to the (Hausdorff) completion A_{q} of A in the seminorm q, which is a unital C^{∗}-algebra. If p, q are two C^{∗}-seminorms on A, then max{p, q} is a C^{∗}-seminorm as well. Thus the set N (A) of C^{∗}-seminorms on A is directed. For q, q′ ∈ N (A) with q ≤ q′, let φ_{q,q′}: A_{q′} → A_{q} be the ^{∗}-homomorphism induced by the identity map on A. The C^{∗}-algebras A_{q} and the ^{∗}-homomorphisms φ_{q,q′} for q ≤ q′ in N (A) form a projective system of C^{∗}-algebras. Each ^{∗}-homomorphism φ_{q,q′} is unital and surjective because its image contains A, which is unital and dense in A_{q′}. The C^{∗}-seminorms in N (A) define a locally convex topology on A, where a net converges if and only if it converges in any C^{∗}-seminorm. Let A with the canonical map j: A → A be the completion of A in this topology. This is a C^{∗}-algebra if and only if there is a largest C^{∗}-seminorm on A. In general, A is the projective limit of the diagram of unital C^{∗}-algebras (A_{q}, φ_{q,q′}) described above. Thus A is a unital pro-C^{∗}-algebra, see [22]. As a concrete example, we describe A for a commutative ^{∗}-algebra A.

Definition 7.1. Let Â be the set of ^{∗}-homomorphisms A → ℂ, which we briefly call characters. Each a ∈ A gives a function â: Â → ℂ, â(χ) := χ(a). We equip Â with the coarsest topology making these functions continuous. That is, a net (χ_{i})_{i∈I} in Â converges to χ ∈ Â if and only if limχ_{i}(a) = χ(a) for all a ∈ A. Let τ_{c} be the compactly generated topology associated to this topology, that is, a subset in Â is closed in τ_{c} if and only if its intersection with any compact subset in Â is closed.

If a ∈ A, then its Gelfand transform â is a continuous function on Â. This defines a ^{∗}-homomorphism A → C(Â). If the usual topology on Â is locally compact or metrisable, then it is already compactly generated and hence equal to τ_{c}. The topology τ_{c} may have more closed subsets and hence more continuous functions to ℂ. So C(Â) ⊆ C(Â, τ_{c}).

Proposition 7.2. Let A be a commutative ^{∗}-algebra. The directed set N (A) of C^{∗}-seminorms on A is isomorphic to the directed set of compact subsets of Â, where K ⊆ Â corresponds to the C^{∗}-seminorm

∥a∥_{K} := sup{|â(χ)| | χ ∈ K}.

The C^{∗}-completion of A in this C^{∗}-seminorm is C(K). And A ≅ C(Â, τ_{c}), where the inclusion map j: A → A is the Gelfand transform A → C(Â, τ_{c}), a ↦ â.

Proof. Let q be a C^{∗}-seminorm on A. Let Â_{q} ⊆ Â be the subspace of all q-bounded characters, that is, χ ∈ Â_{q} if and only if |χ(a)| ≤ q(a) for all a ∈ A. These are precisely the characters that extend to characters on the C^{∗}-completion A_{q}. Conversely, since A is dense in A_{q}, any character on A_{q} is the unique continuous extension of a q-bounded character on A. And the subspace topology on Â_{q} ⊆ Â is equal to the canonical topology on the spectrum of A_{q}: a net of q-bounded characters that converges uniformly on A also converges uniformly on A_{q}. Thus

A_{q} ≅ C(Â_{q})

by the Gelfand–Naimark Theorem, and so Â_{q} ⊆ Â is compact for each q ∈ N (A). If q ≤ q′, then Â_{q} ⊆ Â_{q′} and φ_{qq′}: A_{q′} ↠ A_{q} is the restriction map for the subspace Â_{q} ⊆ Â_{q′}. The pro-C^{∗}-algebra A is the limit of this diagram of commutative C^{∗}-algebras. Since all the maps Â_{q} ⊆ Â_{q′} are injective, A is the algebra of continuous functions on ⋃ _{q∈N (A)} Â_{q} with the inductive limit topology. That is, a subset of ⋃ _{q∈N (A)} Â_{q} is closed if and only if its intersection with each Â_{q} is closed, where Â_{q} carries the (compact) subspace topology from Â. Any character χ on A is bounded with respect to some C^{∗}-seminorm; for instance, ∥a∥_{χ} := |χ(a)|. Thus ⋃ _{q∈N (A)} Â_{q} = Â as a set. If K ⊆ Â is compact, then â ∈ C(Â) for a ∈ A must be uniformly bounded on K, so that

∥a∥_{K} := sup{|â(χ)| | χ ∈ K}

is a C^{∗}-seminorm on A. Thus K ⊆ Â_{q} for some q ∈ N (A). Hence the inductive limit topology on ⋃ _{q∈N (A)} Â_{q} is τ_{c}.

We return to the general noncommutative case. The class of q-bounded representations for a fixed q ∈ N (A) is easily seen to be weakly admissible. The class of bounded representations with variable q is not weakly admissible unless A has a largest C^{∗}-seminorm because it is not closed under direct sums. We are going to define the larger class of “locally bounded” representations to rectify this. Roughly speaking, a representation is locally bounded if and only if it comes from a representation of the pro-C^{∗}-algebra A. Before we define locally bounded representations, we characterise q-bounded representations by some slightly weaker estimates.

Proposition 7.3. Let A be a ^{∗}-algebra and let q be a C^{∗}-seminorm on A. Let (𝔈, π) be a representation of A on a Hilbert module E over some C^{∗}-algebra D and let ξ ∈ 𝔈. The following are equivalent:

(1) there is C > 0 with ∥⟨ξ, π(a)ξ⟩∥ ≤ Cq(a) for all a ∈ A;

(2) there is C > 0 with ∥π(a)ξ∥ ≤ Cq(a) for all a ∈ A;

(3) ∥π(a)ξ∥ ≤ ∥ξ∥q(a) for all a ∈ A.

The set of vectors ξ with these equivalent properties is a norm-closed A, Dsubmodule of E. The representation of A on this submodule extends to the C^{∗}-completion A_{q}.

Proof. The implications (3)⇒(2)⇒(1) are trivial. Conversely, assume (1) and let a ∈ A. Let (b_{n})_{n∈ℕ} be a sequence in A that converges in A_{q} towards the positive square-root of q(a)^{2} − a^{∗}a. Then the sequence (a^{∗}a + b_{n}^{∗}b_{n}) in A converges in the norm q to q(a)^{2} ∈ A. If (1) holds, then

lim_{n→∞} ⟨ξ, π(a^{∗}a + b_{n}^{∗}b_{n})ξ⟩ = q(a)^{2}⟨ξ, ξ⟩.

Since 0 ≤ ⟨π(a)ξ, π(a)ξ⟩ ≤ ⟨π(a)ξ, π(a)ξ⟩ + ⟨π(b_{n})ξ, π(b_{n})ξ⟩ = ⟨ξ, π(a^{∗}a + b_{n}^{∗}b_{n})ξ⟩ for all n, this implies ∥π(a)ξ∥ ≤ lim∥⟨ξ, π(a^{∗}a + b_{n}^{∗}b_{n})ξ⟩∥ = q(a)^{2}∥ξ∥^{2}. Thus (1) implies (3). The set E_{q} of vectors ξ ∈ 𝔈 satisfying (2) is a vector subspace and closed under left multiplication by elements of A and right multiplication by elements of D. On this subspace, the graph and norm topologies coincide because of (3). The subspace E_{q} is closed in the norm topology by the Principle of Uniform Boundedness. The ^{∗}-representation of A on this submodule is globally defined and bounded by the C^{∗}-seminorm q. Hence it extends to a representation of A_{q}.

Definition 7.4. Let (𝔈, π) be a representation of A on a Hilbert module E. A vector ξ ∈ 𝔈 is bounded if it satisfies the equivalent conditions in Proposition 7.3 for some q ∈ N (A). The representation is locally bounded if the bounded vectors are dense in 𝔈 in the graph topology.

By Proposition 7.3, the q-bounded vectors in 𝔈 for a fixed q ∈ N (A) form a closed A, D-submodule E_{q} ⊆ E, on which the representation of A extends to the C^{∗}-completion A_{q} and hence to a representation of A. Since N (A) is directed and E_{q} ⊆ E_{q′} if q ≤ q′, the family of sub-bimodules E_{q} ⊆ 𝔈 is directed. The set of bounded vectors is the increasing union

𝔈_{b} := ⋃_{q∈N (A)} E_{q}.

Since π|_{E_{q}} extends to A for each q, there is a representation π̄ of the pro-C^{∗}-algebra A on 𝔈_{b} ⊆ E. The representation (𝔈, π) is locally bounded if and only if (𝔈_{b}, π̄ ∘ j) is a core for it. Thus (𝔈, π) is the closure of the “restriction” π̄ ∘ j of π̄ to A. We do not claim that π̄ is closed, and neither do we claim that π̄ ∘ j is locally bounded for any representation of A: we need the representation of A to be locally bounded as well:

Definition 7.5. A representation (π, 𝔈) of a pro-C^{∗}-algebra A is locally bounded if the vectors ξ ∈ 𝔈 for which A → E, a ↦ π(a)ξ, is continuous form a core.

Proposition 7.6. Composition with j: A → A induces an equivalence between the categories of locally bounded representations of A and A which is compatible with isometric intertwiners and interior tensor products.

Proof. The ^{∗}-homomorphism j induces an isomorphism between the directed sets of C^{∗}-seminorms on A and A. Therefore, a representation π̄ of A is locally bounded if and only if the vectors ξ with ∥π̄(a)ξ∥ ≤ q(a)∥ξ∥ for all a ∈ A, for some q ∈ N (A), form a core. Since j(A) is dense in A, this is equivalent to ∥π(a)ξ∥ ≤ q(a)∥ξ∥ for all a ∈ A. Thus the closure of π̄ ∘ j is locally bounded if and only if π̄ is. An isometric intertwiner π̄_{1} ↪ π̄_{2} also intertwines the closures of π̄_{1} ∘ j and π̄_{2} ∘j by Lemma 2.14. Conversely, an isometric intertwiner between two locally bounded representations of A must map q-bounded vectors to q-bounded vectors for any q ∈ N (A). Thus it remains an isometric intertwiner between the canonical extensions of the representations to A. So the equivalence between the locally bounded representations of A and A is compatible with isometric intertwiners. It is also compatible with interior tensor products, that is, the closure of (π̄ ⊗_{D} 1_{G}) ∘ j is \overline{π̄ ∘ j} ⊗_{D} 1_{G}.

Proposition 7.7. All irreducible, locally bounded Hilbert space representations are bounded.

Proof. If π is irreducible, then the closed A-submodule E_{q} for a C^{∗}-seminorm q is either {0} or E. The latter must happen for some q if π is locally bounded.

Thus local boundedness is not an interesting notion for irreducible representations. If A has no C^{∗}-seminorms, then A = {0} and A has no locally bounded representations, so that the following discussion will be empty. Even if the map j: A → A is not injective, there are examples where all “integrable” representations of A come from A. An important case is the unit fibre for the canonical ℤ-grading on the Virasoro algebra studied in [27, §9.3]. In this case, A is not commutative, but all irreducible, integrable representations are characters and hence locally bounded.

Proposition 7.8. If π is a locally bounded representation, then \overline{π(a)} is regular and self-adjoint for each a ∈ A_{h}.

Proof. The map of left multiplication by j(a) ± i on A is invertible because j(a) ∈ A is symmetric and A is a pro-C^{∗}-algebra. Therefore, π̄(j(a)) ± i ⊆ π(a) ± i has dense range on E. Thus \overline{π(a)} is regular and self-adjoint, see [17, Chapter 10].

Corollary 7.9. Let A be a ^{∗}-algebra. The class Rep_{b}(A) of locally bounded representations of A is admissible.

Proof. Being locally bounded is clearly invariant under unitary ^{∗}-intertwiners and direct sums. It is also invariant under direct summands because a ∗-intertwiner maps bounded vectors to bounded vectors. If ξ ∈ E is bounded, then ξ ⊗ η ∈ E ⊗_{D} F is bounded for any C^{∗}-correspondence F. Thus a locally bounded representation on E induces one on E ⊗_{D} F. Since A_{h} is a strong generating set for A by Example 2.8, the class of representations for which all a ∈ A_{h} act by a regular and self-adjoint operator is admissible by Theorem 5.17. This class contains the locally bounded representations by Proposition 7.8. Hence this subclass is also admissible.

Any pro-C^{∗}-algebra A contains a dense unital C^{∗}-subalgebra A_{b} of bounded elements, see [22, Proposition 1.11]. For instance, if A is commutative, so that A ≅ C(Â, τ_{c}) by Proposition 7.2, then A_{b} = C_{b}(Â, τ_{c}) consists of the bounded continuous functions. Let (𝔈, π) be a locally bounded representation of A. This comes from a locally bounded representation (𝔈_{b}, π̄) of A by Proposition 7.6. The closure of the restriction of π̄ to A_{b} is a representation of a unital C^{∗}-algebra. Hence it is a unital ^{∗}-homomorphism ϱ: A_{b} → 𝔹(E) by Lemma 2.12.

Proposition 7.10. Two locally bounded representations π_{1} and π_{2} of A on a Hilbert module E are equal if and only if they induce the same representation of A_{b}.

Proof. Of course, π_{1} and π_{2} induce the same representation of A_{b} if π_{1} = π_{2}. Conversely, assume that π_{1} and π_{2} induce the same representation ϱ of A_{b}. If a ∈ A_{h}, then the Cayley transform c_{a} of j(a) ∈ A is a unitary element of A_{b}. The Cayley transforms of \overline{π_{1}(a)} and \overline{π_{2}(a)} are both equal to ϱ(c_{a}). Hence \overline{π_{1}(a)} = \overline{π_{2}(a)}. Since this holds for all a ∈ A_{h}, Proposition 2.9 gives π_{1} = π_{2}.

The C^{∗}-algebra A_{b} usually has many representations that do not come from locally bounded representations of A. Hence it is not a C^{∗}-hull. It is, however, a useful tool to decide when a representation μ of A on a C^{∗}-algebra B is a weak C^{∗}-hull, that is, when A separates the Hilbert space representations of B:

Proposition 7.11. Let μ be a locally bounded representation of A on a C^{∗}-algebra B and let ϱ: A_{b} → M(B) = 𝔹(B) be the associated representation of A_{b}. The image of ϱ is dense in M(B) in the strict topology if and only if B is a weak C^{∗}-hull for the class of B-integrable representations of A defined by μ.

Proof. Combine Proposition 7.10 and the following proposition for D = A_{b}.

Proposition 7.12. Let μ be a representation of A on a C^{∗}-algebra B. Let D be a C^{∗}-algebra and φ: D → M(B) a ^{∗}-homomorphism. Assume that two representations ϱ_{1}, ϱ_{2} of B on a Hilbert space H satisfy μ ⊗_{ϱ_{1}} 1_{H} = μ ⊗_{ϱ_{2}} 1_{H} if and only if ϱ̄_{1} ∘ φ = ϱ̄_{2} ∘ φ, where ϱ̄_{1} and ϱ̄_{2} denote the unique strictly continuous extensions of ϱ_{1}, ϱ_{2} to M(B). Then B is a weak C^{∗}-hull for a class of integrable representations of A if and only if φ(D) is dense in M(B) in the strict topology.

Proof. We use the criterion for weak C^{∗}-hulls in (1) in Proposition 3.8. Assume first that φ(D) is strictly dense in M(B). Let ϱ_{1}, ϱ_{2} be two Hilbert space representations of B that satisfy μ ⊗_{ϱ_{1}} 1_{H} = μ ⊗_{ϱ_{2}} 1_{H}. Extend ϱ_{1}, ϱ_{2} to strictly continuous representations ϱ̄_{1}, ϱ̄_{2} of M(B). By assumption, ϱ̄_{1}∘φ = ϱ̄_{2}∘φ, that is, ϱ̄_{1} and ϱ̄_{2} are equal on φ(D). Since they are strictly continuous and φ(D) is strictly dense, we get ϱ̄_{1} = ϱ̄_{2} and hence ϱ_{1} = ϱ_{2}. Thus the condition (1) in Proposition 3.8 is satisfied, making B a weak C^{∗}-hull of A. Conversely, assume that φ(D) is not strictly dense in M(B). We claim that the image of D is not weakly dense in the bidual W^{∗}-algebra B^{∗∗}. Any positive linear functional on B extends to a strictly continuous, positive linear functional on M(B) by extending its GNS-representation to a strictly continuous representation of M(B). By the Jordan decomposition, the same remains true for self-adjoint linear functionals and hence for all bounded linear functionals on B. Furthermore, such extensions are unique because B is strictly dense in M(B). Hence restriction to B maps the space of strictly continuous linear functionals on M(B) isomorphically onto the dual space B^{∗} of B, which is also the space of weakly continuous linear functionals on B^{∗∗}. If φ(D) is not strictly dense in M(B), then the Hahn–Banach Theorem gives a non-zero functional in B^{∗} that vanishes on the image of D. When viewed as a weakly continuous functional on B^{∗∗}, it witnesses that φ(D) is not weakly dense in B^{∗∗}. Let ϱ: B → 𝔹(H) be the direct sum of all cyclic representations of B. Then ϱ extends to an isomorphism of W^{∗}-algebras from B^{∗∗} onto the double commutant ϱ(B)′′of B in 𝔹(H). The extension of ϱ to M(B) restricts to a representation ϱ̄ ∘ φ: D → 𝔹(H). Since we assume that the image of D is not strictly dense in M(B), our claim shows that ϱ̄ ∘ φ(D) is not weakly dense in ϱ(B)′′. By the bicommutant theorem, this is equivalent to ϱ̄∘ φ(D)′ ≠ ϱ(B)′. Since these commutants are C^{∗}-algebras, they are the linear spans of the unitaries that they contain. So there is a unitary operator U in ϱ̄ ∘ φ(D)′ that is not contained in ϱ(B)′. So ϱ_{2} := UϱU^{∗} ≠ ϱ, but ϱ̄_{2} ∘ φ = ϱ̄∘ φ. By assumption, the latter implies μ⊗_{ϱ}1_{H} = μ⊗_{ϱ_{2}} 1_{H}. So A fails to separate the representations ϱ, ϱ_{2} of B although they are not equal. Hence B is not a weak C^{∗}-hull of A.

Remark 7.13. Proposition 7.12 applies whenever we can somehow produce enough bounded operators from a representation of A so that these bounded operators and the original representation have the same unitary ^{∗}-intertwiners. For instance, it applies if the elements of a strong generating set for A act by regular operators, so that we may take their bounded transforms.

The quotient map A_{q} ↠ A_{q′} for q ≥ q′ in N (A) identifies the primitive ideal space Prim(A_{q′}) with a closed subspace of Prim(A_{q}). Let PrimA := ⋃ _{q∈N (A)} Prim(A_{q}). Let a ∈ A and 𝔭 ∈ Prim(A). Then the norm ∥a∥_{𝔭} of the image of a in A_{q}/𝔭 is the same for all q ∈ N (A) with 𝔭 ∈ Prim(A_{q}). Hence the function 𝔭 ↦ ∥a∥_{𝔭} on Prim(A) is well defined.

Definition 7.14. An element a ∈ A vanishes at ∞ if for every ε > 0 there is q ∈ N (A) such that ∥a∥_{𝔭} < ε for 𝔭 ∈ Prim(A) \ Prim(A_{q}). An element a ∈ A is compactly supported if there is q ∈ N (A) with a ∈ 𝔭 for all 𝔭 ∈ Prim(A) \ Prim(A_{q}). Let C_{0}(A) and C_{c}(A) be the subsets of elements that vanish at ∞ and have compact support, respectively.

It may happen that C_{0}(A) = {0}. In the following, we are interested in the case where C_{0}(A) is dense in A. For instance, C_{0}(ℝ) is dense in C(ℝ).

Lemma 7.15. The subset C_{0}(A) is a closed ideal in A_{b}. The subspace C_{c}(A) is a two-sided ^{∗}-ideal in A. It is norm-dense in C_{0}(A). More generally, if D is a C^{∗}-algebra and φ: D → A is a ^{∗}-homomorphism, then φ^{−1} (C_{c}(A)) is dense in φ^{−1} (C_{0}(A)).

Proof. The quotient maps A ↠ A_{q} ↠ A_{q}/𝔭 for 𝔭 ∈ Prim(A_{q}) are ∗-homomorphisms. Thus C_{0}(A) is a ^{∗}-subalgebra of A. An element a ∈ A is bounded if and only if the norms of its images in A_{q} for q ∈ N (A) are uniformly bounded. The norm of a in A_{q} is the maximum of ∥a∥_{𝔭} for 𝔭 ∈ Prim(A_{q}). Hence a is bounded if and only if the function ∥a∥_{𝔭} on Prim(A) is bounded. Thus C_{0}(A) consists of bounded elements, and it is an ideal in A_{b}. We claim that the limit a of a norm-convergent sequence (a_{n})_{n∈ℕ} in C_{0}(A) again vanishes at ∞. Given ε > 0, there is n_{0} ∈ ℕ so that ∥a − a_{n}∥_{𝔭} ≤ ∥a − a_{n}∥ < ε/2 for all n ≥ n_{0} and all 𝔭 ∈ Prim(A). Since a_{n} vanishes at ∞, there is q ∈ N (A) with ∥a_{n}∥_{𝔭} < ε/2 for 𝔭 ∉ Prim(A_{q}). Thus ∥a∥_{𝔭} < ε for 𝔭 ∉ Prim(A_{q}). Thus C_{0}(A) is a closed ideal in A_{b}. The condition a ∈ 𝔭 for fixed 𝔭 ∈ Prim(A) defines a closed two-sided ^{∗}-ideal in A. Hence C_{c}(A) is a two-sided ^{∗}-ideal in A. Let a ∈ C_{0}(A) and ε > 0.

Let f_{ε} ∈ C_{b}([0, ∞)) be increasing and satisfy f_{ε}(t) = 0 for 0 ≤ t < ε and f_{ε}(t) = 1 for 2ε ≤ t. Then ∥a − af_{ε}(a^{∗}a)∥ ≤ 2ε, and f_{ε}(a^{∗}a) ∈ 𝔭 if ∥a^{∗}a∥_{𝔭} ≤ ε. Hence af_{ε}(a^{∗}a) ∈ C_{c}(A) for all ε > 0. Thus C_{c}(A) is dense in C_{0}(A). Similarly, if φ: D → A is a ^{∗}-homomorphism, x ∈ D, and φ(x) ∈ C_{0}(A), then φ(xf_{ε}(x^{∗}x)) ∈ C_{c}(A) and lim_{ε→0} xf_{ε}(x^{∗}x) = x in the norm topology on D.

Theorem 7.16. Let A be a ^{∗}-algebra and let A be its pro-C^{∗}-algebra completion. If C_{0}(A) is dense in A, then C_{0}(A) is a C^{∗}-hull for the class of locally bounded representations of A.

We shall prove a more general theorem that still applies if C_{0}(A) is not dense in A. Then probably there is no C^{∗}-hull for the class of all locally bounded representations. We may, however, find C^{∗}-hulls for smaller classes of representations. We describe such classes of representations by a generalisation of the spectral conditions in Example 5.14. The spectral condition for a locally closed subset in ℝ^{n} implicitly uses a subquotient of C_{0}(ℝ^{n}). We are going to describe subquotients of the pro-C^{∗}-algebra A. We then associate a class Rep_{b}(A, K) of locally bounded representations of A to a subquotient K. If C_{0}(K) is dense in K, then C_{0}(K) is a C^{∗}-hull for Rep_{b}(A, K). Theorem 7.16 is the special case K = A. Let J ◁ A be a closed, two-sided ^{∗}-ideal in the pro-C^{∗}-algebra A. Being closed, the ideal J is complete in the subspace topology, so it is also a pro-C^{∗}-algebra. Thus J = \underleftarrow{lim}J_{q}, where J_{q} ◁ A_{q} is the image of J in the quotient A_{q}. The quotient A/J is complete if A is metrisable, that is, its topology is defined by a sequence of C^{∗}-seminorms. It need not be complete in general, however. Therefore, we replace the quotient A/J by its completion B, which is a pro-C^{∗}-algebra as well. It is the projective limit of the quotients A_{q}/J_{q} for all q ∈ N (A). A subquotient of A is a closed, two-sided ^{∗}-ideal K ◁ B with B as above. Let Rep_{b}(A, K) consist of all representations π of A with the following properties:

(1) π is locally bounded, so it comes from a locally bounded representation π′ of A;

(2) the representation π′ annihilates J;

(3) the representation π̄ of B induced by π′ is nondegenerate on K, that is, π̄(K)(𝔈) is a core for π̄.

Define the C^{∗}-algebra C_{0}(K) and its dense ideal C_{c}(K) by replacing A by K in Definition 7.14. Equivalently, C_{0}(K) = C_{0}(B) ∩ K. We may choose J = 0 and K = A. Then Rep_{b}(A, A) = Rep_{b}(A) simply consists of all locally bounded representations of A. Hence Theorem 7.16 is the special case K = A of the following theorem:

Theorem 7.17. If C_{0}(K) is dense in K, then Rep_{b}(A, K) has the C^{∗}- hull C_{0}(K).

Proof. First we claim that Rep_{b}(A, K) is equivalent to the class of nondegenerate, locally bounded representations of the pro-C^{∗}-algebra K as in Definition 7.5. If K = A, this is Proposition 7.6. A locally bounded representation π′ of A descends to a representation π′′of the quotient A/J if and only if it annihilates J; the induced representation of A/J remains locally bounded with respect to the family of C^{∗}-seminorms from the quotient mappings A/J ↠ A_{q}/J_{q}. Hence it extends uniquely to a locally bounded representation π̄′′ of the completion B. Thus locally bounded representations of A for which the corresponding representation of A annihilates J are equivalent to locally bounded representations of B. We claim that a nondegenerate, locally bounded representation ϱ of K extends uniquely to B. Let q be a continuous seminorm on B, also write q for its restriction to K. The q-bounded vectors for ϱ form a nondegenerate K_{q}-module. The module structure extends uniquely to the multiplier algebra of K_{q}, and B_{q} maps to this multiplier algebra because K_{q} ◁ B_{q}. Letting q vary gives a locally bounded representation of B that remains nondegenerate on K. Conversely, any such representation of B is obtained in this way from its restriction to K. Thus representations of A that belong to Rep_{b}(A, K) are equivalent to nondegenerate, locally bounded representations of the pro-C^{∗}-algebra K. The equivalence above is compatible with isometric intertwiners, direct sums and interior tensor products, compare Proposition 7.6. Lemma 7.15 shows that C_{c}(K) := C_{c}(B) ∩ K is dense in K. This is an ideal in B as an intersection of two ideals. Hence left multiplication defines a representation of B on K with core C_{c}(K), which is locally bounded by construction. Through the canonical homomorphisms A → A → B, this becomes a representation of A. This representation clearly belongs to Rep_{b}(A, K). We claim that it is the universal representation for the class Rep_{b}(A, K). So let π be any representation in Rep_{b}(A, K). Then π comes from a unique nondegenerate, locally bounded representation π̄ of K. We must show that it comes from a unique nondegenerate representation ϱ of C_{0}(K). Let ϱ be the restriction of π̄ to C_{0}(K). Then ϱ(C_{c}(K))E ⊆ E_{b} ⊆ 𝔈. We are going to prove that this is a core. The bilinear map K ⊙ E_{b} → 𝔈 is separately continuous with respect to the pro-C^{∗}-algebra topology on K and the inductive limit topology on E_{b} = ⋃ _{q∈N (A)} E_{q}. We have assumed that it has dense range. Since C_{c}(K) is dense in K, the image of C_{c}(K) ⊙ E_{b} is a core. Thus ϱ(C_{c}(K))E is dense in 𝔈 in the graph topology. The representation ϱ is nondegenerate, and the associated representation of A is π. So π comes from a representation of C_{0}(K). The uniqueness of ϱ means that C_{0}(K) is a weak C^{∗}-hull for some class of integrable representations of A. We check this using Proposition 7.11. For q ∈ N (A), the image of A_{b} in M(K_{q}) contains K_{q} and hence is strictly dense. This implies that the image of A_{b} in M(C_{0}(K)) is strictly dense. So C_{0}(K) is a weak C^{∗}-hull for a class of representations of A by Proposition 7.11 It is even a C^{∗}-hull because the class of locally bounded representations is admissible by Corollary 7.9.

8 Commutative C^{∗}-hulls

Let A be a commutative ^{∗}-algebra. We are going to describe all commutative weak C^{∗}-hulls for A. Actually, we describe all locally bounded weak C^{∗}-hulls, and these turn out to be the same as the commutative ones. We study when a C^{∗}-hull satisfies the (Strong) Local–Global Principle and when the class of all locally bounded representations has a C^{∗}-hull. We compare the class of locally bounded representations with the class of representations defined by requiring all a ∈ A_{h} to act by a regular, self-adjoint operator.

Proposition 8.1. Let A be a ^{∗}-algebra and let B = C_{0}(X) be a commutative C^{∗}-algebra. Any representation of A on B has C_{c}(X) as a core and is locally bounded. There is a natural bijection between representations of A on B, ∗-homomorphisms A → C(X), and continuous maps Â → X.

Proof. Let (𝔅, μ) be a representation. Since 𝔅 is dense in B, for any x ∈ X there is f ∈ 𝔅 with f(x) ≠ 0. Then there is an open neighbourhood of x on which f is non-zero. A compact subset K of X may be covered by finitely many such open neighbourhoods. This gives finitely many functions f_{1}, …, f_{n} ∈ 𝔅 so that ∑ f_{i} · \overline{f_{i}} (x) > 0 for all x ∈ K. This sum again belongs to the right ideal 𝔅, and hence 𝔅 contains C_{c}(X). There is an approximate unit (u_{i})_{i∈I} for C_{0}(X) that belongs to C_{c}(X). If b ∈ 𝔅, then limμ(a)bu_{i} = μ(a)b for all a ∈ A. That is, limbu_{i} = b in the graph topology. Since bu_{i} ∈ C_{c}(X), C_{c}(X) is a core for (𝔅, μ). Given a ∈ A, we define a function f_{a}: X → ℂ by f_{a}(x) := (μ(a)b)(x)·b(x)^{−1} for any b ∈ C_{c}(X) with b(x) ≠ 0. This does not depend on the choice of b, and f_{a} is continuous in the open subset where b ≠ 0. Thus f_{a} ∈ C(X). The map A → C(X), a ↦ f_{a}, is a ^{∗}-homomorphism. Conversely, any ^{∗}-homomorphism A → C(X) gives a representation of A on C_{0}(X) with core C_{c}(X) by μ(a)b = f_{a} · b for all a ∈ A, b ∈ C_{c}(X). The maps that go back and forth between representations on C_{0}(X) and ^{∗}-homomorphisms A → C(X) are inverse to each other. A ^{∗}-homomorphism f: A → C(X) gives a continuous map X → Â by mapping x ∈ X to the character a ↦ f(a)(x). Conversely, a continuous map g: X → Â induces a ^{∗}-homomorphism g^{∗}: A → C(X), g^{∗}(a)(x) := g(x)(a), and these two constructions are inverse to each other. Let f: X → Â be a continuous map. Then f maps compact subsets in X to compact subsets of Â. If K ⊆ X is compact, then any element in C_{0}(K \∂K) ⊆ C_{0}(X) is ∥␣∥_{f(K)} -bounded for the C^{∗}-seminorm on A associated to the compact subset f(K) ⊆ Â. Thus all elements in C_{c}(X) are bounded. Since C_{c}(X) is a core for the representation of A associated to f, this representation is locally bounded.

Theorem 8.2. Let A be a commutative ^{∗}-algebra, let B = C_{0}(X) be a commutative C^{∗}-algebra, let f: X → Â be a continuous map, and let (𝔅, μ) be the corresponding representation of A on B. Call a representation of A on a Hilbert module E X-integrable if it is isomorphic to (𝔅, μ) ⊗_{ϱ} E for a representation ϱ of B on E. The following are equivalent:

(1) f: X → Â is injective;

(2) B is a weak C^{∗}-hull for the X-integrable representations;

(3) B is a C^{∗}-hull for the X-integrable representations.

Furthermore, any locally bounded weak C^{∗}-hull of A is commutative.

Proof. If f is not injective, then there are x ≠ y in X with f(x) = f(y). The evaluation maps at x and y are different 1-dimensional representations of B that induce the same representation of A. Hence the condition (1) in Proposition 3.8 is violated and so B is not a weak C^{∗}-hull. Conversely, assume that f is injective. The representation of A on B associated to f is locally bounded by Proposition 8.1 and hence induces a representation of the unital C^{∗}algebra C_{b}(Â, τ_{c}) of bounded elements in A ≅ C(Â, τ_{c}), see Proposition 7.2. Explicitly, this representation composes functions with f. Since f is injective, D := f^{∗}(C_{b}(Â, τ_{c})) ⊆ C_{b}(X) separates the points of X. We show that D is strictly dense in C_{b}(X) ≅ M(B). If K ⊆ X is compact, then the image of f^{∗}(C_{b}(Â, τ_{c}))|_{K} in the quotient C(K) of C_{0}(X) separates the points of K. Since this image is again a C^{∗}-algebra, it is equal to C(K) by the Stone–Weierstraß Theorem. Let f ∈ C_{b}(X). For any compact subset K ⊆ X, there is d_{K} ∈ D with d_{K} |_{K} = f. By functional calculus, we may arrange that ∥d_{K} ∥_{∞} ≤ ∥f∥. The net (d_{K}) indexed by the directed set of compact subsets K ⊆ X is uniformly bounded and converges towards f in the topology of uniform convergence on compact subsets. Hence it converges towards f in the strict topology (compare [11, Lemma A.1]). This finishes the proof that f^{∗}(A_{b}) is strictly dense in M(C_{0}(X)). Proposition 7.11 shows that B is a weak C^{∗}-hull for the B-integrable representations of A. Any X-integrable representation of A is locally bounded. The class Rep_{b}(A) of locally bounded representations of A is admissible by Corollary 7.9. Hence the smaller class of X-integrable representations inherits the equivalent conditions (2)–(4) in Proposition 3.8. Thus C_{0}(X) is even a C^{∗}-hull. Let B with the universal representation (𝔅, μ) be a locally bounded weak C^{∗}-hull. Then the image of C_{b}(Â, τ_{0}) in the multiplier algebra of B is strictly dense by Proposition 7.11. Thus M(B) is commutative, and then so is B. Thus a locally bounded weak C^{∗}-hull is commutative.

Theorem 8.3. Let A be a commutative ^{∗}-algebra, let B = C_{0}(X) be a commutative C^{∗}-algebra, and let f: X → Â be an injective continuous map. Let Rep_{int}(A, X) be the class of X-integrable representations. The following statements are equivalent if Â is metrisable:

(1) f: X → Â is a homeomorphism onto its image; (2) Rep_{int}(A, X) is defined by submodule conditions;

(3) Rep_{int}(A, X) satisfies the Strong Local–Global Principle;

(4) Rep_{int}(A, X) satisfies the Local–Global Principle;

(5) if limf(x_{n}) = f(x) for a sequence (x_{n})_{n∈ℕ} in X and x ∈ X, then already limx_{n} = x.

The implications (1)⇒(2)⇒(3)⇒(4)⇒(5) hold without assumptions on Â.

I do not know whether (1)–(4) are equivalent in general. The condition (5) is there to allow to go back from (4) to (1) at least for metrisable Â.

Proof. First we check (5)⇒(1) if Â is metrisable. If f is not a homeomorphism onto its image, then there is a subset U ⊆ X that is open, such that f(U) is not open in the subspace topology on f(X) ⊆ Â. Since Â is metrisable, there is x ∈ U and a sequence in f(X) \ f(U) that converges towards f(x). This lifts to a sequence (x_{n})_{n∈ℕ} in X \ U such that limf(x_{n}) = f(x). We cannot have limx_{n} = x because x_{n} never enters the open neighbourhood U of x. The implication (2)⇒(3) is Theorem 5.9, and (3)⇒(4) is trivial. We are going to verify (1)⇒(2) and (4)⇒(5). This will finish the proof of the theorem. Assume (1). Let π be a representation in Rep_{int}(A, X). Then π is locally bounded, and the operators \overline{π(a)} for a ∈ A_{h} are regular and self-adjoint by Proposition 7.8. Furthermore, their Cayley transforms belong to the image of A_{b} ≅ C_{b}(Â, τ_{c}), which is commutative. Hence the operators \overline{π(a)} for a ∈ A_{h} strongly commute with each other. The class Rep_{int}(A) of representations of A with the property that all \overline{π(a)}, a ∈ A_{h}, are regular and self-adjoint and strongly commute with each other is defined by submodule conditions by Examples 5.10 and Example 5.12. Let Y := ∏ _{a∈A_{h}} S^{1}. Given a representation in Rep_{int}(A), there is a unique representation ϱ: C(Y) → 𝔹(E) that maps the ath coordinate projection to the Cayley transform of \overline{π(a)}. We map Â to Y by sending χ ∈ Â to the point (c_{χ(a)} )_{a∈A_{h}} ∈ Y. Here c_{χ(a)} is the Cayley transform of the number χ(a) ∈ ℝ or, equivalently, the value of the Cayley transform of the unbounded function â ∈ C(Â) at χ. This is a homeomorphism onto its image because for a net of characters (χ_{i}) and a character χ on A, we have limχ_{i}(a) = χ(a) if and only if limc_{χ_{i} (a)} = c_{χ(a)} . Thus the composite map X → Â → Y is a homeomorphism onto its image as well. This forces the image to be locally closed because Y is compact and X locally compact, and a subspace of a locally compact space is locally compact if and only if its underlying subset is locally closed (see [2, I.9.7, Propositions 12 and 13]). Let \overline{X} ⊆ Y be the closure of the image of X in Y. Then X is open in \overline{X}. All representations in Rep_{int}(A) carry a unital ^{∗}-homomorphism C(Y) → 𝔹(E). Asking for this to factor through the quotient C(\overline{X}) of C(Y) is a submodule condition as in Example 5.14. Asking for the induced ^{∗}-homomorphism C(\overline{X}) → 𝔹(E) to remain nondegenerate on C_{0}(X) is another submodule condition as in Example 5.14. The class Rep′(A) defined by these two more submodule conditions is weakly admissible by Lemma 5.8. The universal X-integrable representation belongs to Rep′(A); by weak admissibility, this is inherited by all X-integrable representations. Conversely, we claim that any representation in Rep′(A) is X-integrable. If π ∈ Rep′(A), then the unital ^{∗}-homomorphism C(Y) → 𝔹(E) descends to a nondegenerate ^{∗}-homomorphism ϱ: C_{0}(X) → 𝔹(E). By construction, the extension of ϱ to multipliers maps the Cayley transform of f^{∗}(a) ∈ C(X) for a ∈ A_{h} to the Cayley transform of \overline{π(a)}. Let π′ be the X-integrable representation of A associated to ϱ. The regular, self-adjoint operators \overline{π′(a)} and \overline{π(a)} have the same Cayley transform for all a ∈ A_{h}. Hence \overline{π′(a)} = \overline{π(a)} for all a ∈ A_{h}. The subset A_{h} is a strong generating set for A by Example 2.8. Hence Proposition 2.9 gives π′ = π. Thus Rep_{int}(A, X) is the class of representations defined by the submodule conditions above. This finishes the proof that (1)⇒(2). Now we prove (4)⇒(5) by contradiction. Let (x_{n})_{n∈ℕ} and x be as in (5). Let ℕ̄ = ℕ ∪ {∞} be the one-point compactification of ℕ and view the sequence (x_{n}) and x as a map ξ: ℕ̄ → X. This map is not continuous, but composition with f gives a continuous map ℕ̄ → Â. Hence Proposition 8.1 gives a representation (𝔇, μ) of A on C(ℕ̄). This is not X-integrable because the map ℕ̄ → X is not continuous. We claim, however, that the representation (𝔇, μ) ⊗_{ϱ} H is X-integrable for any GNS-representation ϱ on a Hilbert space H. A state on C(ℕ̄) is the same as a Radon measure on ℕ̄. Since ℕ̄ is countable, any Radon measure is atomic. Thus the resulting GNS-representation is a direct sum of irreducible representations associated to characters. Each character on C(ℕ̄) gives an X-integrable representation because ξ(ℕ̄) ⊆ f(X). Hence (𝔇, μ) is a counterexample to the Local–Global Principle. So (4) cannot hold if (5) fails.

Example 8.4. Let A = ℂ[x], so that Â = ℝ. Let X be ℝ with the discrete topology, and let f: X → ℝ be the identity map. This is a continuous bijection, but not open. Hence the class of X-integrable representations violates the Local–Global Principle by Theorem 8.3. Nevertheless, C_{0}(X) is a C^{∗}-hull for the class of X-integrable representations of A by Theorem 8.2. An X-integrable representation of A is integrable as in Theorem 4.4, and so it comes from a single regular, self-adjoint operator T := \overline{π(x)}. The representation of ℂ[x] associated to T is X-integrable if and only if E = ⨁ _{λ∈ℝ} E_{λ}, where E_{λ} := {ξ ∈ E | T ξ = λξ} for λ ∈ ℝ is the λ-eigenspace of T.

Another example of a C^{∗}-hull for ℂ[x] where X → ℝ is bijective but not a homeomorphism onto its image is discussed in Example 4.9.

Theorem 8.5. There is a C^{∗}-hull for Rep_{b}(A) if and only if the compactly generated topology τ_{c} on Â is locally compact, and then the C^{∗}-hull is C_{0}(Â, τ_{c}).

Proof. Assume first that (Â, τ_{c}) is locally compact. The pro-C^{∗}-algebra completion A that acts on locally bounded representations of A is C(Â, τ_{c}) by Proposition 7.2. The primitive ideal space of C(K) for a compact subspace K ⊆ Â is simply K, and ∥a∥_{𝔭} = |a(𝔭)| for a ∈ C(Â, τ_{c}) and 𝔭 ∈ PrimC(K) ≅ K. Therefore, a function f ∈ C(Â, τ_{c}) vanishes at ∞ in the sense of Definition 7.14 if and only if it vanishes at ∞ in the usual sense. The subalgebra C_{0}(A) = C_{0}(Â, τ_{c}) is dense in A because τ_{c} is locally compact. Now Theorem 7.16 shows that C_{0}(A) = C_{0}(Â, τ_{c}) is a C^{∗}-hull for the class of locally bounded representations of A. Conversely, let B be a (weak) C^{∗}-hull for the locally bounded representations of A. Then B is commutative by Theorem 8.2. Let Y be the spectrum of B. The representation of A on B ≅ C_{0}(Y) corresponds to a continuous map f: Y → Â by Proposition 8.1. Let D = C_{0}(X) be a commutative C^{∗}-algebra. Any representation of A on D is locally bounded. So the bijection Rep_{b}(A, D) ≅ Rep(B, D) is a bijection between the spaces of continuous maps X → Â and X → Y. More precisely, this bijection is composition with f. For the one-point space X, this bijection says that f: Y → Â is bijective. The bijection for all compact X means that f becomes a homeomorphism if we replace the topologies on Y and Â by the associated compactly generated ones. The topology on Y is already compactly generated because Y is locally compact. Hence f is a homeomorphism from Y to (Â, τ_{c}). So τ_{c} is locally compact.

Let Rep_{int}(A) be the class of all representations with the property that \overline{π(a)} is regular and self-adjoint for all a ∈ A_{h}. We are going to compare Rep_{int}(A) and Rep_{b}(A). Proposition 7.8 gives Rep_{b}(A) ⊆ Rep_{int}(A).

Theorem 8.6. The class Rep_{int}(A) is admissible and defined by submodule conditions. Hence it satisfies the Strong Local–Global Principle. The operators \overline{π(a)} for a ∈ A_{h} strongly commute for all π ∈ Rep_{int}(A). Let S ⊆ A_{h} be a strong generating set for A. If \overline{π(a)} is regular and self-adjoint for all a ∈ S, then already π ∈ Rep_{int}(A).

Proof. The class Rep^{S}(A) of representations defined by requiring \overline{π(a)} to be regular and self-adjoint for all a ∈ S for a strong generating set S is admissible and defined by submodule conditions by Theorem 5.17. The class Rep_{int}(A) is defined by submodule conditions as well by Example 5.10. So is the subclass Rep′(A) of all representations in Rep_{int}(A) for which the operators \overline{π(a)} for all a ∈ A_{h} strongly commute (Example 5.12). Hence our three classes of representations satisfy the Strong Local–Global Principle by Theorem 5.9. The classes Rep_{int}(A) and Rep′(A) have the same Hilbert space representations by [28, Theorem 9.1.2]. Since S is a strong generating set, the domain of any representation π in Rep^{S}(A) is ⋂ _{a∈S} dom(\overline{π(a)}) by (2.10). This contains ⋂ _{a∈S,n∈ℕ} dom(\overline{π(a)}^{n}). Now [28, Theorem 9.1.3] shows that Rep^{S}(A) and Rep_{int}(A) contain the same Hilbert space representations. Since our three classes of representations satisfy the (Strong) Local–Global Principle and have the same Hilbert space representations, they are equal.

Theorem 8.7. If A is commutative and countably generated, then

Rep_{int}(A) = Rep_{b}(A).

Proof. Proposition 7.8 gives Rep_{b}(A) ⊆ Rep_{int}(A). Conversely, let (𝔈, π) be a representation on a Hilbert module E in Rep_{int}(A); that is, \overline{π(a)} is regular and self-adjoint for each a ∈ A. Let (a_{i})_{i∈ℕ} be a countable generating set for A. We may assume without loss of generality that a_{i} = a_{i}^{∗} for all i ∈ ℕ and that (a_{i}) is a basis for A and hence a strong generating set. Let ξ ∈ 𝔈. We are going to approximate ξ by bounded vectors for π. This will show that π is locally bounded. For each i ∈ ℕ, there is a canonical homomorphism α_{i}: ℂ[x] → A mapping x ↦ a_{i}. The closure of π ∘ α_{i} is an integrable representation of ℂ[x] as in condition (2) in Theorem 4.4. Hence it corresponds to a representation ϱ_{i}: C_{0}(ℝ) → 𝔹(E), the functional calculus of \overline{π(a_{i})}. The operators \overline{π(a)} for a ∈ A_{h} strongly commute by Theorem 8.6. Thus the Cayley transform of a_{i} commutes with \overline{π(a)} and, in particular, maps the domain of \overline{π(a)} to itself. The same remains true for ϱ_{i}(f) for all f ∈ C_{0}(ℝ) because we get them by the (bounded) functional calculus for the Cayley transform of \overline{π(a_{i})}. So ϱ_{i}(f)(𝔈) ⊆ 𝔈 by (2.6) and ϱ_{i}(f)π(a) = π(a)ϱ_{i}(f) for all f ∈ C_{0}(ℝ), a ∈ A as operators on 𝔈. Now we show that π ∘ α_{i} is locally bounded. If f ∈ C_{c}(ℝ) is supported in a compact subset K ⊆ ℝ, then

∥π(h(a_{i}))ϱ_{i}(f)ξ∥ = ∥ϱ_{i}(h · f)ξ∥ ≤ C sup{|h(x)| | x ∈ K}

for all h ∈ ℂ[x]; thus ϱ_{i}(f)ξ is bounded for the representation π ∘ α_{i}. There is an approximate unit (f_{n}) for C_{0}(ℝ) that lies in C_{c}(ℝ). Then limϱ_{i}(f_{n})ξ = ξ for all ξ ∈ 𝔈, even in the graph topology for π because π(a)ϱ_{i}(f_{n})ξ = ϱ_{i}(f_{n})π(a)ξ for all a ∈ A, f_{n} ∈ C_{0}(ℝ), ξ ∈ 𝔈. Therefore, the bounded vectors of the form ϱ(f)ξ with f ∈ C_{c}(ℝ), ξ ∈ 𝔈 form a core for π ∘ α_{i}. So π ∘ α_{i} is locally bounded. We now refine this construction to approximate ξ by bounded vectors for the whole representation π. We construct ϱ_{i} as above. Fix i, k ∈ ℕ and let ξ′ := (1 + π(a_{0}^{2}) + ⋯+ π(a_{k}^{2}))ξ ∈ 𝔈. The argument above gives f_{i,k} ∈ C_{c}(ℝ) with 0 ≤ f_{i,k} ≤ 1 and ∥ϱ_{i}(f_{i,k} )ξ′− ξ′∥ < 2^{−k} . Thus ∥ϱ_{i}(f_{i,k} )ξ − ξ∥_{a_{j}} < 2^{−k} in the graph norm for a_{j} for 0 ≤ j ≤ k. For k, l ∈ ℕ, let

ξ_{k,l} := ϱ_{0}(f_{0,k})ϱ_{1}(f_{1,k+1} ) ⋯ ϱ_{l}(f_{l,k+l} )ξ.

The operators ϱ_{i}(f_{i,j} ) are norm-contracting, map 𝔈 into itself, and commute with each other and with the unbounded operators π(a) for all a ∈ A. Hence

∥ξ_{k,l} − ξ_{k,l+d} ∥_{a_{j}} ≤ ∑_{i=1}^{d} ∥ξ_{k,l+i−1} − ξ_{k,l+i} ∥_{a_{j}}

≤ ∑_{i=1}^{d} ∥ϱ_{l+i} (f_{l+i,k+l+i} )ξ − ξ∥_{a_{j}} ≤ ∑_{i=1}^{d} 2^{−k−l−i} = 2^{−k−l}

for all k, l, d ∈ ℕ, 0 ≤ j ≤ k + l + 1. Since we assumed (a_{j}) to be a strong generating set, the graph norms for a_{j} generate the graph topology. So the estimate above shows that (ξ_{k,l})_{l∈ℕ} with fixed k is a Cauchy sequence in 𝔈 in the graph topology. Thus it converges to some ξ_{k} ∈ 𝔈. Letting ξ_{k,−1} := ξ, the above estimate remains true for l = −1 and gives ∥ξ_{k,l} − ξ∥_{a_{j}} ≤ 2^{−k+1} for all j ≤ k, uniformly in l ∈ ℕ. This implies ∥ξ_{k} − ξ∥_{a_{j}} ≤ 2^{−k+1} for j ≤ k, so that limξ_{k} = ξ in the graph topology. It remains to show that each ξ_{k} is a bounded vector. Fix k, i ∈ ℕ and let b ∈ A. Choose R_{i} > 0 so that f_{i,k+i} is supported in [−R_{i}, R_{i}]. If l ≥ i, then π(b)ξ_{k,l} ∈ ϱ_{i}(C_{0}(−R_{i}, R_{i}))E because ϱ_{i}(f_{i,k+i} ) occurs in the definition of ξ_{k,l}. As above, this implies ∥π(a_{i})π(b)ξ_{k}∥ ≤ R_{i}∥π(b)ξ_{k}∥ for all b ∈ A. Thus

q(a) := sup_{b∈A} \frac{∥π(a)π(b)ξ_{k}∥}{∥π(b)ξ_{k}∥}

is finite for a = a_{i}. Since a_{i} is a basis for A and q is subadditive, we get q(a) < ∞ for all a ∈ A. Since q(a) is the operator norm of π(a)|_{π(A)ξ_{k}} , it is a C^{∗}-seminorm on A. By construction, ∥π(a)ξ_{k}∥ ≤ q(a) for all a ∈ A, that is, ξ_{k} is bounded.

Proposition 8.8. If Rep_{int}(A) has a weak C^{∗}-hull, then Rep_{int}(A) = Rep_{b}(A).

Proof. Let B with the universal representation (𝔅, μ) be a weak C^{∗}-hull for Rep_{int}(A). First we claim that B is commutative. Let ω: B ↪ 𝔹(H) be a faithful representation. This corresponds to an integrable representation π of A. Since the equivalence Rep_{int}(A, H) ≅ Rep(B, H) is compatible with unitary ∗-intertwiners, the commutant of ω(B) is the C^{∗}-algebra of ^{∗}-intertwiners of π by Proposition 3.3. The commutant of this is a commutative von Neumann algebra by [28, Theorem 9.1.7]. So the bicommutant of ω(B) is commutative. This forces B to be commutative. Any representation of A on a commutative C^{∗}-algebra is locally bounded by Theorem 8.2. If the universal representation for Rep_{int}(A) is locally bounded, then all representations in Rep_{int}(A) are locally bounded, so that Rep_{int}(A) = Rep_{b}(A). Thus Rep_{int}(A) only has a weak C^{∗}-hull if Rep_{int}(A) = Rep_{b}(A).

Example 8.9. Let A be the ^{∗}-algebra ℂ[(x_{i})_{i∈ℕ} ] of polynomials in countably many symmetric generators. Then Â ≅ ∏ _{ℕ} ℝ with the product topology. This is metrisable. So τ_{c} is the usual product topology. Since this is not locally compact, Rep_{b}(A) has no C^{∗}-hull, not even a weak one (Theorem 8.5). Since A is countably generated, Rep_{int}(A) = Rep_{b}(A) by Theorem 8.7. A commutative (weak) C^{∗}-hull for some class of representations of A is equivalent to an injective, continuous map X → Â for a locally compact space X by Theorem 8.2. Let G be a topological group. A host algebra for a G is defined in [12] as a C^{∗}-algebra B with a continuous representation λ of G by unitary multipliers, such that for each Hilbert space H, the map that sends a representation ϱ: B → 𝔹(H) to a unitary representation ϱ ∘ λ of G is injective. We claim that commutative C^{∗}-hulls for the polynomial algebra ℂ[(x_{i})_{i∈ℕ} ] are equivalent to host algebras of the topological group ℝ^{(ℕ)} := ⨁ _{ℕ} ℝ. Let C^{∗}(G_{d}) be the C^{∗}-algebra of G viewed as a discrete group. Representations of C^{∗}(G_{d}) are equivalent to representations of the discrete group underlying G by unitary multipliers. Since any representation of C^{∗}(G_{d}) is bounded, any weakly admissible class of representations of C^{∗}(G_{d}) is admissible by Corollary 7.9. Call a representation of C^{∗}(G_{d}) continuous if the corresponding representation of G is continuous. This class is easily seen to be weakly admissible, hence admissible. The unital ^{∗}-homomorphism C^{∗}(G_{d}) → M(B) associated to the unitary representation λ for a host algebra B is continuous by assumption. Thus B-integrable representations of C^{∗}(G_{d}) are continuous. The injectivity requirement in the definition of a host algebra is exactly the condition (1) in Proposition 3.8, and this is equivalent to B being a C^{∗}-hull. Thus a host algebra for G is the same as a C^{∗}-hull or weak C^{∗}-hull for a class of continuous representations of C^{∗}(G_{d}). In applications, we would rather study continuous representations of G through the Lie algebra of G instead of through the inseparable C^{∗}-algebra C^{∗}(G_{d}). The Lie algebra of G = ℝ^{(ℕ)} is the Abelian Lie algebra ℝ^{(ℕ)}, and its universal enveloping algebra is the polynomial algebra A = ℂ[(x_{i})_{i∈ℕ} ]. Call a representation of A integrable if it belongs to Rep_{int}(A) = Rep_{b}(A). Let E be a Hilbert module. We claim that an integrable representation of A on E is equivalent to a strictly continuous, unitary representation of the group ℝ^{(ℕ)} on E. Indeed, a unitary representation of ℝ is equivalent to a representation of C^{∗}(ℝ) ≅ C_{0}(ℝ), and these are equivalent to integrable representations of ℂ[x] as in Theorem 4.4. In an integrable representation of ℂ[(x_{i})_{i∈ℕ} ], the operators \overline{π(x_{i})} for i ∈ ℕ strongly commute by Theorem 8.6. Hence the resulting representations of C_{0}(ℝ) commute. Equivalently, the resulting continuous representations of ℝ commute, so that we may combine them to a representation of the Abelian group ℝ^{(ℕ)}. Conversely, a continuous unitary representation of ℝ^{(ℕ)} provides nondegenerate representations of C_{0}(ℝ^{m}) for all m ∈ ℕ by restricting the representation to ℝ^{m} ⊆ ℝ^{(ℕ)}. These correspond to a compatible family of representations of the polynomial algebras ℂ[x_{1}, …, x_{m}] for m ∈ ℕ. The intersection of their domains is dense by [28, Lemma 1.1.2]. So these representations combine to a representation of A = ℂ[(x_{i})_{i∈ℕ} ]. Hence an integrable representation of A on a Hilbert module as in Theorem 8.6 is equivalent to a continuous representation of ℝ^{(ℕ)}.

9 From graded ^{∗}-algebras to Fell bundles

Let G be a discrete group with unit element e.

Definition 9.1. A G-graded ^{∗}-algebra is a unital algebra A with a linear direct sum decomposition A = ⨁ _{g∈G} A_{g} with A_{g} · A_{h} ⊆ A_{gh}, A_{g}^{∗} = A_{g^{−1}} , and 1 ∈ A_{e} for all g, h ∈ G. Thus A_{e} ⊆ A is a unital ^{∗}-subalgebra.

The articles [7, 27] study many examples of G-graded ^{∗}-algebras. We fix some notation used throughout this section. Let E be a Hilbert module over a C^{∗}-algebra D. Let (𝔈, π) be a representation of A on E. Let π_{g}: A_{g} → End_{D}(𝔈) for g ∈ G be the restrictions of π, so π = ⨁ _{g∈G} π_{g}. Since π is a ∗-homomorphism,

π_{g}(a_{g})π_{h}(a_{h}) = π_{gh}(a_{g} · a_{h}), π_{g^{−1}} (a_{g}^{∗}) ⊆ π_{g}(a_{g})^{∗}

for all a_{g} ∈ A_{g}, a_{h} ∈ A_{h}. The last condition means that ⟨ξ, π_{g}(a_{g})η⟩ = ⟨π_{g^{−1}} (a_{g}^{∗})ξ, η⟩ for all ξ, η ∈ 𝔈. In particular, π_{e}: A_{e} → End(𝔈) is a representation of A_{e}.

Lemma 9.2 (compare [27, Lemma 12]). The families of norms ∥ξ∥_{a} := ∥π(a)ξ∥ for a ∈ A and for a ∈ A_{e} generate equivalent topologies on 𝔈. Hence the representation π_{e}: A_{e} → End_{D}(𝔈) is closed if and only if π is closed.

Proof. Any element of A is a sum a = ∑ _{g∈G} a_{g} with a_{g} ∈ A_{g} and only finitely many non-zero terms. We estimate ∥ξ∥_{a} ≤ ∑ _{g∈G} ∥ξ∥_{a_{g}} , and ∥ξ∥_{a_{g}} ≤ \frac{5}{4}∥ξ∥_{a_{g}^{∗} a_{g}} by the proof of Lemma 2.2. Since a_{g}^{∗}a_{g} ∈ A_{e}, the graph topologies for π_{e} and π are equivalent.

9.1 Integrability by restriction

Definition 9.3. Let a weakly admissible class of integrable representations of A_{e} on Hilbert modules be given. We call a representation of A on a Hilbert module integrable if its restriction to A_{e} is integrable.

Here “restriction of π” means the representation π_{e} with the same domain 𝔈 as π. This is closed by Lemma 9.2.

Proposition 9.4. If integrability for representations of A_{e} is defined by submodule conditions, then the same holds for A. If the Local–Global Principle holds for the integrable representations of A_{e}, it also holds for the integrable representations of A. If the class of integrable representations of A_{e} is admissible or weakly admissible, the same holds for A.

Proof. The first two statements and the claim about weak admissibility are trivial because integrability for a representation of A only involves its restriction to A_{e}. Lemma 9.2 shows that restriction from A to A_{e} does not change the domain. Hence (2) in Proposition 3.8 is inherited by A if it holds for A_{e}. That is, admissibility of the integrable representations passes from A_{e} to A.

It is unclear whether A also inherits the Strong Local–Global Principle from A_{e}. This may often be bypassed using Theorem 5.9.

9.2 Inducible representations and induction

Let F be a Hilbert D-module and let 𝔉 ⊆ F and φ_{e}: A_{e} → End_{D}(𝔉) be a representation of A_{e} on F. We try to induce φ_{e} to a representation of A as in [27]. Thus we consider the algebraic tensor product A ⊙ 𝔉 and equip it with the obvious right D-module structure and the unique sesquilinear map that satisfies

⟨a_{1} ⊗ ξ_{1}, a_{2} ⊗ ξ_{2}⟩ = δ_{g,h} ⟨ξ_{1}, φ_{e}(a_{1}^{∗}a_{2})ξ_{2}⟩

for all g, h ∈ G, a_{1} ∈ A_{g}, a_{2} ∈ A_{h}, ξ_{1}, ξ_{2} ∈ 𝔉. This map is sesquilinear and descends to the quotient space A ⊙_{A_{e}} 𝔉. It is symmetric and D-linear in the sense that ⟨x, y⟩ = ⟨y, x⟩^{∗} and ⟨x, yd⟩ = ⟨x, y⟩d. Let π be the action of A on A ⊙_{A_{e}} 𝔉 by left multiplication. This representation is formally a ∗-homomorphism in the sense that ⟨x, π(a)y⟩ = ⟨π(a^{∗})x, y⟩ for all a ∈ A, x, y ∈ A ⊙_{A_{e}} 𝔉. The only thing that is missing to get a representation of A on a Hilbert D-module is positivity of the inner product. This requires a subtle extra condition.

Proposition 9.5. The following are equivalent:

(1) the sesquilinear map on A ⊙_{A_{e}} 𝔉 defined above is positive semidefinite;

(2) for all g ∈ G, n ∈ ℕ and all a_{1}, …, a_{n} ∈ A_{g}, ξ_{1}, …, ξ_{n} ∈ 𝔉, the element ∑ _{k,l=1}^{n} ⟨ξ_{k}, φ_{e}(a_{k}^{∗}a_{l})ξ_{l}⟩ ∈ D is positive;

(3) for all g ∈ G, n ∈ ℕ and all a_{1}, …, a_{n} ∈ A_{g}, ξ_{1}, …, ξ_{n} ∈ 𝔉, the matrix (⟨ξ_{k}, φ_{e}(a_{k}^{∗}a_{l})ξ_{l}⟩)_{k,l} ∈ 𝕄_{n}(D) is positive.

Proof. The condition (2) for fixed g ∈ G says that the sesquilinear map on A_{g} ⊙_{A_{e}} 𝔉 is positive semidefinite. Since the subspaces A_{g} ⊙_{A_{e}} 𝔉 for different g are orthogonal, this is equivalent to positive semidefiniteness on A⊙_{A_{e}} 𝔉. Thus (1) ⟺ (2). We prove (2) ⟺ (3). Fix g ∈ G, n ∈ ℕ, a_{1}, …, a_{n} ∈ A_{g} and ξ_{1}, …, ξ_{n} ∈ 𝔉. Let y = (y_{kl}) ∈ 𝕄_{n}(D) be the matrix in (3). By [17, Lemma 4.1], y ≥ 0 in 𝕄_{n}(D) ⊆ 𝔹(D^{n}) if and only if ⟨d, yd⟩ ≥ 0 for all d = (d_{1}, …, d_{n}) ∈ D^{n}. That is, ∑ _{k,l=1}^{n} d_{k}^{∗}y_{kl}d_{l} ≥ 0 for all d_{1}, …, d_{n} ∈ D. Since 𝔉 is a right D-module, this condition for all ξ_{i} ∈ 𝔉, d_{i} ∈ D is equivalent to (2).

Definition 9.6. A representation φ_{e} of A_{e} is inducible (to A) if it satisfies the equivalent conditions in Proposition 9.5.

If A_{e} were a C^{∗}-algebra, it would be enough to assume ⟨ξ, φ_{e}(a^{∗}a)ξ⟩ ≥ 0 for all g ∈ G, a ∈ A_{g}, ξ ∈ 𝔉, which amounts to the condition a^{∗}a ≥ 0 in A_{e} for all g ∈ G, a ∈ A_{g}. This is part of the definition of a Fell bundle over a group.

For more general ^{∗}-algebras, the positivity conditions for different n ∈ ℕ in Proposition 9.5 may differ, compare [9]. Let A⊗_{A_{e}} F be the Hilbert module completion of A⊙_{A_{e}} 𝔉 for the inner product above. The ^{∗}-algebra A acts on A ⊙_{A_{e}} 𝔉 by left multiplication, a_{1} · (a_{2} ⊗ ξ) := (a_{1}a_{2}) ⊗ ξ for a_{1}, a_{2} ∈ A, ξ ∈ 𝔉. As in the proof of Lemma 2.18, this module structure descends to the image of A ⊙_{A_{e}} 𝔉 in A ⊗_{A_{e}} F and gives a well defined representation π of A on A⊗_{A_{e}} F. Its closure is called the induced representation from φ_{e}, and its domain is denoted by A ⊗_{A_{e}} 𝔉. The decomposition A ⊙_{A_{e}} 𝔉 = ⨁ _{g∈G} A_{g} ⊙_{A_{e}} 𝔉 is A_{e}-invariant and orthogonal for the above inner product. Hence

A ⊗_{A_{e}} F ≅ ⨁_{g∈G} A_{g} ⊗_{A_{e}} F,  (9.7)

where A_{g} ⊗_{A_{e}} F is the closure of the image of A_{g} ⊙_{A_{e}} 𝔉 or, equivalently, the Hilbert D-module completion of A_{g} ⊙_{A_{e}} 𝔉 with respect to the restriction of the inner product. Each summand A_{g} ⊗_{A_{e}} F carries a closed representation of A_{e} with domain A_{g} ⊗_{A_{e}} 𝔉, and π|_{A_{e}} is the direct sum of these representations.

Lemma 9.8. Let π be any representation of A. Then π|_{A_{e}} is inducible.

Proof. For g ∈ G, a_{1}, …, a_{n} ∈ A_{g}, ξ_{1}, …, ξ_{n} ∈ 𝔈, let y := ∑ _{k=1}^{n} π(a_{k})ξ_{k}. Then

∑_{k,l=1}^{n} ⟨ξ_{k}, π|_{A_{e}} (a_{k}^{∗}a_{l})ξ_{l}⟩ = ∑_{k,l=1}^{n} ⟨π(a_{k})ξ_{k}, π(a_{l})ξ_{l}⟩ = ⟨y, y⟩ ≥ 0.

Lemma 2.24 about the associativity of ⊗ has a variant for induction:

Lemma 9.9. Let D_{1}, D_{2} be C^{∗}-algebras, let E be a Hilbert D_{1}-module and let F be a C^{∗}-correspondence between D_{1}, D_{2}. Let (φ_{e}, 𝔈) be an inducible representation of A on E. Then the representation φ_{e} ⊗_{D_{1}} F on E ⊗_{D_{1}} F is inducible and there is a canonical unitary ^{∗}-intertwiner of representations of A,

(A ⊗_{A_{e}} E) ⊗_{D_{1}} F ≅ A ⊗_{A_{e}} (E ⊗_{D_{1}} F).

Proof. Let 𝔈 ⊗_{D_{1}} F ⊆ E ⊗_{D_{1}} F be the domain of φ_{e} ⊗_{D_{1}} F. Let g_{1}, …, g_{n} ∈ G, a_{1}, …, a_{n} ∈ A_{g_{i}} , and ω_{1}, …, ω_{n} ∈ 𝔈 ⊗_{D_{1}} F. Let ζ := ∑ _{k=1}^{n} a_{k} ⊗ ω_{k} ∈ A ⊙ (𝔈 ⊗_{D_{1}} F). To show that φ_{e} ⊗_{D_{1}} F is inducible, we must prove that ⟨ζ, ζ⟩ ∈ D_{2} is positive. Vectors in 𝔈 ⊙ F form a core for φ_{e} ⊗_{D_{1}} F by construction. Hence there is a sequence of vectors of the form

ω_{j,τ} := ∑_{i=1}^{ℓ_{j}} ξ_{τ,j,i} ⊗ η_{τ,j,i} , ξ_{τ,j,i} ∈ 𝔈, η_{τ,j,i} ∈ F,

which, for τ → ∞, converges to ω_{j} in the graph norms of the elements δ_{g_{m} ,g_{k}} a_{m}^{∗}a_{k} ∈ A_{e} for all m, k = 1, …, n. Let ζ_{τ} := ∑ _{j=1}^{n} a_{j} ⊗ ω_{j,τ} . Then

lim_{τ→∞} ⟨ζ_{τ}, ζ_{τ}⟩ = lim_{τ→∞} ⟨ζ_{τ}, ζ⟩ = lim_{τ→∞} ⟨ζ, ζ_{τ}⟩ = ⟨ζ, ζ⟩

in norm and

⟨ζ_{τ}, ζ_{τ}⟩ = ⟨ ∑_{i,j} a_{j} ⊗ ξ_{τ,j,i} ⊗ η_{τ,j,i} , ∑_{m,k} a_{k} ⊗ ξ_{τ,k,m} ⊗ η_{τ,k,m} ⟩

= ∑_{i,j,k,m} δ_{g_{j} ,g_{k}} ⟨η_{τ,j,i} , ⟨ξ_{τ,j,i} , φ_{e}(a_{j}^{∗}a_{k})ξ_{τ,k,m} ⟩_{D_{1}} · η_{τ,k,m} ⟩_{D_{2}} .

This is also the inner product of ζ_{τ} with itself in the tensor product (A⊗_{A_{e}} E)⊗F. This is positive because φ_{e} is inducible and the usual tensor product of the Hilbert D_{1}-module A ⊗_{A_{e}} E with the D_{1}, D_{2}-correspondence F is a Hilbert D_{2}-module. Hence ⟨ζ_{τ}, ζ_{τ}⟩ ≥ 0 for all τ. Since the positive elements in D_{2} form a closed subset, this implies ⟨ζ, ζ⟩ ≥ 0. Thus φ_{e} ⊗_{D_{1}} F is inducible. The argument above also shows that the linear span of vectors of the form a ⊗ ξ ⊗ η with a ∈ A, ξ ∈ 𝔈, η ∈ F is a core for the representation of A_{e} on A ⊗_{A_{e}} (E ⊗_{D_{1}} F). Such vectors also form a core for the representation of A on (A ⊗_{A_{e}} E) ⊗_{D_{1}} F. The left actions of A and the D_{2}-valued inner products coincide on such vectors. Hence there is a unique unitary ^{∗}-intertwiner that maps the image of a⊗ξ⊗η in (A⊗_{A_{e}} E)⊗_{D_{1}} F to its image in A⊗_{A_{e}} (E⊗_{D_{1}} F).

9.3 C^{∗}-Hulls for the unit fibre

We assume that the chosen class of integrable representations of A_{e} has a (weak) C^{∗}-hull B_{e}. We want to construct a Fell bundle whose section C^{∗}-algebra is a (weak) C^{∗}-hull for the integrable representations of A. At some point, we need B_{e} to be a full C^{∗}-hull (compatible with isometric intertwiners) and one more extra condition. But we may begin the construction without these assumptions. First we build the unit fibre B_{e}^{+} of the Fell bundle. It is a (weak) C^{∗}-hull for the inducible, integrable representations of A_{e}. Let (𝔅_{e}, μ_{e}) be the universal integrable representation of A_{e} on B_{e}. Let x_{−} for a self-adjoint element x in a C^{∗}-algebra denote its negative part.

Definition 9.10. Let B_{e}^{+} be the quotient C^{∗}-algebra of B_{e} by the closed twosided ideal generated by elements of the form

( ∑_{k,l=1}^{n} b_{k}^{∗} · μ_{e}(a_{k}^{∗}a_{l}) · b_{l}) _{−} for g ∈ G, a_{1}, …, a_{n} ∈ A_{g}, b_{1}, …, b_{n} ∈ 𝔅_{e}.

(9.11)

Let 𝔅_{e}^{+} be the image of 𝔅_{e} in B_{e}^{+} and let μ_{e}^{+} : A_{e} → End_{B_{e}^{+}} (𝔅_{e}^{+} ) be the induced representation of A_{e} on this quotient.

The following proposition shows that the representation (𝔅_{e}^{+} , μ_{e}^{+} ) of A_{e} on B_{e}^{+} is the universal inducible, integrable representation of A_{e}.

Proposition 9.12. Let (𝔉, φ_{e}) be an integrable representation of A_{e} on a Hilbert module F. Let φ̄_{e} : B_{e} → 𝔹(F) be the corresponding representation of B_{e}. Then φ_{e} is inducible if and only if φ̄_{e} factors through the quotient map B_{e} ↠ B_{e}^{+} . Thus B_{e}^{+} is a C^{∗}-hull for the inducible, integrable representations of A_{e}.

Proof. Assume first that φ_{e} is inducible. Let ξ ∈ F and let g ∈ G, a_{1}, …, a_{n} ∈ A_{g} and b_{1}, …, b_{n} ∈ 𝔅_{e} be as in (9.11). Let ξ_{k} := φ_{e}(b_{k})ξ. Since φ_{e} is inducible, Proposition 9.5 implies

0 ≤ ∑_{k,l=1}^{n} ⟨ξ_{k}, φ_{e}(a_{k}^{∗}a_{l})ξ_{l}⟩ = ∑_{k,l=1}^{n} ⟨ξ, φ̄_{e} (b_{k})^{∗}φ_{e}(a_{k}^{∗}a_{l})φ̄_{e} (b_{l})ξ⟩

= ⟨ ξ, φ̄_{e} ( ∑_{k,l=1}^{n} b_{k}^{∗}μ_{e}(a_{k}^{∗}a_{l})b_{l} ) ξ⟩ .

Since ξ ∈ F is arbitrary, this means that φ̄_{e} ( ∑ _{k,l=1}^{n} b_{k}μ_{e}(a_{k}^{∗}a_{l})b_{l}) ≥ 0 in 𝔹(F). Equivalently, φ̄_{e} annihilates the negative part of ∑ _{k,l=1}^{n} b_{k}μ_{e}(a_{k}^{∗}a_{l})b_{l}. So φ̄_{e} descends to a homomorphism on the quotient B_{e}^{+} . Conversely, the representation (𝔅_{e}^{+} , μ_{e}^{+} ) is inducible by Proposition 9.5. If φ̄_{e}^{+} : B_{e}^{+} → 𝔹(F) is a representation, then the representation μ_{e}^{+} ⊗_{}_{φ̄_{e}^{+}} 1_{F} ≅ φ_{e} on B_{e}^{+} ⊗_{B_{e}^{+}} F ≅ F is inducible by Lemma 9.9. That is, φ_{e} is inducible if φ̄_{e} factors through the quotient map B_{e} ↠ B_{e}^{+} . Summing up, the representation φ̄_{e} associated to an integrable representation φ_{e} of A_{e} descends to B_{e}^{+} if and only if φ_{e} is inducible. The quotient map induces a fully faithful embedding Rep(B_{e}^{+} , D) ↪ Rep(B_{e}, D). The argument above shows that its image consists of those representations of B_{e} that correspond to inducible, integrable representations of A_{e} under the correspondence Rep(B_{e}, D) ≅ Rep_{int}(A_{e}, D). Hence B_{e}^{+} is a (weak) C^{∗}-hull for the class of inducible, integrable representations of A_{e}.

Definition 9.13. Let B_{g}^{+} := A_{g} ⊗_{A_{e}} B_{e}^{+} . This is a well defined Hilbert B_{e}^{+} -module because the representation (𝔅_{e}^{+} , μ_{e}^{+} ) of A_{e} on B_{e}^{+} is inducible. Let (𝔅_{g}^{+} , μ_{e,g}^{+} ) be the induced representation of A_{e} on B_{g}^{+} . It has the image of A_{g} ⊙_{A_{e}} 𝔅_{e}^{+} as a core, with the representation μ_{e,g}^{+} (a_{e})(a_{g} ⊗ b) := (a_{e}a_{g}) ⊗ b for a_{e} ∈ A_{e}, a_{g} ∈ A_{g}, b ∈ 𝔅_{e}^{+} .

By definition, the right B_{e}^{+} -module structure and the inner product on B_{g}^{+} are the unique extensions of the following pre-Hilbert module structure on A_{g} ⊙_{A_{e}} 𝔅_{e}^{+} : (a_{g} ⊗ b_{1}) · b_{2} := a_{g} ⊗ (b_{1} · b_{2}) for all a_{g} ∈ A_{g}, b_{1} ∈ 𝔅_{e}^{+} , b_{2} ∈ B_{e}^{+} , and

⟨a_{1} ⊗ b_{1}, a_{2} ⊗ b_{2}⟩ := b_{1}^{∗}μ_{e}^{+} (a_{1}^{∗}a_{2})b_{2}  (9.14)

for a_{1}, a_{2} ∈ A_{g}, b_{1}, b_{2} ∈ 𝔅_{e}^{+} . This is positive definite by Proposition 9.12. By definition, B_{g}^{+} is the norm completion of this pre-Hilbert B_{e}^{+} -module, and 𝔅_{g}^{+} is the completion of A_{g}⊙_{A_{e}} 𝔅_{e}^{+} in the graph topology for the representation μ_{e,g}^{+} of A_{e}.

The Hilbert B_{e}^{+} -modules B_{g}^{+} are the fibres of our Fell bundle. The Fell bundle structure on (B_{g}^{+} )_{g∈G} only exists under extra assumptions. Before we turn to these, we construct representations of the Hilbert B_{e}^{+} -modules B_{g}^{+} from an integrable representation π of A on E. Let π_{g} := π|_{A_{g}} and let π̄_{e} : B_{e} → 𝔹(E) be the representation of the C^{∗}-hull corresponding to π_{e}. Since π_{e} is inducible by Lemma 9.8, π̄_{e} descends to a representation π̄_{e}^{+} : B_{e}^{+} → 𝔹(E) by Proposition 9.12. Let a ∈ A_{g} and b ∈ 𝔅_{e}^{+} . The operator π_{g}(a)π̄_{e}^{+} (b) is defined on all of E because π̄_{e}^{+} (b) maps E into the domain 𝔈 of π_{e}, which is also the domain of π_{g}(a) by Lemma 9.2. Its adjoint contains the densely defined operator π̄_{e}^{+} (b^{∗})π_{g^{−1}} (a^{∗}), and the operator

π̄_{e}^{+} (b^{∗})π_{g^{−1}} (a^{∗})π_{g}(a)π̄_{e}^{+} (b) = π̄_{e}^{+} (b^{∗})π_{e}(a^{∗}a)π̄_{e}^{+} (b) = π̄_{e}^{+} (b^{∗} · μ_{e}^{+} (a^{∗}a) · b)

is bounded. Hence π̄_{e}^{+} (b^{∗})π_{g^{−1}} (a^{∗}) extends to a bounded operator on E, which is adjoint to π_{g}(a)π̄_{e}^{+} (b). Thus π_{g}(a)π̄_{e}^{+} (b) ∈ 𝔹(E). Define

π̄_{g}^{+} : A_{g} ⊙ 𝔅_{e}^{+} → 𝔹(E), a ⊗ b ↦ π_{g}(a)π̄_{e}^{+} (b).

As above, we check that

π̄_{g}^{+} (x_{1})^{∗} π̄_{g}^{+} (x_{2}) = π̄_{e}^{+} (⟨x_{1}, x_{2}⟩),  π̄_{g}^{+} (x · b) = π̄_{g}^{+} (x)π̄_{e}^{+} (b)  (9.15)

for all x_{1}, x_{2}, x ∈ A_{g} ⊙ 𝔅_{e}^{+} , b ∈ B_{e}^{+} , where the inner product is the one that defines B_{g}^{+} . Thus π̄_{g}^{+} extends uniquely to a bounded linear map

π̄_{g}^{+} : B_{g}^{+} → 𝔹(E),

which still satisfies (9.15). That is, it is a representation of the Hilbert module B_{g}^{+} with respect to π̄_{e}^{+} .

Lemma 9.16. If π̄_{e}^{+} : B_{e}^{+} ↪ 𝔹(E) is faithful (hence isometric), then so is π̄_{g}^{+} .

Proof. Let ξ ∈ B_{g}^{+} . Then

∥ξ∥ = ∥⟨ξ, ξ⟩_{B_{e}^{+}} ∥^{^{1}/2} = ∥π̄_{e}^{+} (⟨ξ, ξ⟩_{B_{e}^{+}} )∥^{^{1}/2} = ∥π̄_{g}^{+} (ξ)^{∗} π̄_{g}^{+} (ξ)∥^{^{1}/2} = ∥π̄_{g}^{+} (ξ)∥.

Next we want to prove that

π̄_{g}^{+} (B_{g}^{+} ) · π̄_{h}^{+} (B_{h}^{+} ) ⊆ π̄_{gh}^{+} (B_{gh}^{+}) and π̄_{g}^{+} (B_{g}^{+} )^{∗} = π̄_{g^{−1}}^{+} (B_{g^{−1}}^{+} )  (9.17)

for all g, h ∈ G and for all integrable representations π of A. This would give (π̄_{g}^{+} (B_{g}^{+} ))_{g∈G} a Fell bundle structure, which would lift to (B_{g}^{+} )_{g∈G} itself if π̄_{e}^{+} is faithful. Lemma 9.23 below gives (9.17) provided the closed linear span of π̄_{e}^{+} (B_{e}^{+} )·π̄_{g}^{+} (B_{g}^{+} ) is π̄_{g}^{+} (B_{g}^{+} ) for all g ∈ G. But this only holds if we impose two extra assumptions. First, compatibility of integrability and induction gives B_{g}^{+} a canonical left B_{e}^{+} -module structure. Secondly, compatibility of the weak C^{∗}-hull B_{e}^{+} with isometric intertwiners ensures that the representation π̄_{g}^{+} is compatible with this left B_{e}^{+} -module structure.

9.4 Integrability and induction

Definition 9.18. We say that integrability is compatible with induction if induction of inducible representations preserves integrability; that is, if φ_{e} is an inducible, integrable representation of A_{e} on E and π is the representation of A on A ⊗_{A_{e}} E induced by φ_{e}, then the representation π_{e} := π|_{A_{e}} of A_{e} is again integrable.

We shall use this assumption in Section 9.5 to prove (9.17). But first, we study some sufficient conditions for integrability to be compatible with induction. A direct sum of representations is integrable if and only if each summand is integrable by Corollary 3.4. Hence integrability is compatible with induction if and only if an inducible, integrable representation φ_{e} on F induces integrable representations of A_{e} on A_{g} ⊗_{A_{e}} F for all g ∈ G.

Proposition 9.19. Integrability is compatible with induction if and only if the representations (𝔅_{g}^{+} , μ_{e,g}^{+} ) of A_{e} on B_{g}^{+} are integrable for all g ∈ G.

Proof. The representations (𝔅_{g}^{+} , μ_{e,g}^{+} ) of A_{e} on B_{g}^{+} are integrable for all g ∈ G if and only if their direct sum is integrable. Denote this by (A ⊗_{A_{e}} 𝔅_{e}^{+} , μ^{+} ). If integrability is compatible with induction, then (A ⊗_{A_{e}} 𝔅_{e}^{+} , μ^{+} ) must be integrable because it is the induced representation of the universal integrable (inducible) representation (𝔅_{e}^{+} , μ_{e}^{+} ) of A_{e} on B_{e}^{+} . Conversely, by Lemma 9.9, induction maps the representation (𝔅_{e}^{+} , μ_{e}^{+} ) ⊗_{ϱ} F of A_{e} associated to a representation ϱ: B_{e}^{+} → 𝔹(F) to the representation (A ⊗_{A_{e}} 𝔅_{e}^{+} , μ^{+} ) ⊗_{ϱ} F, which is integrable if (A ⊗_{A_{e}} 𝔅_{e}^{+} , μ^{+} ) is, see Definition 3.11.(2).

The (Strong) Local–Global Principle is useful to check that integrability is compatible with induction:

Proposition 9.20. Assume that the integrable representations of A_{e} satisfy the Strong Local–Global Principle and that induction maps irreducible, inducible, integrable Hilbert space representations of A_{e} to integrable Hilbert space representations of A. Then integrability is compatible with induction. The same conclusion holds if the integrable representations of A_{e} satisfy the Local–Global Principle and induction maps all inducible, integrable Hilbert space representations of A_{e} to integrable Hilbert space representations of A.

Proof. Let B_{e}^{+} with the representation (𝔅_{e}^{+} , μ_{e}^{+} ) of A_{e} be the C^{∗}-hull for the inducible, integrable representations of A_{e}. By Proposition 9.19, it suffices to prove that the canonical representation of A_{e} on A ⊗_{A_{e}} 𝔅_{e}^{+} is integrable. By the Strong Local–Global Principle, this follows if the induced representation π̃ of A_{e} on (A ⊗_{A_{e}} 𝔅_{e}^{+} ) ⊗_{ϱ} H is integrable for each irreducible representation ϱ of B_{e}^{+} on a Hilbert space H. The representation ϱ is equivalent to an irreducible, inducible, integrable representation π of A_{e} on H, and π̃ is the representation induced by π. By assumption, π̃ is integrable. This finishes the proof in the case of the Strong Local–Global Principle. The argument in the other case is the same without the word “irreducible.”

Proposition 9.21. Assume the following. First, the integrable representations of A_{e} satisfy the Strong Local–Global Principle. Secondly, all irreducible, integrable Hilbert space representations of A_{e} are finite-dimensional. Third, all finite-dimensional inducible representations of A_{e} are integrable. And fourth, each A_{g} is finitely generated as a right A_{e}-module. Then integrability is compatible with induction.

Proof. First, since B_{e}^{+} is a quotient of B_{e}, its irreducible representations form a subset of the irreducible representations of B_{e}. Thus the irreducible, inducible, integrable Hilbert space representations of A_{e} are finite-dimensional as well. By Proposition 9.20, it suffices to check that the induced representation of A_{e} on A_{g} ⊗_{A_{e}} H is integrable when H is a Hilbert space with an irreducible, inducible, integrable representation. By our assumptions, H is finite-dimensional and A_{g} is finitely generated as an A_{e}-module. Hence A_{g} ⊗_{A_{e}} H is finite-dimensional. This representation is a direct summand in a representation of A on A ⊗_{A_{e}} H and hence inducible by Lemma 9.8. By assumption, the induced representation of A_{e} on A_{g} ⊗_{A_{e}} H is integrable.

9.5 The Fell bundle structure

If integrability is compatible with induction, the representation μ_{e,g}^{+} of A_{e} on B_{g}^{+} is integrable. It is inducible as well by Lemma 9.8 because it is a direct summand in a representation of A. Hence there is a unique (nondegenerate) representation μ̄_{e,g}^{+} of B_{e}^{+} on B_{g}^{+} such that μ̄_{e,g}^{+} (𝔅_{e}^{+} )B_{g}^{+} is a core for μ_{e,g}^{+} , and μ_{e,g}^{+} (a_{e})(μ̄_{e,g}^{+} (b)x) = μ̄_{e,g}^{+} (μ_{e}^{+} (a_{e})b)x for all a ∈ A_{e}, b ∈ 𝔅_{e}^{+} , x ∈ B_{g}^{+} . Our next goal is to show that the representations π̄_{e}^{+} : B_{e}^{+} → 𝔹(E) and π̄_{g}^{+} : B_{g}^{+} → 𝔹(E) constructed using (9.15) are compatible in the sense that

π̄_{e}^{+} (b_{e}) · π̄_{g}^{+} (b_{g}) = π̄_{g}^{+} (μ̄_{e,g}^{+} (b_{e})b_{g})  for all b_{e} ∈ B_{e}^{+} , b_{g} ∈ B_{g}^{+} . (9.22)

This is not automatic. The following lemma is the most subtle point in the proof of the Induction Theorem.

Lemma 9.23. Equation (9.22) holds if B_{e}^{+} is a C^{∗}-hull, not just a weak C^{∗}-hull. Then also π̄_{e}^{+} (B_{e}^{+} ) · π̄_{g}^{+} (B_{g}^{+} ) = π̄_{g}^{+} (B_{g}^{+} ) for all g ∈ G.

Proof. Let F := B_{g}^{+} ⊗_{B_{e}^{+}} E. The linear map B_{g}^{+} ⊙ E → E, b ⊗ ξ ↦ π̄_{g}^{+} (b)ξ, for b ∈ B_{g}^{+} , ξ ∈ E, preserves the inner products by (9.15). Hence it extends to a well defined isometry I: F ↪ E. The representation μ̄_{e,g}^{+} of B_{e}^{+} on B_{g}^{+} induces a representation μ̄_{e,g}^{+} ⊗ 1_{E} of B_{e}^{+} on F. The meaning of (9.22) is that I intertwines the representations μ̄_{e,g}^{+} ⊗ 1 and π̄_{e}^{+} of B_{e}^{+} on F and E. These representations correspond to the integrable representations μ_{e,g}^{+} ⊗ 1 and π_{e} of A_{e} on F and E, respectively. Since B_{e}^{+} is a C^{∗}-hull, it suffices to prove that I intertwines these representations of A_{e}. We identify E ≅ B_{e}^{+} ⊗_{}_{π̄_{e}^{+}} E and describe π_{e} as μ_{e}^{+} ⊗_{}_{π̄_{e}^{+}} 1_{E} as in Proposition 3.6. Then Lemma 9.9 gives a canonical unitary ^{∗}-intertwiner

F := (A_{g} ⊗_{A_{e}} B_{e}^{+} ) ⊗_{B_{e}^{+}} E ≅ A_{g} ⊗_{A_{e}} (B_{e}^{+} ⊗_{B_{e}^{+}} E) ≅ A_{g} ⊗_{A_{e}} E

of representations of A_{e}. An inspection of the proof shows that I corresponds to the isometry I′: A_{g} ⊗_{A_{e}} E ↪ E defined by I′(a ⊗ ξ) := π_{g}(a)ξ for all a ∈ A_{g}, ξ ∈ 𝔈. Since I′ is an A_{e}-intertwiner, so is I. This finishes the proof of (9.22). Then π̄_{e}^{+} (B_{e}^{+} ) · π̄_{g}^{+} (B_{g}^{+} ) = π̄_{g}^{+} (B_{g}^{+} ) follows because μ̄_{e,g}^{+} is nondegenerate.

Lemma 9.24. Assume π̄_{e}^{+} (B_{e}^{+} ) · π̄_{g}^{+} (B_{g}^{+} ) = π̄_{g}^{+} (B_{g}^{+} ) for all g ∈ G. Then (9.17) holds.

Proof. We write ≐ to denote that two sets of operators have the same closed linear span. By definition, π̄_{g}^{+} (B_{g}^{+} ) ≐ π_{g}(A_{g})π̄_{e}^{+} (𝔅_{e}^{+} ), and π̄_{e}^{+} (𝔅_{e}^{+} )^{∗} ≐ π̄_{e}^{+} (B_{e}^{+} ) because 𝔅_{e}^{+} is dense in B_{e}^{+} . Our assumption π̄_{g}^{+} (B_{g}^{+} ) ≐ π̄_{e}^{+} (B_{e}^{+} ) · π̄_{g}^{+} (B_{g}^{+} ) implies π̄_{g}^{+} (B_{g}^{+} ) ≐ π̄_{e}^{+} (𝔅_{e}^{+} )^{∗}π_{g}(A_{g})π̄_{e}^{+} (𝔅_{e}^{+} ). We have seen above (9.15) that

π̄_{e}^{+} (b^{∗})π_{g^{−1}} (a^{∗}) = π̄_{e}^{+} (b)^{∗}π_{g^{−1}} (a^{∗})

for b ∈ 𝔅_{e}^{+} , a ∈ A_{g} extends to a bounded operator on E that is adjoint to the bounded operator π_{g}(a)π̄_{e}^{+} (b). Therefore,

(π̄_{e}^{+} (b_{1})^{∗}π_{g}(a)π̄_{e}^{+} (b_{2}))^{∗} = π̄_{e}^{+} (b_{2})^{∗}π_{g^{−1}} (a^{∗})π̄_{e}^{+} (b_{1})

for all b_{1}, b_{2} ∈ 𝔅_{e}^{+} , a ∈ A_{g}; both sides are globally defined bounded operators because π̄_{e}^{+} (𝔅_{e}^{+} ) maps E into 𝔈. The closed linear spans on the two sides of this equality are π̄_{g}^{+} (B_{g}^{+} )^{∗} and π̄_{g^{−1}}^{+} (B_{g^{−1}}^{+} ), respectively. Thus π̄_{g}^{+} (B_{g}^{+} )^{∗} =

π̄_{g^{−1}}^{+} (B_{g^{−1}}^{+} ). As above, the operators \overline{π̄_{e}^{+} (b)π_{g}(a)} for b ∈ (𝔅_{e}^{+} )^{∗}, g ∈ G, a ∈ A_{g} are bounded and generate (B_{g^{−1}}^{+} )^{∗} = B_{g}^{+} . Hence

π̄_{g}^{+} (B_{g}^{+} ) · π̄_{h}^{+} (B_{h}^{+} ) ≐ π̄_{e}^{+} ((𝔅_{e}^{+} )^{∗})π_{g}(A_{g}) · π_{h}(A_{h})π̄_{e}^{+} (𝔅_{e}^{+} )

⊆ π̄_{e}^{+} (𝔅_{e}^{+} )^{∗} · π_{gh}(A_{gh})π̄_{e}^{+} (𝔅_{e}^{+} ) ≐ π̄_{gh}^{+} (B_{gh}^{+}).

We used here that π is a homomorphism on A and that A_{g} · A_{h} ⊆ A_{gh}.

Lemma 9.25. Assume that B_{e} is a C^{∗}-hull and that integrability is compatible with induction. There is a unique Fell bundle structure on (B_{g}^{+} )_{g∈G} such that the maps π̄_{g}^{+} : B_{g}^{+} → 𝔹(E) form a Fell bundle representation for any integrable representation π of A on a Hilbert module E.

Proof. Lemmas 9.23 and 9.24 show that (9.17) holds under our assumptions. Hence the multiplication and involution in 𝔹(E) restrict to a Fell bundle structure on the subspaces π̄_{g}^{+} (B_{g}^{+} ) ⊆ 𝔹(E) for g ∈ G, such that the inclusions π̄_{g}^{+} (B_{g}^{+} ) ↪ 𝔹(E) give a Fell bundle representation. The induced representation λ of A on the Hilbert B_{e}^{+} -module A ⊗_{A_{e}} B_{e}^{+} gives a faithful representation of B_{e}^{+} because A⊗_{A_{e}} B_{e}^{+} ⊇ A_{e} ⊗_{A_{e}} B_{e}^{+} = B_{e}^{+} contains the identity representation. Hence the resulting representations λ̄_{g} of B_{g}^{+} are also faithful, even isometric, by Lemma 9.16. So the Fell bundle structure on λ̄_{g}(B_{g}^{+} ) lifts to B_{g}^{+} , so that the maps λ̄_{g} : B_{g}^{+} → 𝔹(E) form a Fell bundle representation.

Let π be any integrable representation of A. The exterior direct sum π ⊕ λ on the Hilbert D ⊕ B_{e}^{+} -module E′ := E ⊕ (A ⊗_{A_{e}} B_{e}^{+} ) is still integrable. The resulting maps from B_{g}^{+} to 𝔹(E′) simply give block matrices π̄_{g}^{+} (b) ⊕ λ̄_{g}(b) for b ∈ B_{g}^{+} . The compressions to the direct summands E and A ⊗_{A_{e}} B_{e}^{+} therefore restrict to Fell bundle representations with respect to the Fell bundle structure on (π̄_{g}^{+} ⊕ λ̄_{g})(B_{g}^{+} ) defined above. Since λ is faithful, the projection (π̄_{g}^{+} ⊕ λ̄_{g})(B_{g}^{+} ) → λ̄_{g}(B_{g}^{+} ) ≅ B_{g}^{+} is a Fell bundle isomorphism. Hence the map B_{g}^{+} ⟶^{∼} (π̄_{g}^{+} ⊕ λ̄_{g})(B_{g}^{+} ) → π̄_{g}^{+} (B_{g}^{+} ) is a Fell bundle representation.

Let (β_{g})_{g∈G} be a Fell bundle over a discrete group G (see [8]). Then β := ⨁ _{g∈G} β_{g} is a G-graded ^{∗}-algebra using the given multiplications and involutions among the subspaces β_{g}. The section C^{∗}-algebra C^{∗}(β) of the Fell bundle is defined as the completion of β in the maximal C^{∗}-seminorm. By construction, a representation of C^{∗}(β) is equivalent to a representation of the Fell bundle. This holds also for representations on Hilbert modules.

Theorem 9.26. Let A be a graded ^{∗}-algebra for which A_{e} has a C^{∗}-hull. Assume that integrability is compatible with induction as in Definition 9.18. The section C^{∗}-algebra B of the Fell bundle (B_{g}^{+} )_{g∈G} constructed above is a C^{∗}-hull for the integrable representations of A.

Proof. Representations of B are in natural bijection with Fell bundle representations: restricting a representation of B to the subspaces B_{g}^{+} gives a Fell bundle representation, and conversely a Fell bundle representation gives a representation of the ^{∗}-algebra ⨁ _{g∈G} B_{g}^{+} , which extends uniquely to the C^{∗}-completion. Lemma 9.25 says that any integrable representation π = ⨁ _{g∈G} π_{g} of A induces a Fell bundle representation (π̄_{g}^{+} )_{g∈G} of (B_{g}^{+} )_{g∈G} and thus a representation of B. By construction, this family of maps Rep_{int}(A) → Rep(B) is compatible with interior tensor products and unitary ^{∗}-intertwiners. We are going to show that this is a family of bijections. First we describe an integrable representation (𝔅, μ) of A on B. By construction, A ⊗_{A_{e}} 𝔅_{e}^{+} = ⨁ _{g∈G} 𝔅_{g}^{+} is dense in B. This subspace carries a representation of A by left multiplication. We extend this to the right ideal in B generated by A ⊗_{A_{e}} 𝔅_{e}^{+} to get a representation of A on B. Let (𝔅, μ) be its closure. The representations μ̄_{e,g}^{+} of B_{e}^{+} on B_{g}^{+} are defined so that μ̄_{e,g}^{+} (𝔅_{e}^{+} )B_{g}^{+} is another core for the representation (𝔅_{g}^{+} , μ_{e,g}^{+} ) of A_{e} on B_{g}^{+} . Therefore, 𝔅_{e}^{+} · B is a core for the restriction of the representation (𝔅, μ) to A_{e}. This core shows that (𝔅, μ|_{A_{e}} ) = (𝔅_{e}^{+} , μ_{e}^{+} ) ⊗_{B_{e}} B, where the interior tensor product is with respect to the canonical embedding B_{e}^{+} ↪ B. Therefore, the restriction of (𝔅, μ) to A_{e} is integrable and the corresponding representation μ̄_{e}^{+} of B_{e}^{+} is simply the inclusion map B_{e}^{+} ↪ B. Thus the representation (𝔅, μ) of A on B is also integrable. The integrable representation (𝔅, μ) of A on B yields a representation μ̄_{g}^{+} of the Fell bundle (B_{g}^{+} )_{g∈G} in M(B) = 𝔹(B). By construction, the image of a_{g} ⊗ b ∈ A_{g} ⊙_{A_{e}} 𝔅_{e}^{+} in B_{g}^{+} acts by μ(a_{g})μ̄_{e}^{+} (b) = μ(a_{g}) · b. That is, B_{g}^{+} is represented by the canonical inclusion map B_{g}^{+} ↪ B. The representation of B associated to this Fell bundle representation is the identity map on B. Interior tensor product with (𝔅, μ) gives a family of maps Rep(B) → Rep_{int}(A) that is compatible with unitary ^{∗}-intertwiners and interior tensor products. Since the composite family of maps Rep(B) → Rep_{int}(A) → Rep(B) is compatible with interior tensor products and maps the identity representation of B to itself, the composite map on Rep(B) is the identity. Let (𝔈, π) be an integrable representation of A on a Hilbert D-module E for some C^{∗}-algebra D. This yields a representation (π̄_{g}^{+} )_{g∈G} of the Fell bundle (B_{g}^{+} )_{g∈G} and an associated representation π̄ of B. We claim that the integrable representation (𝔈′, π′) := (𝔅, μ) ⊗π̄ E is equal to (𝔈, π). Both representations have the same restriction to A_{e} because

(𝔅, μ|_{A_{e}} ) ⊗π̄ E ≅ (𝔅_{e}^{+} , μ_{e}^{+} ) ⊗_{B_{e}} B ⊗π̄ E ≅ (𝔅_{e}^{+} , μ_{e}^{+} ) ⊗_{π̄|_{Be}} E ≅ (𝔈, π).

Hence both representations have the same domain by Lemma 9.2. And π̄_{e}^{+} (𝔅_{e}^{+} )E is a core for both. On π̄_{e}^{+} (𝔅_{e}^{+} )E, a_{g} ∈ A_{g} acts by mapping π̄_{e}^{+} (b_{e})ξ to π_{g}(a_{g})π̄_{e}^{+} (b_{e})ξ = π̄(a_{g} ⊗ b_{e})ξ in both representations, where we view a_{g} ⊗ b_{e} ∈ B_{g}^{+} ⊆ B. Since (𝔈, π) and (𝔈′, π′) have a common core, they are equal. This finishes the proof that our two families of maps Rep_{int}(A) ↔ Rep(B) are inverse to each other. Thus B is a weak C^{∗}-hull for the integrable representations of A. Since A_{e} is a C^{∗}-hull, the integrable representations of A_{e} are admissible. So are the integrable representations of A by Proposition 9.4. Thus B is a C^{∗}-hull.

Remark 9.27. The fibres B_{e}^{+} of the Fell bundle in Theorem 9.26 are described in Definitions 9.10 and 9.13, including the right Hilbert B_{e}^{+} -module structure on B_{g}^{+} . The rest of the Fell bundle structure needs technical extra assumptions. The simplest way to get it is by inducing the universal inducible, integrable representation of A on B_{e}^{+} to an integrable representation of A on the Hilbert B_{e}^{+} -module A ⊗_{A_{e}} B_{e}^{+} . The Fell bundle (B_{g}^{+} )_{g∈G} is represented faithfully in 𝔹(A ⊗_{A_{e}} B_{e}^{+} ) by Lemma 9.16. The multiplication, involution, and norm in our Fell bundle are simply the multiplication, involution and norm in the C^{∗}-algebra 𝔹(A ⊗_{A_{e}} B_{e}^{+} ). The dense image of A_{g} ⊙_{A_{e}} 𝔅_{e}^{+} in B_{g}^{+} acts on A ⊗_{A_{e}} B_{e}^{+} by a_{g} ⊗ b ↦ π_{g}(a_{g}) · π̄_{e}^{+} (b), where π̄_{e}^{+} (b) is the representation of the C^{∗}-hull B_{e}^{+} associated to the induced representation of A_{e} on A⊗_{A_{e}} B_{e}^{+} , which is integrable by assumption.

9.6 Two counterexamples

Two assumptions limit the generality of the Induction Theorem 9.26. First, integrability must be compatible with induction. Secondly, B_{e} should be a C^{∗}-hull and not a weak C^{∗}-hull. Equivalently, all isometric intertwiners between integrable Hilbert space representations of A_{e} are ^{∗}-intertwiners. We show by two simple counterexamples that both assumptions are needed. In particular, there is no version of the Induction Theorem for weak C^{∗}-hulls. Both counterexamples involve the group G = ℤ/2 = {0, 1}. A G-graded ∗-algebra is a ^{∗}-superalgebra, that is, a ^{∗}-algebra with a decomposition A = A_{0} ⊕ A_{1} such that

A_{0}·A_{0}+A_{1}·A_{1} ⊆ A_{0}, A_{0}·A_{1}+A_{1}·A_{0} ⊆ A_{1}, A_{0}^{∗} = A_{0}, A_{1}^{∗} = A_{1}, 1 ∈ A_{0}.

In both examples, A_{0} = ℂ[x] with x = x^{∗}. In the first example, A is the crossed product for the action of ℤ/2 on A_{0} = ℂ[x] through the involution x ↦ −x. That is,

A = ℂ⟨x, ε | ε^{2} = 1, xε = −εx, x = x^{∗}, ε = ε^{∗}⟩, x ∈ A_{0}, ε ∈ A_{1}.

Since A_{1} = εA_{0} ≅ A_{0} as a right A_{0}-module, any representation of A_{0} is inducible. Let B_{0} = C_{0}((0, ∞)) with the representation of A_{0} from the inclusion map (0, ∞) ↪ ℝ = Â_{0} (see Proposition 8.1). This gives a C^{∗}-hull for a class of representations of A_{0} that is defined by submodule conditions and satisfies the Strong Local–Global Principle by Theorems 8.2 and 8.3. The class of (0, ∞)- integrable representations consists of those representations of ℂ[x] that are generated by a regular, self-adjoint, strictly positive operator. In a representation of A, the element ε ∈ A acts by a unitary involution that conjugates \overline{π(x)} to −\overline{π(x)}. Hence \overline{π(x)} cannot be strictly positive. Thus the zero-dimensional representation is the only representation of A whose restriction to A_{0} is C_{0}((0, ∞))-integrable. The C^{∗}-hull for this class is {0}. Theorem 9.26 does not apply here because induced representations of inducible, integrable representations of A_{0} are never integrable when they are non-zero. The second example is the commutative ^{∗}-superalgebra

A = ℂ⟨x, ε | ε^{2} = 1 + x^{2}, xε = εx, x = x^{∗}, ε = ε^{∗}⟩, x ∈ A_{0}, ε ∈ A_{1}.

Thus A_{1} = εℂ[x] ≅ A_{0} with the usual A_{0}-bimodule structure and the inner product ⟨εa_{1}, εa_{2}⟩ = (1 + x^{2}) · \overline{a_{1}} · a_{2}. Since (1 + x^{2})|a|^{2} is positive in ℂ[x] for any a ∈ ℂ[x], any representation of A_{0} is inducible. Let (𝔈, π) be a representation of A_{0} on a Hilbert module E over a C^{∗}-algebra D. The induced representation A_{1} ⊗_{A_{0}} (𝔈, π) lives on the Hilbert D-module completion E_{1} of 𝔈 for the inner product ⟨ξ_{1}, ξ_{2}⟩_{1} := ⟨ξ_{1}, π(1 + x^{2})ξ_{2}⟩. Its domain is 𝔈, viewed as a dense D-submodule in E_{1}, and the representation of A_{0} is π again. The operator π(x + i) on 𝔈 extends to an isometry I: E_{1} ↪ E because

⟨π(x + i)ξ_{1}, π(x + i)ξ_{2}⟩ = ⟨ξ_{1}, π(x − i)π(x + i)ξ_{2}⟩ = ⟨ξ_{1}, π(1 + x^{2})ξ_{2}⟩ = ⟨ξ_{1}, ξ_{2}⟩_{1}

for all ξ_{1}, ξ_{2} ∈ 𝔈. This isometry commutes with π(a) for all a ∈ A, so it is an isometric intertwiner A_{1} ⊗_{A_{0}} (𝔈, π) ↪ (𝔈, π). Now let B_{0} with the universal representation (𝔅_{0}, μ_{0}) be one of the two noncommutative weak C^{∗}-hulls T_{0} or 𝕂(ℓ^{2}ℕ) of ℂ[x] described in §6. In a Toeplitz integrable representation, π(x + i) has dense range. Even more, π(x + i)𝔈 is dense in 𝔈 in the graph topology. Thus I is a unitary ^{∗}-intertwiner A_{1} ⊗_{A_{0}} (𝔈, π) ⟶^{∼} (𝔈, π) for any integrable representation (𝔈, π) of A_{0}. Since all representations of A_{0} are inducible, the unit fibre of the Fell bundle should be B_{0}. The other fibre B_{1} is A_{1}⊗_{A_{0}} B_{0}, which we have identified with B_{0}. The unitary A_{1} ⊗_{A_{0}} B_{0} ≅ B_{0} is a ^{∗}-intertwiner between the representations of A_{0} as well. Therefore, integrability is compatible with induction. And the left B_{0}-module structure μ̄_{0,1} on B_{1} in (9.22) is simply left multiplication. Next we describe the induced representation of A on the Hilbert B_{0}-module

A ⊗_{A_{0}} B_{0} = A_{0} ⊗_{A_{0}} B_{0} ⊕ A_{1} ⊗_{A_{0}} B_{0} ≅ B_{0} ⊕ B_{0}.

The representations of A and A_{0} on A ⊗_{A_{0}} B_{0} have the same domain by Lemma 9.2, and for A_{0} the domain is 𝔅_{0} ⊕ 𝔅_{0}. We claim that A acts on this domain by

x ↦ ( \begin{matrix}μ_{0}(x) & 0 \\ 0 & μ_{0}(x) \end{matrix}) , ε ↦ ( \begin{matrix}0 & μ_{0}(x − i) \\ μ_{0}(x + i) & 0 \end{matrix} ) .

We have already seen this for x ∈ A_{0}. Left multiplication by ε maps b ∈ 𝔅_{0} ⊆ B_{0} first to ε ⊗ b ∈ A_{1} ⊗_{A_{0}} B_{0}, which is mapped by the isometry I to μ_{0}(x + i)b ∈ 𝔅_{0} ⊆ B_{0}. And it maps the element μ_{0}(x + i)b ∈ B_{0} for b ∈ 𝔅_{0}, which corresponds to ε ⊗ b in the odd fibre, to ε^{2} ⊗ b = μ_{0}(x^{2} + 1)b = μ_{0}(x − i)μ_{0}(x + i)b ∈ B_{0}. This proves the formula for the action of ε. The representation μ̄_{0} of B_{0} on A ⊗_{A_{0}} B_{0} is the representation of the weak C^{∗}-hull that corresponds to the representation of A_{0} ⊆ A described above. This is

μ̄_{0} : B_{0} → 𝕄_{2}(B_{0}), b ↦ ( \begin{matrix}b & 0 \\ 0 & b \end{matrix}) .

Hence ε ⊗ b ∈ A_{1} ⊗_{A_{0}} B_{0} for b ∈ 𝔅_{0} acts by the matrix

( \begin{matrix}0 & μ_{0}(x − i) \\ μ_{0}(x + i) & 0 \end{matrix} ) ( \begin{matrix}b & 0 \\ 0 & b \end{matrix}) = ( \begin{matrix}0 & μ_{0}(x − i)b \\ μ_{0}(x + i)b & 0 \end{matrix} ) .

The map μ_{0}(x + i)b ↦ μ_{0}(x − i)b is the Cayley transform of μ_{0}(x). For our two weak C^{∗}-hulls, this is the unilateral shift S ∈ M(B_{0}) by construction. Thus the odd fibre B_{1} ≅ B_{0} of our Fell bundle should act by

μ̄_{1} : B_{0} → 𝕄_{2}(B_{0}), b ↦ ( \begin{matrix}0 & Sb \\ b & 0 \end{matrix}) .

The map μ̄_{0} is a ^{∗}-representation, and (9.15) gives

μ̄_{1} (b_{1})^{∗} μ̄_{1} (b_{2}) = μ̄_{0} (b_{1}^{∗}b_{2}),  μ̄_{1} (b_{1})μ̄_{0} (b_{2}) = μ̄_{1} (b_{1}b_{2})

for all b_{1}, b_{2} ∈ B_{0}. This is also obvious from our explicit formulas. But

μ̄_{0} (b_{1})μ̄_{1} (b_{2}) = ( \begin{matrix}0 & b_{1}Sb_{2} \\ b_{1}b_{2} & 0 \end{matrix}) and μ̄_{0} (b_{1}b_{2}) = ( \begin{matrix}0 & Sb_{1}b_{2} \\ b_{1}b_{2} & 0 \end{matrix})

differ if, say, b_{1} = S^{∗}, b_{2} = 1. In fact, μ̄_{0} (B_{0})·μ̄_{1} (B_{0}) is not contained in μ̄_{1} (B_{0}). Hence there is no Fell bundle structure on (B_{g})_{g∈ℤ/2} for which (μ̄_{g})_{g∈ℤ/2} would be a Fell bundle representation.

10 Locally bounded unit fibre representations

We now specialise the Induction Theorem 9.26 to the case where the universal integrable representation of the unit fibre A_{e} is locally bounded. In this case, we may first construct a pro-C^{∗}-algebraic Fell bundle whose unit fibre is the pro-C^{∗}-algebra completion of A_{e}. This is relevant because pro-C^{∗}-algebras are much closer to ordinary C^{∗}-algebras than general ^{∗}-algebras. We will see the importance of this in the commutative case, where the pro-C^{∗}-algebraic Fell bundle gives us a twisted partial group action on the space Â_{e}^{+} of positive characters. As before, let G be a group and let A = ⨁ _{g∈G} A_{g} be a G-graded ^{∗}-algebra. We are interested in the locally bounded representations of A_{e}, and representations of A that restrict to locally bounded representations on A_{e}. The class Rep_{b}(A_{e}) of locally bounded representations of A_{e} is admissible by Corollary 7.9. So any weak C^{∗}-hull for some smaller class of representations will be an ordinary C^{∗}-hull. Let A_{e} be the pro-C^{∗}-algebra completion of the unit fibre A_{e}, that is, the completion of A_{e} in the topology defined by the directed set N (A_{e}) of all C^{∗}-seminorms on A_{e}. Locally bounded representations of A_{e} are equivalent to locally bounded representations of A_{e} by Proposition 7.6. When is a locally bounded representation inducible?

Proposition 10.1. A locally bounded representation (𝔈, φ) of A_{e} on a Hilbert module E is inducible if and only if φ(a^{∗}a) ≥ 0 for all a ∈ A_{g}, g ∈ G.

The difference to the general criterion for inducibility in Proposition 9.5 is that we do not consider matrices.

Proof. The subspace E_{b} ⊆ 𝔈 of bounded vectors is a core for φ. As in the proof of Proposition 9.12, it suffices to prove the positivity of the inner product for a finite linear combination ∑ _{k=1}^{n} a_{k} ⊗ ξ_{k} with a_{k} ∈ A_{g}, ξ_{k} ∈ E_{b} for a fixed g ∈ G. Since there are only finitely many ξ_{k}, there is a C^{∗}-seminorm q on A_{e} so that all ξ_{k} are q-bounded. Thus we may replace E by the Hilbert submodule E_{q} of q-bounded vectors, where the representation of A_{e} extends to the C^{∗}-completion D := (A_{e})_{q} for q. Since we assume φ(a^{∗}a) ≥ 0 for all a ∈ A_{g}, this representation factors through the quotient of D by the closed ideal I generated by the negative parts (a^{∗}a)_{−} for all a ∈ A_{g}, g ∈ G. The D/I-valued inner product ⟨a_{1}, a_{2}⟩ := a_{1}^{∗}a_{2} mod I on A_{g} is positive definite by construction; since D/I is a C^{∗}-algebra, we may use the usual notion of positivity here, which does not involve matrices. Then the inner product on the tensor product A_{g} ⊗_{D/I} E_{q} is also positive definite. This is what we had to prove.

A pro-C^{∗}-algebra has a functional calculus for self-adjoint elements. Hence we may construct the negative parts (a^{∗}a)_{−} ∈ A_{e} for a ∈ A_{g}, g ∈ G. We let A_{e}^{+} be the completed quotient of A_{e} by the closed two-sided ideal generated by these elements. This is another pro-C^{∗}-algebra, and it is the largest quotient in which a^{∗}a ≥ 0 for all a ∈ A_{g}, g ∈ G. By Proposition 10.1, a locally bounded representation of A_{e} is inducible if and only if the corresponding locally bounded representation of A_{e} factors through A_{e}^{+} .

Corollary 10.2. There is an equivalence between the inducible, locally bounded representations of A_{e} and the locally bounded representations of the pro-C^{∗}-algebra A_{e}^{+} , which is compatible with isometric intertwiners and interior tensor products.

Proof. Proposition 10.1 says that the equivalence in Proposition 7.6 maps the subclass in Rep_{b}(A_{e}) of inducible, locally bounded representations of A_{e} onto the subclass Rep_{b}(A_{e}^{+} ) in Rep_{b}(A_{e}).

Let N (A_{e})^{+} be the directed set of C^{∗}-seminorms on A_{e}^{+} . This is isomorphic to the subset of N (A_{e}) consisting of all C^{∗}-seminorms q on A_{e} for which a^{∗}a ≥ 0 holds in the C^{∗}-completion (A_{e})_{q} for all a ∈ A_{g}, g ∈ G. We would like to complete A to a ^{∗}-algebra ⨁ _{g∈G} A_{g}^{+} with unit fibre A_{e}^{+} , where each A_{g}^{+} is a Hilbert bimodule over A_{e}^{+} . But such a construction does not work in the following example.

Example 10.3. It can happen that the class of locally bounded representations of A_{e} is not compatible with induction. Let End^{∗}(ℂ[ℕ]) be the ^{∗}-algebra of all ∞ × ∞-matrix with only finitely many entries in each row and each column, with the usual matrix multiplication and involution. Let A be the ℤ/2-graded ∗-algebra of block 2 × 2-matrices

( \begin{matrix}a & b \\ c & d \end{matrix}) , a ∈ ℂ, b ∈ ℂ[ℕ], c ∈ ℂ[ℕ], d ∈ End^{∗}(ℂ[ℕ]),

with the grading where a, d are even and b, c are odd. Here b and c are infinite column and row vectors with only finitely many non-zero entries, respectively. Thus A ≅ End^{∗}(ℂ[ℕ]) with the grading induced by the grading on ℂ[ℕ] where ℂ · δ_{0} is the even part and the span of δ_{i} for i > 0 is the odd part. The character (a, d) ↦ a is a bounded representation of the unit fibre A_{0}. Induction gives the standard representation of A on the Hilbert space ℂ ⊕ ℓ^{2}(ℕ) ≅ ℓ^{2}(ℕ) by matrix-vector multiplication. This representation is irreducible because already the ideal of finite matrices 𝕄_{∞} (ℂ) in A acts irreducibly. It is not bounded, that is, some elements in End^{∗}(ℂ[ℕ]) act by unbounded operators on ℓ^{2}(ℕ). Hence it is not locally bounded by Proposition 7.7.

To rule out this problem, we now assume that induction from A_{e} to A and restriction back to A_{e} maps bounded representations of A_{e} again to bounded representations of A_{e}, briefly, that boundedness is compatible with induction.

This implies that local boundedness is compatible with induction because a locally bounded representation contains bounded subrepresentations whose union is a core for it. Our assumption is equivalent to the boundedness of the induced representations of A_{e} on the Hilbert (A_{e}^{+} )_{q}-modules A_{g} ⊗_{A_{e}} (A_{e}^{+} )_{q} for all g ∈ G and q ∈ N (A_{e})^{+} . That is, there is another norm q′ ∈ N (A_{e})^{+} such that

q(a^{∗}b^{∗}ba) = ∥ba∥_{q}^{2} ≤ ∥b∥_{q′}^{2}∥a∥_{q}^{2} = q′(b)^{2} · q(a^{∗}a)

for all a ∈ A_{g}, b ∈ A_{e}. Let A_{g}^{+} be the completion of A_{g} in the topology generated by the family of norms q(a^{∗}a) for q ∈ N (A_{e})^{+} .

Lemma 10.4. The multiplication maps and the involutions in (A_{g})_{g∈G} extend to continuous maps A_{g}^{+} × A_{h}^{+} → A_{gh}^{+} and A_{g}^{+} → A_{g^{−1}}^{+} for g, h ∈ G.

Proof. Given q ∈ N (A_{e})^{+} , let q′ ∈ N (A_{e})^{+} be such that q(a^{∗}b^{∗}ba) ≤ q′(b)^{2} · q(a^{∗}a) for all a ∈ A_{h}, b ∈ A_{e}. If b ∈ A_{g}, a ∈ A_{h}, then

∥ba∥_{q}^{2} := q(a^{∗}b^{∗}ba) = q(a^{∗}(b^{∗}b)^{^{1}/2} (b^{∗}b)^{^{1}/2} a) ≤ q′((b^{∗}b)^{^{1}/2} )^{2} · q(a^{∗}a) = ∥b∥_{q′}^{2}∥a∥_{q}^{2}

That is, the multiplication is jointly continuous with respect to the topology defining (A_{g}^{+} )_{g∈G} and hence extends to a jointly continuous map A_{g}^{+} × A_{h}^{+} → A_{gh}^{+}. Furthermore, q(aa^{∗})^{2} = q(aa^{∗}aa^{∗}) ≤ q′(a^{∗}a)·q(aa^{∗}) and hence q(aa^{∗}) ≤ q′(a^{∗}a) for all a ∈ A_{h}. That is, ∥a^{∗}∥_{q}^{2} ≤ ∥a∥_{q′}^{2} for all a ∈ A_{h}. Thus the involution is continuous as well.

The completion A^{+} := ⨁ _{g∈G} A_{g}^{+} of A is again a ^{∗}-algebra by Lemma 10.4. By construction of A_{e}^{+} , the inner products a^{∗}a ∈ A_{e}^{+} are positive for a ∈ A_{g}, g ∈ G; this remains so for a ∈ A_{g}^{+} because the subset of positive elements in A_{e}^{+} is closed. Thus (A_{g}^{+} )_{g∈G} has the usual properties of a Fell bundle over G, except that the fibres are only Hilbert bimodules over a pro-C^{∗}-algebra. We interpret (A_{g}^{+} )_{g∈G} as a partial action of G on A_{e}^{+} by Hilbert bimodules as in [4]. Usually, the norms q(a^{∗}a) and q(aa^{∗}) on A_{g} are not equivalent for a fixed q ∈ N (A)^{+} . This prevents us from completing A^{+} to a pro-C^{∗}-algebra. It also means that the integrable representations of A are not locally bounded on A, but only on A_{e}. This happens in interesting examples such as the Weyl algebra discussed in §13. This phenomenon for Fell bundles is related to the known problem that crossed products for group actions on pro-C^{∗}-algebras only work well if the action is strongly bounded, that is, the invariant continuous C^{∗}-seminorms are cofinal in the set of all continuous C^{∗}-seminorms, see [14].

Proposition 10.5. Suppose that boundedness for representations of A_{e} is compatible with induction to A. Representations of A that restrict to locally bounded representations on A_{e} are equivalent to representations of the ∗-algebra A^{+} = ⨁ _{g∈G} A_{g}^{+} that restrict to locally bounded representations on A_{e}^{+} ; this equivalence is compatible with isometric intertwiners and interior tensor products.

Proof. Let π be a representation of A for which π_{e} is a locally bounded representation of A_{e}. The representation π_{e} is inducible by Lemma 9.8. Hence π_{e} is the closure of the restriction of a locally bounded representation π̄_{e}^{+} of A_{e}^{+} by Corollary 10.2. The representation π_{g} of A_{g} for g ∈ G is continuous with respect to the topology defining A_{g}^{+} and the graph topology on the domain of π_{g} because π_{g}(a)^{∗}π_{g}(a) = π_{e}(a^{∗}a). Hence it extends uniquely to A_{g}^{+} , and this gives a representation π̄^{+} of ⨁ A_{g}^{+} such that π is the closure of π̄^{+} ∘j. It is easy to see that this equivalence between the locally bounded representations of A and the representations of ⨁ _{g∈G} A_{g}^{+} that are locally bounded on A_{e}^{+} is compatible with isometric intertwiners and interior tensor products.

We will explore the consequences of this in the case of commutative A_{e} in §11. In that case, boundedness is automatically compatible with induction, and the pro-C^{∗}-algebraic Fell bundle A_{e}^{+} gives rise to a twisted groupoid with object space Â_{e}^{+} . Thus the C^{∗}-hull produced by the Induction Theorem 9.26 is a twisted groupoid C^{∗}-algebra when A_{e} is commutative and the integrable representations of A_{e} are locally bounded. Here we briefly consider the situation of Theorem 7.16 where C_{0}(A_{e}^{+} ) is dense in A_{e}^{+} and provides a C^{∗}-hull for the class of locally bounded representations. Then we define

C_{0}(A_{g}^{+} ) := {a ∈ A_{g} | a^{∗}a ∈ C_{0}(A_{e}^{+} )}.

That is, a ∈ C_{0}(A_{g}^{+} ) if and only if for all ε > 0 there is q ∈ N (A_{e})^{+} such that ∥a^{∗}a∥_{𝔭} < ε for all 𝔭 ∈ Prim(A_{e}^{+} ) \ Prim(A_{e}^{+} )_{q}. Since the involutions A_{g} → A_{g^{−1}} and A_{g^{−1}} → A_{g} are both continuous, they are homeomorphisms. Thus a ∈ C_{0}(A_{g}^{+} ) if and only if aa^{∗} ∈ C_{0}(A_{e}^{+} ). The proof of Lemma 7.15 shows that A_{g}^{+} · C_{c}(A_{e}^{+} ) and C_{c}(A_{e}^{+} ) · A_{g}^{+} are dense in C_{0}(A_{g}^{+} ).

Theorem 10.6. Assume that boundedness is compatible with induction from A_{e} to A and that C_{0}(A_{e}^{+} ) is dense in A_{e}^{+} . Then C_{0}(A_{g}^{+} )_{g∈G} is a Fell bundle over G whose section C^{∗}-algebra is a C^{∗}-hull for the class of all representations of A that restrict to a locally bounded representation of A_{e}.

Proof. The assumption that boundedness is compatible with induction allows us to build the pro-C^{∗}-algebraic Fell bundle (A_{g}^{+} )_{g∈G} . Call a representation of A = ⨁ _{g∈G} A_{g} or A^{+} := ⨁ _{g∈G} A_{g}^{+} integrable if the restriction to the unit fibre A_{e} or A_{e}^{+} is locally bounded, respectively. These classes of integrable representations are equivalent by Proposition 10.5. Since C_{0}(A_{e}^{+} ) is dense in A_{e}^{+} , it is a C^{∗}-hull for the locally bounded representations of A_{e}^{+} by Theorem 7.17. Equivalently, it is a C^{∗}-hull for the inducible, locally bounded representations of A_{e}. Let C_{0}(A^{+} ) := ⨁ _{g∈G} C_{0}(A_{g}^{+} ). Representations of C_{0}(A^{+} ) are equivalent to representations of the Fell bundle C_{0}(A_{g}^{+} ). Thus we must prove that the class of all representations of C_{0}(A^{+} ) is equivalent to the class of integrable representations of A^{+} . More precisely, the equivalence maps a representation ϱ of C_{0}(A^{+} ) on a Hilbert module E to the representation π of A^{+} with the core ϱ(C_{c}(A_{e}^{+} ))E and π(a)ϱ(b)ξ := ϱ(a · b)ξ for all a ∈ A^{+} , b ∈ C_{c}(A_{e}^{+} ), ξ ∈ E; here a · b is the product in A^{+} , which belongs to C_{0}(A^{+} ) if b ∈ C_{c}(A_{e}^{+} ). In the converse direction, we may simply restrict a locally bounded representation of A^{+} to the ^{∗}-subalgebra C_{0}(A^{+} ). This restriction is nondegenerate because C_{0}(A_{e}^{+} ) ⊆ C_{0}(A^{+} ) acts nondegenerately in any integrable representation of A^{+} : this is part of the equivalence between representations of C_{0}(A_{e}^{+} ) and locally bounded representations of A_{e}^{+} in Theorem 7.17. We claim that the maps from representations of C_{0}(A^{+} ) to integrable representations of A^{+} and back are inverse to each other. Let π be an integrable representation of A^{+} on a Hilbert module E. The representations π and π|_{A_{e}^{+}} have the same domain by Lemma 9.2. Since π|_{A_{e}^{+}} is locally bounded, π(C_{c}(A_{e}^{+} ))𝔈 is a core for π|_{A_{e}^{+}} . Since C_{c}(A_{e}^{+} )·A_{g}^{+} = A_{g}^{+} ·C_{c}(A_{e}^{+} ) for all g ∈ G, this subspace is π(A^{+} )-invariant and thus a core for π. The representation ϱ of C_{0}(A^{+} ) is the closure of the restriction of π to C_{0}(A^{+} ) ⊆ A^{+} . By definition, the representation of A^{+} has the core ϱ(C_{c}(A_{e}^{+} ))E and acts there by π′(a)ϱ(b)ξ = ϱ(a·b)ξ. The subspace ϱ(C_{c}(A_{e}^{+} ))𝔈 is a core for this representation because the map ξ ↦ π′(a)ϱ(b)ξ is continuous in the norm topology on E and 𝔈 is dense in E. If ξ ∈ 𝔈, then ϱ(b)ξ = π(b)ξ and hence π′(a)π(b)ξ = π(a)π(b)ξ for all a ∈ A^{+} , b ∈ C_{c}(A_{e}^{+} ), ξ ∈ 𝔈. This implies π = π′, as desired. Now start with a representation ϱ of C_{0}(A^{+} ). Let π be the associated integrable representation of A^{+} . It has the core ϱ(C_{c}(A_{e}^{+} ))E and acts there by π(a)ϱ(b)ξ = ϱ(a · b)ξ for all a ∈ A^{+} , b ∈ C_{c}(A_{e}^{+} ), ξ ∈ E. In particular, if a ∈ C_{0}(A^{+} ), then π(a)ϱ(b)ξ = ϱ(a · b)ξ = ϱ(a)ϱ(b)ξ. Since C_{0}(A_{e}^{+} ) · C_{0}(A_{g}^{+} ) is dense in C_{0}(A_{g}^{+} ) for all g ∈ G, the restriction of ϱ to C_{0}(A_{e}^{+} ) remains nondegenerate. Therefore, the set of ϱ(b)ξ for b ∈ C_{c}(A_{e}^{+} ), ξ ∈ E is dense in E. Hence ϱ is the restriction of π to C_{0}(A^{+} ) ⊆ A^{+} , as desired.

The proof of Theorem 10.6 does not use the constructions in Section 9 and so provides an alternative proof of the Induction Theorem in case the chosen class of integrable representations of A_{e} is the class of all locally bounded representations.

11 Fell bundles with commutative unit fibre

In this section, we apply the Induction Theorem in the case where A_{e} and the chosen C^{∗}-hull B_{e} are commutative. This is the only case considered in [27]. Extra assumptions in [27] ensure that the C^{∗}-hull for the integrable representations of A is the crossed product for a partial action of G on the space Â_{e}^{+} ⊆ Â_{e} of positive characters. Without these assumptions, we shall get a “twisted” crossed product for a partial action. So let G be a discrete group and A = ⨁ _{g∈G} A_{g} a G-graded ^{∗}-algebra such that A_{e} is commutative. We have already classified the possible commutative C^{∗}-hulls for A_{e} in §8. In particular, all commutative weak C^{∗}-hulls are already C^{∗}-hulls by Theorem 8.2, and they correspond to injective, continuous maps from locally compact spaces to the spectrum Â_{e} of A_{e}.

Explicitly, let X be a locally compact space and let j: X → Â_{e} be an injective, continuous map. Let B_{e} = C_{0}(X) and define a representation of A_{e} on B_{e} with domain C_{c}(X) by (a · f)(x) = â(j(x)) · f(x) for all a ∈ A_{e}, f ∈ C_{c}(X), x ∈ X, where â(χ) = χ(a) for χ ∈ Â_{e}. Let μ_{e} be the closure of this representation of A_{e} on B_{e}. The C^{∗}-algebra B_{e} with the universal representation μ_{e} is a C^{∗}-hull for a class Rep_{int}(A_{e}, X) of representations of A_{e} by Theorem 8.2, and any commutative C^{∗}-hull is of this form. Let Rep_{int}(A, X) be the class of representations of A that restrict to a representation in Rep_{int}(A_{e}, X) on A_{e}, as in Definition 9.3. If Rep_{int}(A_{e}, X) is compatible with induction to A as in Definition 9.18, then Theorem 9.26 gives a Fell bundle whose section C^{∗}-algebra is a C^{∗}-hull for Rep_{int}(A, X). We are going to characterise exactly when this happens and describe the C^{∗}-hull for Rep_{int}(A, X) as a twisted groupoid C^{∗}-algebra. Any representation of A_{e} on a commutative C^{∗}-algebra is locally bounded by Proposition 8.1. Hence the constructions in §10 specialise to our commutative case. Actually, we shall make these results more explicit through independent proofs. First we describe the C^{∗}-hull B_{e}^{+} for the inducible representations in Rep_{int}(A_{e}, X) as in Proposition 10.1:

Lemma 11.1. Call a character χ ∈ Â_{e} positive if χ(a^{∗}a) ≥ 0 for all a ∈ A_{g} and all g ∈ G. These form a closed subset Â_{e}^{+} of Â_{e}, and B_{e}^{+} = C_{0}(j^{−1} (Â_{e}^{+} )).

Proof. The positive characters form a closed subset in Â_{e} by definition of the topology on Â_{e}. We have constructed B_{e}^{+} in Proposition 9.12 as a quotient of B_{e}, such that a representation is inducible if and only if it factors through B_{e}^{+} . Thus B_{e}^{+} corresponds to a certain closed subset of Â_{e}. Its points are the inducible characters of A_{e}. Let χ be a character. Any vector in A_{g} ⊗_{A_{e} ,χ} ℂ is of the form a ⊗ 1 for some a ∈ A_{g}, that is, there is no need to take linear combinations. Hence the sesquilinear form on A_{g} ⊗_{A_{e} ,χ} ℂ for all g ∈ G is positive semidefinite if and only if χ(a^{∗}a) ≥ 0 for all a ∈ A_{g} and all g ∈ G, that is, χ is positive. Thus B_{e}^{+} is the quotient corresponding to those x ∈ Â_{e} for which j(x) ∈ Â_{e} is positive.

Theorem 11.2. Let g ∈ G and χ ∈ Â_{e}^{+} . Then dimA_{g} ⊗_{A_{e} ,χ} ℂ ≤ 1. The set

D_{g^{−1}} := {χ ∈ Â_{e}^{+} | dimA_{g} ⊗_{A_{e} ,χ} ℂ = 1}

is relatively open in Â_{e}^{+} . The left A_{e}-module structure on A_{g} ⊗_{A_{e} ,χ} ℂ ≅ ℂ for χ ∈ D_{g^{−1}} is by a character ϑ_{g}(χ) that belongs to D_{g}. The map ϑ_{g} is a homeomorphism from D_{g^{−1}} onto D_{g}, and these maps form a partial action of G on Â_{e}^{+} , that is, ϑ_{e} = idÂ_{e}^{+} and ϑ_{g} ∘ ϑ_{h} ⊆ ϑ_{gh} for all g, h ∈ G.

Proof. As in the proof of Lemma 11.1, A_{g} ⊗_{A_{e} ,χ} ℂ is the Hausdorff completion of A_{g} in the norm coming from the inner product ⟨a_{1}, a_{2}⟩ := χ(a_{1}^{∗}a_{2}). We write λ · a for a ⊗ λ for a ∈ A_{g}, λ ∈ ℂ throughout this proof, and we write a ≡ b if a, b ∈ A have the same image in A_{g} ⊗_{A_{e} ,χ} ℂ. Let a, b ∈ A_{g} satisfy χ(a^{∗}a) ≠ 0 and χ(b^{∗}b) ≠ 0. We must show that a and b are parallel in A_{g} ⊗_{A_{e} ,χ} ℂ.

The following computation makes [7, Footnote 3] explicit:

(a^{∗}ab^{∗}b)^{2} = a^{∗}ab^{∗}(ba^{∗})(ab^{∗})b = a^{∗}a(b^{∗}a)(b^{∗}ba^{∗}b) = a^{∗}ab^{∗}ba^{∗}bb^{∗}a

because A_{e} is commutative and the terms in parentheses belong to A_{e}. Hence

χ(a^{∗}a)^{2}χ(b^{∗}b)^{2} = χ(a^{∗}a)χ(b^{∗}b)χ(a^{∗}b)χ(b^{∗}a).

Since χ(a^{∗}a) ≠ 0 and χ(b^{∗}b) ≠ 0, this implies

χ(a^{∗}a)χ(b^{∗}b) = χ(a^{∗}b)χ(b^{∗}a) = |χ(a^{∗}b)|^{2} ≠ 0.  (11.3)

The inner product on A_{g} ⊗_{A_{e} ,χ} ℂ annihilates a · c ⊗ 1 − a ⊗ χ(c), which we write as a · c − χ(c)a, for all a ∈ A_{g}, c ∈ A_{e}. Hence

a = \frac{χ(a^{∗}b)χ(b^{∗}a)}{χ(a^{∗}a)χ(b^{∗}b)}a ≡ \frac{aa^{∗}bb^{∗}a}{χ(a^{∗}a)χ(b^{∗}b)} = \frac{bb^{∗}aa^{∗}a}{χ(a^{∗}a)χ(b^{∗}b)}

= \frac{χ(b^{∗}a)χ(a^{∗}a)}{χ(a^{∗}a)χ(b^{∗}b)} b = \frac{χ(b^{∗}a)}{χ(b^{∗}b)}b. (11.4)

Thus all non-zero a, b ∈ A_{g} ⊗_{A_{e} ,χ} ℂ are parallel, that is, dimA_{g} ⊗_{A_{e} ,χ} ℂ ≤ 1. The space A_{g} ⊗_{A_{e} ,χ} ℂ is non-zero if and only if there is a ∈ A_{g} with χ(a^{∗}a) ≠ 0. Thus

D_{g^{−1}} = {χ ∈ Â_{e}^{+} | χ(a^{∗}a) ≠ 0 for some a ∈ A_{g}}.  (11.5)

The latter set is relatively open in Â_{e}^{+} . Let χ ∈ D_{g^{−1}} . Then dimA_{g} ⊗_{A_{e} ,χ} ℂ = 1. Hence the representation of A_{e} on it is by a character, which we denote by ϑ_{g}(χ). This character is an inducible representation by Lemma 9.8, and hence positive by Lemma 11.1. There is b ∈ A_{g} with χ(b^{∗}b) > 0. If a ∈ A_{e}, then (11.4) implies ab ≡ \frac{χ(b^{∗} ab)}{χ(b^{∗} b)} b. Thus

ϑ_{g}(χ)(a) = \frac{χ(b^{∗}ab)}{χ(b^{∗}b)}  (11.6)

for all a ∈ A_{e}. Hence ϑ_{g}(χ)((b^{∗})^{∗}b^{∗}) ≠ 0, so that ϑ_{g}(χ) ∈ D_{g} by (11.5). Thus ϑ_{g} maps D_{g^{−1}} to D_{g}. Equation (11.6) also implies that the map ϑ_{g} is continuous on the open set of characters in A_{e}^{+} with χ(b^{∗}b) > 0. Since these open sets for different b ∈ A_{g} cover D_{g^{−1}} , the map ϑ_{g} is continuous on all of D_{g^{−1}} . Let g, h ∈ G and let χ ∈ D_{h^{−1}} and ϑ_{h}(χ) ∈ D_{g^{−1}} . Then there is b_{h} ∈ A_{h} with χ(b_{h}^{∗}b_{h}) > 0, and b_{g} ∈ A_{g} with ϑ_{h}(χ)(b_{g}^{∗}b_{g}) > 0. Thus χ(b_{h}^{∗}b_{g}^{∗}b_{g}b_{h}) = χ(b_{h}^{∗}b_{h}) · ϑ_{h}(χ)(b_{g}^{∗}b_{g}) > 0, and so (11.6) for b = b_{g}b_{h} ∈ A_{gh} describes ϑ_{gh}. Hence

ϑ_{gh}(χ)(a) = \frac{χ(b_{h}^{∗}b_{g}^{∗}ab_{g}b_{h})}{χ(b_{h}^{∗}b_{g}^{∗}b_{g}b_{h})}= \frac{ϑ_{h}(χ)(b_{g}^{∗}ab_{g})}{ϑ_{h}(χ)(b_{g}^{∗}b_{g})} = ϑ_{g}(ϑ_{h}(χ))(a).

Thus ϑ_{gh} ⊆ ϑ_{g}ϑ_{h} for all g, h ∈ G. In addition, ϑ_{e} = idÂ_{e}^{+} . So the maps ϑ_{g} form a partial action of G on Â_{e}^{+} , see [8]. In particular, ϑ_{g} is a homeomorphism from D_{g^{−1}} onto D_{g} with inverse ϑ_{g^{−1}} .

In the examples considered in [7, 27], the space Â_{e}^{+} is locally compact and the C^{∗}-hull for the integrable representations of A is the crossed product for the partial action of G on Â_{e}^{+} described above. In general, however, certain twists are possible. The partial action of G on Â_{e}^{+} may be encoded in a transformation groupoid G⋉Â_{e}^{+} , which has object space Â_{e}^{+} , arrow space ⨆ _{g∈G} D_{g^{−1}} with the disjoint union topology, range and source maps s(g, χ) := χ, r(g, χ) := ϑ_{g}(χ) for g ∈ G, χ ∈ D_{g^{−1}} , and multiplication (g, ϑ_{h}(χ)) · (h, χ) := (g · h, χ) for all g, h ∈ G, χ ∈ D_{h^{−1}} ∩ ϑ_{h}^{−1} (D_{g^{−1}} ). The unit arrow on χ is (1, χ), and the inverse of (g, χ) is (g^{−1} , ϑ_{g}(χ)). This is an étale topological groupoid because r and s restrict to homeomorphisms on the open subsets D_{g^{−1}} of the arrow space. The object space Â_{e}^{+} need not be locally compact. We are going to construct another topological groupoid Σ that is a central extension of G ⋉ Â_{e}^{+} by the circle group 𝕋. That is, Σ comes with a canonical functor to G⋉Â_{e}^{+} whose kernel is the group bundle Â_{e}^{+} ×𝕋. Such an extension is also called a twisted groupoid in [25, Section 4], following a definition by Kumjian [16]. A twisted groupoid with locally compact object space has a twisted groupoid C^{∗}-algebra. For a suitable injective continuous map X → Â_{e}^{+} , we are going to identify the C^{∗}-hull of the X-integrable representations of A with the twisted groupoid C^{∗}-algebra of the restriction of Σ to j(X^{+} ) ⊆ Â_{e}^{+} . A point in Σ is a triple (g, χ, [a]), where g ∈ G, χ ∈ D_{g^{−1}} , and [a] is a unit vector in the 1-dimensional Hilbert space A_{g}⊗_{A_{e} ,χ} ℂ. We represent unit vectors in A_{g} ⊗_{A_{e} ,χ} ℂ by elements a ∈ A_{g} with χ(a^{∗}a) = 1; two elements a, b ∈ A_{g} with χ(a^{∗}a) = χ(b^{∗}b) = 1 represent the same unit vector [a] = [b] if and only if χ(a^{∗}b) = 1. We get the same set of equivalence classes if we allow a ∈ A with χ(a^{∗}a) > 0 and set [a] = [b] if χ(a^{∗}b) > 0: then a_{1} := χ(a^{∗}a)^{−^{1}/2} a and b_{1} := χ(b^{∗}b)^{−^{1}/2} b satisfy [a] = [a_{1}], [b] = [b_{1}], and [a] = [b] if and only if χ(a_{1}^{∗}b_{1}) = 1 by (11.3). The circle group 𝕋 acts on Σ by multiplication: λ · (g, χ, [a]) := (g, χ, [λa]). The orbit space projection for this circle action is the coordinate projection F: Σ ↠ G⋉Â_{e}^{+} , (g, χ, [a]) ↦ (g, χ). Next we equip Σ with a topology so that this coordinate projection is a locally trivial principal 𝕋-bundle. For a ∈ A, let U_{a} := {χ ∈ Â_{e}^{+} | χ(a^{∗}a) ≠ 0}. This is an open subset in Â_{e}^{+} , and χ(a^{∗}a) > 0 if χ ∈ U_{a} because χ is positive. The map σ_{a}: {g} × U_{a} → Σ, (g, χ) ↦ (g, χ, [a]), for a ∈ A_{g} is a local section for the coordinate projection F. If a, b ∈ A_{g}, and χ ∈ U_{a} ∩ U_{b}, then

[a] = [ \frac{χ(b^{∗}a)}{χ(a^{∗}a)^{^{1}/2} χ(b^{∗}b)^{^{1}/2}} b],

by (11.4). Since the functions sending χ to χ(b^{∗}a), χ(a^{∗}a) and χ(b^{∗}b) are continuous on A_{e}, the two trivialisations induce the same topology on the restriction of Σ to {g} × (U_{a} ∩ U_{b}). For any χ ∈ D_{g^{−1}} , there is a ∈ A_{g} with χ(a^{∗}a) > 0. Thus the open subsets U_{a} cover D_{g^{−1}} . Consequently, there is a unique topology on Σ that makes the local sections σ_{a} for all a ∈ A_{g} continuous, and this topology turns Σ into a locally trivial 𝕋-bundle over G ⋉ Â_{e}^{+} .

We define a groupoid with object space A_{e}^{+} , arrow space Σ, and

r(g, χ, [a]) := ϑ_{g}(χ), s(g, χ, [a]) := χ,
(g, [ϑ_{h}(χ)], [a]) · (h, χ, [b]) := (g · h, χ, [a · b]);

we must show that this multiplication is well defined. We have ab ∈ A_{gh} and

χ(b^{∗}a^{∗}ab) = ϑ_{h}(χ)(a^{∗}a) · χ(b^{∗}b) ≠ 0

by (11.6), so (g · h, χ, [a · b]) ∈ Σ. If χ(b^{∗}b_{1}) > 0 and ϑ_{h}(χ)(a^{∗}a_{1}) > 0, then χ(b^{∗}a^{∗}a_{1}b_{1}) > 0 by computations as in the proof of Theorem 11.2. Hence the multiplication is well defined. It is clearly associative. The unit arrow on χ is 1_{χ} := (1, χ, [1]), and (g, χ, [a])^{−1} = (g^{−1} , ϑ_{g}(χ), [a^{∗}]). The multiplication, unit map and inversion are continuous and the range and source maps are open surjections (even locally trivial). So Σ is a topological groupoid. The identity map on objects and the coordinate projection F: Σ → G ⋉ Â_{e}^{+} on arrows form a functor, which is a locally trivial, open surjection on arrows. The kernel of F consists of those (g, χ, [a]) ∈ Σ for which F (g, χ, [a]) is a unit arrow in G ⋉ Â_{e}^{+} . Then g = 1, and a ∈ A_{e} is equivalent to [a] = [χ(a) · 1] because χ(a^{∗}χ(a)1) > 0. The map (g, χ, [a]) ↦ (χ, χ(a)) is an isomorphism of topological groupoids from the kernel of F onto the trivial group bundle Â_{e}^{+} × 𝕋. Thus we have an extension of topological groupoids

Â_{e}^{+} × 𝕋 ↣ Σ ↠ G ⋉ Â_{e}^{+} .

The three groupoids above are clearly Hausdorff. To construct C^{∗}-algebras, we need groupoids with a locally compact object space. Therefore, we replace Â_{e} by a locally compact space X with an injective, continuous map j: X → Â_{e}. Then B_{e} = C_{0}(X) is a C^{∗}-hull for a class Rep_{int}(A_{e}, X) of representations of A_{e}. By Lemma 11.1, the C^{∗}-hull for the class of X-integrable, inducible representations of A_{e} is B_{e}^{+} = C(X^{+} ) with X^{+} := j^{−1} (Â_{e}^{+} ) ⊆ X.

Proposition 11.7. Let j: X → Â_{e} be an injective, continuous map. The class Rep_{int}(A, X) is compatible with induction if and only if j(X^{+} ) ⊆ Â_{e}^{+} is invariant under the partial maps ϑ_{g} in Theorem 11.2 and the resulting partial maps on X^{+} are continuous in the topology of X^{+} . We briefly say that the partial action of G on Â_{e}^{+} restricts to X^{+} .

Proof. By Proposition 9.19, it suffices to check that the induced representation of A_{e} on A_{g} ⊗_{A_{e}} B_{e}^{+} is X-integrable for g ∈ G if and only if the partial map ϑ_{g} ∘ j on X factors through j and the resulting partial map j^{−1} ∘ ϑ_{g} ∘ j on X is again continuous. View the Hilbert module A_{g} ⊗_{A_{e}} B_{e}^{+} as a continuous field of Hilbert spaces over X^{+} . The fibres of this field have dimension at most 1 by Theorem 11.2, and the set where the fibre is non-zero is the open subset j^{−1} (D_{g^{−1}} ). Hence 𝕂(A_{g} ⊗_{A_{e}} B_{e}^{+} ) ≅ C_{0}(j^{−1} (D_{g^{−1}} )). The representation of A_{e} on A_{g} ⊗_{A_{e}} B_{e}^{+} is equivalent to a representation on 𝕂(A_{g} ⊗_{A_{e}} B_{e}^{+} ) by Proposition 3.13. This is equivalent to a continuous map j^{−1} (D_{g^{−1}} ) → Â_{e} by Proposition 8.1. This map is ϑ_{g} ∘ j by a fibrewise computation. Hence the induced representation of A_{e} on A_{g} ⊗_{A_{e}} B_{e}^{+} is X-integrable if and only if ϑ_{g} ∘ j has values in j(X) and the partial maps j^{−1} ∘ ϑ_{g} ∘ j on X are continuous.

From now on, we assume that the partial action of G on Â_{e}^{+} restricts to X^{+} . By Proposition 11.7, this assumption is necessary and sufficient for X-integrability to be compatible with induction. The “restriction” of the partial action on Â_{e}^{+} to X^{+} is a partial action of G on X^{+} by partial homeomorphisms. Its transformation groupoid G ⋉ X^{+} is constructed like G ⋉ Â_{e}^{+} . Its set of arrows is the subset of G ⋉ Â_{e}^{+} of arrows with range and/or source in j(X^{+} ), and the topology on the arrow space is the unique one that makes the inclusion G ⋉ X^{+} → G ⋉ Â_{e}^{+} and the range and source maps G ⋉ X^{+} → X^{+} continuous. There is also a unique topology on the restriction Σ_{X} of Σ to j(X^{+} ) so that there is an extension of topological groupoids

X^{+} × 𝕋 ↣ Σ_{X} ↠ G ⋉ X^{+} .

Since X^{+} is locally compact, the groupoids in this extension are locally compact, Hausdorff groupoids. Since G ⋉ X^{+} is étale, it carries a canonical Haar system, namely, the family of counting measures. There is also a unique normalised Haar system on X^{+} × 𝕋. These produce a unique Haar system on Σ_{X} by [5, Theorem 5.1], so that the groupoid C^{∗}-algebra C^{∗}(Σ_{X}) is defined. The twisted groupoid C^{∗}-algebra C^{∗}(G ⋉ X^{+} , Σ_{X}) of G ⋉ X^{+} with respect to the twist Σ_{X} is defined in [24]. It is related to the groupoid C^{∗}-algebra of Σ_{X} in [5, Corollary 7.2].

Theorem 11.8. Let G be a discrete group and let A be a G-graded ^{∗}-algebra with commutative A_{e}. Let j: X → Â_{e} be an injective, continuous map, such that the partial action of G on Â_{e}^{+} in Theorem 11.2 restricts to X^{+} as in Proposition 11.7. Then C^{∗}(G ⋉ X^{+} , Σ_{X}) is a C^{∗}-hull for Rep_{int}(A, X).

Proof. The C^{∗}-algebra C^{∗}(G ⋉ X^{+} , Σ_{X}) may be defined as the full section C^{∗}-algebra of a certain Fell line bundle over the étale, locally compact groupoid G ⋉ X^{+} . The Fell line bundle involves the space of sections of the Hermitian complex line bundle L := Σ_{X} ×_{𝕋} ℂ associated to the principal 𝕋-bundle Σ_{X} ↠ G ⋉ X^{+} and the multiplication maps L_{g} × L_{h} → L_{gh} induced by the multiplication of Σ_{X} (see [5]). By construction, the Hilbert B_{e}^{+} -module B_{g}^{+} = A_{g} ⊗_{A_{e}} B_{e}^{+} is isomorphic to the continuous sections of this line bundle L over the subset {g} × D_{g^{−1}}^{X} of {g} × X^{+} : an element a ⊗ b is mapped to the continuous section that sends (g, x) for x ∈ X with j(x) ∈ D_{g^{−1}} to b(x) · χ(a^{∗}a)^{^{1}/2} [a]. The multiplication in Σ_{X} is defined so that the multiplication maps B_{g}^{+} ⊗_{B_{e}^{+}} B_{h}^{+} → B_{gh}^{+} are exactly the multiplication maps in the Fell line bundle associated to Σ_{X}. Thus the Fell bundle (B_{g}^{+} )_{g∈G} constructed in Theorem 9.26 is isomorphic to the Fell bundle (β_{g})_{g∈G} , where β_{g} is the space of C_{0}-sections of L over {g} × D_{g^{−1}} and the multiplication and involution come from the Fell line bundle structure on L over the groupoid G ⋉ X^{+} . The full section C^{∗}-algebra of this Fell bundle is canonically isomorphic to the section C^{∗}-algebra of the corresponding Fell bundle over the groupoid G ⋉ X^{+} by results of [3]. The small issue to check here is that it makes no difference whether we use C_{0}-sections or compactly supported continuous sections of L over {g} × D_{g^{−1}} . Both have the same C^{∗}-completion. This is a special case of general results about Fell bundles over étale locally compact groupoids.

If (B_{g})_{g∈G} is any Fell bundle over G, then ⨁ _{g∈G} B_{g} is a ^{∗}-algebra, to which we may apply our machinery although all its representations are bounded. Thus any Fell bundle over G may come up for some choice of the G-graded ∗-algebra A. Thus the section C^{∗}-algebra of a Fell bundle (B_{g})_{g∈G} with commutative unit fibre is always a twisted groupoid C^{∗}-algebras of a twist of an étale groupoid, namely, the transformation groupoid of a certain partial action on the spectrum of the unit fibre associated to the Fell bundle. This result is already known, even for Fell bundles over inverse semigroups with commutative unit fibre, see [3]. If Σ_{X} ≅ (G ⋉ X^{+} ) × 𝕋 as a groupoid, then C^{∗}(G ⋉ X^{+} , Σ_{X}) ≅ C^{∗}(G ⋉ X^{+} ). This is the same as the crossed product for the partial action of G on X^{+} . This happens in all the examples in [7,27]. The possible twists have two levels. First, Σ_{X} may be non-trivial as a principal circle bundle over G ⋉ X^{+} . Secondly, if it is trivial as a principal circle bundle, the multiplication may create a non-trivial twist. The circle bundle Σ_{X} ↠ G ⋉ X^{+} is trivial if and only if its restriction to {g} × D_{g^{−1}} is trivial for each g ∈ G. For a circle bundle, this means that there is a nowhere vanishing section. For instance, if there is a ∈ A_{g} that generates A_{g} as a right A_{e}-module, then U_{a} = D_{g^{−1}} and σ_{a} is a global trivialisation of Σ_{X} |_{{g}×D_{g^{−1}}} . The complex line bundles over a space X are classified by the second cohomology group H^{2}(X, ℤ). If L is a line bundle, then the spaces of C_{0}-sections of L^{⊗n} for n ∈ ℤ form a Fell bundle over ℤ, and the direct sum of these spaces of sections is a ℤ-graded ^{∗}-algebra such that the given line bundle L appears in the resulting twisted groupoid. If H^{2}(X, ℤ) ≠ 0, the space X is at least 2-dimensional. There are indeed non-trivial complex line bundles over all compact oriented 2-dimensional manifolds. The resulting ^{∗}-algebra, however, has only ^{∗}-representations by bounded operators if X is compact. Examples where unbounded operators appear must involve a non-trivial line bundle over a noncompact space. These first appear in dimension 3. It is easy to write down a ℤ-graded ^{∗}-algebra A where B_{e}^{+} is, say, S^{2} × ℝ and B_{g}^{+} involves the Bott line bundle over S^{2}. These examples seem artificial, however. Now assume that Σ_{X} is trivial as a principal circle bundle over (G⋉X^{+} )^{1}, that is, Σ_{X} ≅ (G ⋉ X^{+} )^{1} × 𝕋 as a 𝕋-space. We may choose this homeomorphism to be the obvious one on the open subset (1 ⋉ X^{+} ) × 𝕋 corresponding to 1 ∈ G.

The multiplication must be of the form

(g_{1}, ϑ_{g_{2}} (x), λ_{1}) · (g_{2}, x, λ_{2}) = (g_{1} · g_{2}, x, φ(g_{1}, g_{2}, x) · λ_{1} · λ_{2})

for some continuous 𝕋-valued function φ with φ(1, g, x) = 1 = φ(g, 1, x) for all g, x; here φ is defined on the space of all triples (g_{1}, g_{2}, x_{2}) ∈ G × G × X^{+} with x_{2} ∈ D_{g_{2}^{−1}} and ϑ_{g_{2}} (x_{2}) ∈ D_{g_{1}^{−1}} ; this space is homeomorphic to the space (G ⋉ X^{+} )^{2} of pairs of composable arrows in G ⋉ X^{+} . The associativity of the multiplication in Σ_{X} is equivalent to the cocycle condition

φ(g_{1}, g_{2} · g_{3}, x) · φ(g_{2}, g_{3}, x) = φ(g_{1} · g_{2}, g_{3}, x) · φ(g_{1}, g_{2}, ϑ_{g_{3}} (x))  (11.9)

for all g_{1}, g_{2}, g_{3} ∈ G, x ∈ X^{+} for which ϑ_{g_{3}} (x), ϑ_{g_{2}} ∘ϑ_{g_{3}} (x), and ϑ_{g_{1}} ∘ϑ_{g_{2}} ∘ϑ_{g_{3}} (x) are defined. A different trivialisation of the circle bundle Σ_{X} ↠ (G ⋉ X^{+} )^{1} modifies φ by the coboundary

∂ψ(g_{1}, g_{2}, x) := ψ(g_{2}, x)ψ(g_{1} · g_{2}, x)^{−1} ψ(g_{1}, ϑ_{g_{2}} (x))  (11.10)

of a continuous function ψ: (G ⋉ X^{+} )^{1} → 𝕋 normalised by ψ(1, x) = 0 for all x ∈ X^{+} . Thus isomorphism classes of twists of G⋉X^{+} are in bijection with the groupoid cohomology H^{2}(G⋉X, 𝕋), that is, the quotient of the group of continuous maps φ: (G ⋉ X^{+} )^{2} → 𝕋 satisfying (11.9) by the group of 2-coboundaries ∂ψ of continuous 1-cochains ψ: (G⋉X^{+} )^{1} → 𝕋, where ∂ψ is defined in (11.10). In the easiest case, the function φ above does not depend on x. Then φ: G × G → 𝕋 is a normalised 2-cocycle on G in the usual sense. These cocycles appear, for instance, in the classification of projective representations of the group G. This is related to the twists above because the Hilbert space representations of the twisted group algebra for a 2-cocycle φ: G × G → 𝕋 are exactly the projective representations π: G → U(H) with π(g)π(h) = φ(g, h)π(gh) for all g, h ∈ G. The group ℤ has no nontrivial 2-cocycles. They do appear, however, for the group ℤ^{2}. A well known example is the noncommutative torus. Its usual gauge action corresponds to a ℤ^{2}-grading, where U^{n}V^{m} for the canonical generators U, V has degree (n, m) ∈ ℤ^{2}. In this case, A_{e} = ℂ = B_{e} = B_{e}^{+} , and Â_{e}^{+} has only one point. The transformation groupoid G ⋉ Â_{e}^{+} is simply G = ℤ^{2}. This is discrete, so Σ is always trivial as a principal circle bundle. Thus the only non-trivial aspect of Σ is a 2-cocycle φ: ℤ × ℤ → 𝕋. The cohomology group H^{2}(ℤ^{2}, 𝕋) is isomorphic to 𝕋, and the resulting twisted group algebras of ℤ^{2} are exactly the noncommutative tori.

Proposition 11.11. If there are subsets S_{g} ⊆ A_{g} such that S_{g} generates A_{g} as a right A_{e}-module, S_{g} · S_{h} ⊆ S_{gh}, and χ(a^{∗}b) ≥ 0 for all a, b ∈ S_{g}, g ∈ G, χ ∈ j(X) ⊆ Â_{e}, then the twist Σ_{X} is trivial and so the C^{∗}-hull of A is C^{∗}(G⋉X^{+} ).

Proof. If χ ∈ D_{g^{−1}} , then there is b ∈ A_{g} with χ(b^{∗}b) ≠ 0. Since S_{g} generates A_{g} as a right A_{e}-module, we may write b = ∑ _{i=1}^{n} a_{i} ·c_{i} with a_{i} ∈ S_{g}, c_{i} ∈ A_{e}. Then

χ(b^{∗}b) = ∑_{i,j=1}^{n} χ(c_{i}^{∗})χ(c_{j})χ(a_{i}^{∗}a_{j}).

Hence there are i, j with χ(a_{i}^{∗}a_{j}) ≠ 0. Then χ(a_{i}^{∗}a_{i}) ≠ 0 by (11.3). This shows that ⋃ _{a∈S_{g}} U_{a} = D_{g^{−1}} . We have (g, χ, [a]) = (g, χ, [b]) for all a, b ∈ S_{g}, χ ∈ j(X)∩U_{a} ∩U_{b} because χ(a^{∗}b) ≥ 0 for all χ ∈ j(X). Hence the local sections σ_{a} of Σ_{X} |_{{g}×D_{g^{−1}}} for a ∈ S_{g} coincide on the intersections of their domains and thus combine to a global trivialisation. This trivialisation is multiplicative as well.

If Â_{e}^{+} itself is locally compact, then we may take X^{+} = X = Â_{e}^{+} with the inclusion map j. Since Â_{e}^{+} is closed in Â_{e}, this happens if Â_{e} is locally compact.

Theorem 11.12. Assume that Â_{e}^{+} is locally compact in the topology τ_{c}. Call a representation of A integrable if its restriction to A_{e} is locally bounded. Let Σ be the twisted groupoid constructed above. Then C^{∗}(G⋉(Â_{e}^{+} , τ_{c}), Σ) is a C^{∗}-hull for the integrable representations of A.

Proof. If X^{+} = (Â_{e}^{+} , τ_{c}), then integrability is compatible with induction by Proposition 11.7 because the construction of the topology τ_{c} is natural and compatible with restriction to open subsets. Theorem 11.8 shows that C^{∗}(G ⋉ (Â_{e}^{+} , τ_{c}), Σ) is a C^{∗}-hull for the class of representations of A whose restriction to A_{e} is Â_{e}^{+} -integrable. The locally bounded representations of A_{e} are equivalent to the locally bounded representations of the pro-C^{∗}-algebra C(Â_{e}, τ_{c}) by Propositions 7.2 and 7.6. Restrictions of representations of A to A_{e} are automatically inducible by Lemma 9.8. By a pro-C^{∗}-algebraic variant of Lemma 11.1, the inducible, locally bounded representations of A_{e} are equivalent to those representations of C(Â_{e}, τ_{c}) that factor through the quotient C(Â_{e}^{+} , τ_{c}). Since Â_{e}^{+} is locally compact, C_{0}(Â_{e}^{+} , τ_{c}) is dense in the pro-C^{∗}-algebra C(Â_{e}^{+} , τ_{c}). Hence C_{0}(Â_{e}^{+} , τ_{c}) is a C^{∗}-hull for the inducible, locally bounded representations of A_{e} by Theorem 7.17.

Assume that A_{e} is countably generated. Then the usual topology on Â_{e} is metrisable and hence compactly generated, so that τ_{c} is the standard topology on Â_{e}. A representation of A_{e} is locally bounded if and only if all symmetric elements of A_{e} act by regular, self-adjoint operators by Theorem 8.7. Thus a representation π of A is integrable as in Theorem 11.12 if and only if \overline{π(a)} is regular and self-adjoint for all a ∈ A_{e} with a = a^{∗}. This class of integrable representations has the C^{∗}-hull C^{∗}(G ⋉ Â_{e}^{+} , Σ) if Â_{e}^{+} is locally compact. In particular, if A_{e} is finitely generated, then Â_{e} is mapped homeomorphically onto a closed subset of ℝ^{n} for some n ∈ ℕ by evaluating characters on a finite set of symmetric generators. Thus Â_{e} is locally compact. The discussion above gives:

Corollary 11.13. Assume that A_{e} is finitely generated. Call a representation of A integrable if its restriction to A_{e} is locally bounded. Then Â_{e}^{+} is locally compact and C^{∗}(G ⋉ Â_{e}^{+} , Σ) for the twisted groupoid Σ constructed above is a C^{∗}-hull for the integrable representations of A. Moreover, a representation π of A is integrable if and only if \overline{π(a)} is regular and self-adjoint for all a ∈ A_{e} with a = a^{∗}.

Corollary 11.13 covers all the examples considered in [7, 27], except for the enveloping algebra W of the Virasoro algebra that is studied in [27, §9.3]. The ^{∗}-algebra W is ℤ-graded. Its unit fibre W_{0} is noncommutative. The first step in the study of its representations in [27, §9.3] is to replace W by a certain ℤ-graded quotient A := W/I, whose unit fibre A_{0} = W_{0}/(I ∩ W_{0}) is commutative by construction. The motivation is that all “integrable” representations of W factor through A. The main result in [27, §9.3] shows that the partial action of ℤ on Â_{e}^{+} is free and that the disjoint union Y := X_{1} ⊔ X_{2} ⊔ X_{3} of the three families of characters described in (61)–(63) of [27] is a fundamental domain, that is, it meets each orbit of the partial action exactly once. Each subset X_{i} is closed in Â_{e} and locally compact and second countable in the subspace topology. Hence so is Y. Since ℤ acts by partial homeomorphisms and Y is a fundamental domain, there is a continuous bijection

X := ⨆_{n∈ℤ} (D_{−n} ∩ Y) → Â_{e}^{+} , (n, y) ↦ ϑ_{n}(y).

Each D_{−n} ∩ Y is an open subset of Y, so that X is locally compact. I have not checked whether this continuous bijection is a homeomorphism. If so, then Â_{e}^{+} would be locally compact and the results in [27] for the Virasoro algebra would be contained in Theorem 11.12 after passing to the quotient W/I. If not, we would use the locally compact space X. The partial action of ℤ on Â_{e}^{+} is clearly continuous on X as well, so that Theorem 11.8 applies.

12 Rieffel deformation

Let G be a discrete group. Given a normalised 2-cocycle on G, Rieffel deformation is a deformation functor that modifies the multiplication on a G-graded ∗-algebra by the 2-cocycle. There is a similar process for Fell bundles over G, which we may transfer to section C^{∗}-algebras. This is how Rieffel deformation is usually considered. The setting of graded algebras or Fell bundles is easier. We now define Rieffel deformation more precisely and show that it is compatible with the construction of C^{∗}-hulls in Theorem 9.26. This deformation process has also recently been treated in [23]. A normalised 2-cocycle on a group G is a function Λ: G × G → U(1) with Λ(e, g) = 1 = Λ(g, e) for all g ∈ G and

Λ(g, h · k)Λ(h, k) = Λ(g · h, k)Λ(g, h)  (12.1)

for all g, h, k ∈ G. Let A = ⨁ _{g∈G} A_{g} be a G-graded algebra. Let A^{Λ} be the same G-graded vector space with the deformed multiplication and involution

∑_{g∈G} a_{g} ∗ ∑_{h∈G} b_{h} := ∑_{g,h∈G} Λ(g, h)a_{g}b_{h}, (∑ a_{g})^{†} := ∑ \overline{Λ(g^{−1} , g)}a_{g}^{∗},

where a_{g}, b_{g} ∈ A_{g} for all g ∈ G. We call A^{Λ} the Rieffel deformation of A with respect to Λ.

Lemma 12.2. The deformed multiplication and involution on A^{Λ} give a G-graded ^{∗}-algebra with a ∗ b = ab if a ∈ A_{e} or b ∈ A_{e}, and a^{†} ∗ b = a^{∗}b for all g ∈ G, a, b ∈ A_{g}.

Proof. The multiplication remains associative by the 2-cocycle condition (12.1). The normalisation of Λ and (12.1) for g, g^{−1} , g give Λ(g, g^{−1} ) = Λ(g^{−1} , g) for all g ∈ G. Thus

(a_{g}^{†})^{†} = \overline{Λ(g, g^{−1} )} · \overline{\overline{Λ(g^{−1} , g)}}(a_{g}^{∗})^{∗} = a_{g}

for a_{g} ∈ A_{g}. The normalisation condition and (12.1) for g, h, h^{−1} and gh, h^{−1} , g^{−1} for g, h ∈ G give

Λ(gh, h^{−1} )Λ(g, h) = Λ(h, h^{−1} ),
Λ(g, g^{−1} )Λ(gh, h^{−1} ) = Λ(gh, h^{−1} g^{−1} )Λ(h^{−1} , g^{−1} ).

Hence Λ(g, g^{−1} )Λ(h, h^{−1} ) = Λ(g, h)Λ(gh, h^{−1} g^{−1} )Λ(h^{−1} , g^{−1} ). This implies the condition (a_{g} ∗ b_{h})^{†} = b_{h}^{†} ∗ a_{g}^{†} for a_{g} ∈ A_{g}, b_{h} ∈ A_{g}:

(a_{g} ∗ b_{h})^{†} = \overline{Λ(g, h)} · \overline{Λ(gh, (gh)^{−1} )} · (a_{g}b_{h})^{∗}
= \overline{Λ(g, g^{−1} ) · Λ(h, h^{−1} )} · Λ(h^{−1} , g^{−1} ) · b_{h}^{∗}a_{g}^{∗} = b_{h}^{†} ∗ a_{g}^{†}.

Thus the deformed multiplication and involution give a ^{∗}-algebra. The formula a^{†} ∗ b = a^{∗}b for g ∈ G, a, b ∈ A_{g} is trivial, and a ∗ b = ab if a ∈ A_{e} or b ∈ A_{e} follows from the normalisation of Λ.

The same formulas work if (B_{g})_{g∈G} is a Fell bundle over G. Let (B_{g}^{Λ} )_{g∈G} be the same Banach space bundle as B_{g} with the multiplication and involution a_{g} ∗ b_{h} := Λ(g, h)a_{g}b_{h} and a_{g}^{†} = \overline{Λ(g^{−1} , g)}a_{g}^{∗} for g, h ∈ G, a_{g} ∈ B_{g}, b_{h} ∈ B_{h}. By Lemma 12.2, the deformation does not change ab for a ∈ B_{e} or b ∈ B_{e} and a^{∗}b and ab^{∗} for a, b ∈ B_{g}. Hence B_{g}^{Λ} = B_{g} as Hilbert B_{e}-bimodules, so that the positivity and completeness conditions for a Fell bundle are not affected by the deformation. We call (B_{g}^{Λ} )_{g∈G} the Rieffel deformation of the Fell bundle (B_{g})_{g∈G} with respect to Λ. For a C^{∗}-algebra of the form B = C^{∗}(B_{g}) for a Fell bundle (B_{g})_{g∈G} over G, we define its Rieffel deformation with respect to Λ as B^{Λ} := C^{∗}(B_{g}^{Λ} ) for the deformed Fell bundle. If G is an Abelian group, then C^{∗}(B_{g}) for a Fell bundle over G carries a canonical continuous action of Ĝ, called the dual action. Conversely, any C^{∗}-algebra with a continuous Ĝ-action β is of the form B = C^{∗}(B_{g}), where (B_{g})_{g∈G} is the spectral decomposition of the action,

B_{g} = {b ∈ B | β_{χ}(b) = χ(g) · b for all χ ∈ Ĝ}.

Thus Rieffel deformation takes a C^{∗}-algebra with a continuous Ĝ-action to another C^{∗}-algebra with a continuous Ĝ-action. This is how it is usually formulated. Since Ĝ is compact, there are no analytic difficulties with oscillatory integrals as in [26].

Theorem 12.3. Let A = ⨁ _{g∈G} A_{g} be a G-graded ^{∗}-algebra and let B_{e} be a C^{∗}-hull for a class of integrable representations of A_{e}. Assume that integrability is compatible with induction for A. Let Λ be a normalised 2-cocycle on G. Then integrability is also compatible with induction for A^{Λ} , and the C^{∗}-hull for the integrable representations of A^{Λ} is the Rieffel deformation with respect to Λ of the C^{∗}-hull for the integrable representations of A.

Proof. The compatibility condition in Definition 9.18 is equivalent to the integrability of A_{g} ⊗_{A_{e}} (𝔈, π) for all g ∈ G, which only involves a single A_{g} with its A_{e}-bimodule structure and the A_{e}-valued inner product ⟨a, b⟩ = a^{∗}b for a, b ∈ A_{g}. This is not changed by Rieffel deformation by Lemma 12.2. Hence A^{Λ} inherits the compatibility condition from A, and Theorem 9.26 applies to both A and A^{Λ} . The Hilbert B_{e}^{+} -bimodule B_{g}^{+} depends only on A_{g} with the extra structure above and the universal inducible, integrable representation (𝔅_{e}^{+} , μ_{e}^{+} ) of A_{e} by Remark 9.27. Since none of this is changed by Rieffel deformation, the Fell bundle obtained from A^{Λ} has the same fibres (B_{g}^{+} )^{Λ} as B_{g}^{+} . Rieffel deformation changes the multiplication maps A_{g} ×A_{h} → A_{gh} and the involution A_{g} → A_{g^{−1}} for fixed g, h ∈ G only by a scalar. Inspecting the construction above, we see that the multiplication maps B_{g}^{+} × B_{h}^{+} → B_{gh}^{+} and the involution B_{g}^{+} → B_{g^{−1}}^{+} in the Fell bundle are changed by exactly the same scalars. Hence the Fell bundle for A^{Λ} is (B_{g}^{+} )^{Λ} . Now the assertion follows from Theorem 9.26.

13 Twisted Weyl algebras

We illustrate our theory by studying C^{∗}-hulls of twisted n-dimensional Weyl algebras for 1 ≤ n ≤ ∞. We begin with the case n = 1, where no twists occur. Then we consider the case of finite n without twists and with twists. Finally, we consider the case n = ∞ with and without twists. The (1-dimensional) Weyl algebra A is the universal ^{∗}-algebra with one generator a and the relation aa^{∗} = a^{∗}a+1. There is a unique ℤ-grading A = ⨁ _{n∈ℤ} A_{n} with a ∈ A_{1}. The^{∗}-subalgebra A_{0} is isomorphic to the polynomial algebra ℂ[N] with N = a^{∗}a, which is commutative. The other subspaces A_{k} ⊆ A for k ∈ ℕ are isomorphic to A_{0} as left or right A_{0}-modules because A_{k} = A_{0} ·a^{k} = a^{k} ·A_{0} and A_{−k} = (a^{∗})^{k} · A_{0} = A_{0} · (a^{∗})^{k} for all k ≥ 0. The spectrum Â_{0} of A_{0} is ℝ, where the character ℂ[N] → ℂ for t ∈ ℝ evaluates a polynomial at t. A character is positive if and only if it is positive on (a^{∗})^{k}a^{k} and a^{k}(a^{∗})^{k} for all k ≥ 1. This happens if and only if t ∈ ℕ by [27, Example 10]. Since Na = a^{∗}aa = (aa^{∗} − 1)a = a · (a^{∗}a − 1) = a · (N − 1), the partial automorphism ϑ_{1} of Â_{0}^{+} = ℕ associated to the A_{0}-bimodule A_{1} acts on Â_{e}^{+} by the automorphism N ↦ N − 1, which corresponds to translation by −1. By induction, we get N · a^{k} = a · (N − 1) · a^{k−1} = ⋯= a^{k} · (N − k). The domain of ϑ_{k} is as big as it could possibly be, that is, it contains all n ∈ ℕ with n ≥ k by (11.5) (see also [27, Example 16]). For any k, l ∈ ℕ there is a unique n ∈ ℤ with k − n = l. Thus the transformation groupoid ℤ ⋉_{ϑ} ℕ is simply the pair groupoid on ℕ. There can be no twist in this case. First, the pair groupoid simply has no non-trivial twists. And secondly, the generators a^{k}, (a^{∗})^{k} for k ≥ 0 satisfy the positivity condition in Proposition 11.11, which also rules out a twist. Since no proper non-empty subset of ℕ is invariant under the partial action ϑ of ℤ, a commutative C^{∗}-hull for A_{0} for which integrability is compatible with induction gives either B_{0}^{+} = C_{0}(ℕ) or B_{0}^{+} = {0}. In the second case, A has no non-zero integrable representations. In the first case, the C^{∗}-hull for the integrable representations of A is the groupoid C^{∗}-algebra 𝕂(ℓ^{2}ℕ) of the pair groupoid ℕ × ℕ. The universal representation of A on 𝕂(ℓ^{2}ℕ) is equivalent to a representation π of A on ℓ^{2}ℕ by Proposition 3.13. The domain of this representation is the space S(ℕ) of rapidly decreasing sequences, with π(a)(δ_{k}) = \sqrt{k}δ_{k−1} for k ∈ ℕ, so π(a^{∗})(δ_{k}) = \sqrt{k + 1} δ_{k+1} , π(N)(δ_{k}) = kδ_{k}. By Theorem 4.4, a representation π of A_{0} on a Hilbert module E is integrable if and only \overline{π(N^{k})} is regular and self-adjoint for each k ∈ ℕ or, equivalently, π(N) is regular and self-adjoint and \overline{π(N^{k})} = \overline{π(N)}^{k} for all k ∈ ℕ. By definition, a representation of A is integrable if and only if its restriction to A_{0} is integrable. The ℤ-grading on the C^{∗}-hull 𝕂(ℓ^{2}ℕ) is “inner”: it is induced by the ℤ-grading on ℓ^{2}ℕ where δ_{k} has degree k. Equivalently, the dual action of 𝕋 on 𝕂(ℓ^{2}ℕ) associated to the ℤ-grading is the inner action associated to the unitary representation U: 𝕋 → U(ℓ^{2}ℕ), where U_{z}(δ_{k}) := z^{k}δ_{k} for all z ∈ 𝕋, k ∈ ℕ.

Now let m ∈ ℕ and let Θ = (Θ_{jk}) be an antisymmetric m × m-matrix. Let λ_{jk} = exp(2πiΘ_{jk}). Let A^{m,Θ} be the ^{∗}-algebra with generators a_{1}, …, a_{m} and the commutation relations a_{j} a_{j}^{∗} = a_{j}^{∗}a_{j} + 1 for 1 ≤ j ≤ m and

a_{j} a_{k} = λ_{jk} a_{k}a_{j} ,  a_{j}^{∗}a_{k} = λ_{jk}^{−1} a_{k}a_{j}^{∗}  (13.1)

for 1 ≤ j ≠ k ≤ m. Since λ_{jk} = λ_{kj}^{−1} , the relations (13.1) for (j, k) and (k, j) are equivalent; so it suffices to require (13.1) for 1 ≤ j < k ≤ m. The ∗-algebra A^{m,Θ} is ℤ^{m}-graded by giving a_{j} degree e_{j} ∈ ℤ^{m}, where e_{1}, …, e_{m} is the standard basis of ℤ^{m}. We first consider the case Θ = 0 and write A^{m} := A^{m,0} . This is the m-dimensional Weyl algebra, which is the tensor product of m copies of the 1-dimensional Weyl algebra, with the induced ℤ^{m}-grading. Thus the zero fibre A_{0}^{m} for 0 ∈ ℤ^{m} is isomorphic to the polynomial algebra ℂ[N_{1}, …, N_{m}] in the m generators N_{j} = a_{j}^{∗}a_{j}. Its spectrum is ℝ^{m}. Each A_{k}^{m} for k ∈ ℤ^{m} is isomorphic to A_{0}^{m} both as a left and a right A_{0}^{m}-module; the generator is the product of a_{j}^{k_{j}} for k_{j} ≥ 0 or (a_{j}^{∗})^{−k_{j}} for k_{j} < 0 from j = 1, …, m. Here the order of the factors does not matter because Θ = 0. We may identify A_{k}^{m} with the exterior tensor product of the A^{1}-bimodules A_{k_{1}}^{1} ⊗ A_{k_{2}}^{1} ⊗ ⋯⊗ A_{k_{m}}^{1} . Hence the space of positive characters on A^{m} is ℕ^{m}, and the partial action of ℤ^{m} on ℕ^{m} is the exterior product of the partial actions of ℤ on ℕ for the 1-dimensional Weyl algebras. That is, k ∈ ℤ^{m} acts on ℕ^{m} by translation by −k with the maximal possible domain. Thus the transformation groupoid ℤ^{m} ⋉Â_{0}^{+} is isomorphic to the pair groupoid of the discrete set ℕ^{m}. Once again, the only ℤ^{m}-invariant subsets of Â_{0}^{+} are the empty set and ℕ^{m}, so that the only inducible commutative C^{∗}-hulls of A_{0} for which integrability is compatible with induction are {0} and C_{0}(ℕ^{m}). The first case is boring, and the second case leads to the C^{∗}-hull 𝕂(ℓ^{2}ℕ^{m}) of the m-dimensional Weyl algebra. As for m = 1, the universal representation of A^{m} is equivalent to a representation on ℓ^{2}ℕ^{m}. This has the domain S(ℕ^{m}), and the representation is determined by

π(a_{j})(δ_{(k_{1} ,...,k_{m} )} ) = \sqrt{k_{j}}δ_{(k_{1} ,...,k_{j} −1,...,k_{m} )}

for (k_{1}, …, k_{m}) ∈ ℕ^{m} and j = 1, …, m. Hence π(N_{j})(δ_{(k_{1} ,...,k_{m} )} ) = k_{j} δ_{(k_{1} ,...,k_{m} )} . A representation of A is integrable if and only if its restriction to A_{0} is integrable in the sense that it integrates to a representation of C_{0}(ℝ^{m}). This automatically descends to a representation of C_{0}(ℕ^{m}) by Lemma 9.8. There are several ways to characterise when a representation of ℂ[N_{1}, …, N_{m}] integrates to a representation of C_{0}(ℝ^{m}). One is that \overline{π(N_{j})} for j = 1, …, m are strongly commuting, regular, self-adjoint operators and \overline{π(N_{j}^{k})} = \overline{π(N_{j})}^{k} for all j = 1, …, m, k ∈ ℕ, compare [28, Theorem 9.1.13]. The groups ℤ^{m} for m ≥ 2 have non-trivial 2-cocycles, and A^{m,Θ} is, by definition, a Rieffel deformation of A^{m,0} for the normalised 2-cocycle

Λ((x_{1}, …, x_{m}), (y_{1}, …, y_{m})) := ∏_{j=1}^{m} ∏_{k=j+1}^{m} λ_{jk}^{x_{k} y_{j}} .  (13.2)

We could also use the cohomologous antisymmetric 2-cocycle

∏_{j=1}^{m} ∏_{k=j+1}^{m} \sqrt{λ_{jk}} ^{x_{k} y_{j} −x_{j} y_{k}} = ∏_{j,k=1}^{m} exp(−πiΘ_{jk} x_{j} y_{k}).

Theorem 12.3 says that the C^{∗}-hull B^{m,Θ} of A^{m,Θ} is the Rieffel deformation of the C^{∗}-hull B^{m,0} of A^{m,0} for the same 2-cocycle. In the classification of Fell bundles with commutative unit fibre, the important cohomology is that of the transformation groupoid G ⋉ X^{+} , not of G itself. Here G ⋉ X^{+} is the pair groupoid of ℕ^{m}.

Lemma 13.3 (compare [18, Lemma 2.9]). The cohomology of the pair groupoid of a discrete set X with coefficients in an Abelian group H vanishes in all positive degrees.

Proof. The set of composable n-tuples in the pair groupoid of X is X^{n+1} . The groupoid cohomology with coefficients H is the cohomology of the chain complex with cochains C_{n} := H^{X^{n+1}} , the space of all maps X^{n+1} → H, and with the boundary map ∂: C_{n} → C_{n+1} ,

∂φ(x_{0}, …, x_{n}) := ∑_{i=0}^{n} (−1)^{i}φ(x_{0}, …, x̂_{i}, …, x_{n});

here the hat means that the entry x_{i} is deleted. Pick some point x_{0} ∈ X and let hφ(x_{1}, …, x_{n}) := φ(x_{0}, x_{1}, …, x_{n}) for all n ∈ ℕ, x_{1}, …, x_{n} ∈ X, φ ∈ C_{n+1} . Then ∂ ∘ h(φ) + h ∘ ∂(φ) = φ for all φ ∈ C_{n}, n ≥ 1. Thus the cohomology vanishes in positive degrees.

Any twist of the pair groupoid on ℕ^{m} is trivial by Lemma 13.3. Therefore, the C^{∗}-hull B^{m,Θ} is isomorphic to 𝕂(ℓ^{2}ℕ^{m}), the untwisted groupoid C^{∗}-algebra of the pair groupoid. The proof of Lemma 13.3 is explicit and so allows to construct this isomorphism. We explain another way to construct it, using properties of Rieffel deformation. Since the ℤ-grading on the C^{∗}-hull 𝕂(ℓ^{2}ℕ) is inner or, equivalently, the corresponding action of 𝕋 is inner, the same holds for the ℤ^{m}-grading on the C^{∗}-hull 𝕂(ℓ^{2}ℕ^{m}) and the corresponding 𝕋^{m}-action on 𝕂(ℓ^{2}ℕ^{m}). Explicitly, the 𝕋^{m}-action is induced by the unitary representation of 𝕋^{m} on ℓ^{2}ℕ^{m} defined by

U_{(z_{1} ,...,z_{m} )} δ_{(k_{1} ,...,k_{m} )} := z_{1}^{−k_{1}} ⋯ z_{m}^{−k_{m}} δ_{(k_{1} ,...,k_{m} )} .

Rieffel deformation of C^{∗}-algebras for inner actions does not change the isomorphism type of the C^{∗}-algebra. Hence the C^{∗}-hull for the integrable representations of A^{m,Θ} is also isomorphic to 𝕂(ℓ^{2}ℕ^{m}). We make this more explicit in our Fell bundle language. Let U: 𝕋^{m} → UM(B) be a strictly continuous homomorphism to the group of unitary multipliers of a C^{∗}-algebra B and let α_{z}(b) := U_{z}bU_{z}^{∗} for z ∈ 𝕋^{m}, b ∈ B be the resulting inner action. Let (B_{k})_{k∈ℤ^{m}} be the spectral decomposition of this action, that is, b ∈ B_{k} if and only if α_{z}(b) = z^{k} · b for all z ∈ 𝕋^{m}. In particular, U ∈ UM(B_{0}) because 𝕋^{m} is commutative. Assume for simplicity that the 2-cocycle Λ is a bicharacter as above. For fixed k ∈ ℤ^{m}, we view Λ(k, ␣): ℤ^{m} → 𝕋 as an element Λ̃(k) of the dual group 𝕋^{m}. The map Λ̃ : ℤ^{m} → 𝕋^{m} is a group homomorphism. The maps ψ_{k}: B_{k} → B_{k}, b ↦ U_{Λ̃ (k)}^{∗} · b, for k ∈ ℕ are Banach space isomorphisms that modify the multiplication as follows:

ψ_{k}(b_{1})ψ_{l}(b_{2}) = U_{Λ̃ (k)}^{∗}b_{1}U_{Λ̃ (l)}^{∗}b_{2} = U_{Λ̃ (k+l)}^{∗} α_{Λ̃ (l)}(b_{1})b_{2} = ψ_{k+l} (Λ(k, l) · b_{1}b_{2}).

They keep the involution unchanged. This is exactly what Rieffel deformation does. Hence the maps ψ_{k} form an isomorphism between the undeformed and Rieffel deformed Fell bundles. This finishes the proof that the Rieffel deformed algebra for an inner action is canonically isomorphic to the original algebra. The universal representation of A^{m,Θ} on 𝕂(ℓ^{2}ℕ^{m}) again corresponds to a representation of A^{m,Θ} on ℓ^{2}(ℕ^{m}). We may construct it by carrying over the isomorphism B^{m,Θ} ≅ B^{m,0} between the C^{∗}-hulls. This is the inverse of the isomorphism above, so it multiplies on the left by the unitary U_{Λ̃ (k)} of degree 0 on elements of degree k. We do exactly the same on elements of A^{m,Θ} and so let x ∈ A_{k}^{m,Θ} for k ∈ ℤ^{m} act on ℓ^{2}ℕ^{m} by the operator U_{Λ̃ (k)}π^{m,0} (x), where π^{m,0} is the universal representation of the untwisted Weyl algebra A^{m,0} on ℓ^{2}ℕ^{m} described above. The same computation as above shows that this defines a ∗-representation of A^{m,Θ} . We compute it explicitly. First, the action of elements of A_{0}^{m,Θ} on ℓ^{2}ℕ^{m} is not changed. The domain of a representation of A^{m,Θ} is equal to the domain of its restriction to A_{0}^{m,Θ} . Hence the domain of our representation is the Schwartz space S(ℕ^{m}), as in the untwisted case. Let 1 ≤ j ≤ m. The generator a_{j} has degree e_{j} ∈ ℤ^{m}, and

Λ̃(e_{j}) = (λ_{1,j} , …, λ_{j−1,j} , 1, …, 1) ∈ 𝕋^{m}

for our first definition of Λ in (13.2). Thus

π^{m,Θ} (a_{j})δ_{(k_{1} ,...,k_{m} )} = U_{Λ̃ (e_{j} )}π^{m,0} (a_{j})δ_{(k_{1} ,...,k_{m} )}

= ( ∏_{l=1}^{j−1} λ_{j,l}^{k_{l}} ) \sqrt{k_{j}}δ_{(k_{1} ,...,k_{j} −1,...,k_{m} )} .

These operators on S(ℕ^{m}) satisfy the defining relations of A^{m,Θ} .

The infinite-dimensional Weyl algebra A^{∞} is the universal^{∗}-algebra with generators a_{j} for j ∈ ℕ and relations a_{j}^{∗}a_{j} = a_{j} a_{j}^{∗} +1, a_{j} a_{k} = a_{k}a_{j}, and a_{j}^{∗}a_{k} = a_{k}a_{j}^{∗} for 0 ≤ j < k. Let ℤ[ℕ] be the free Abelian group on countably many generators. The Weyl algebra A^{∞} is ℤ[ℕ]-graded, where a_{j} has degree e_{j} ∈ ℤ[ℕ], the jth generator of ℤ[ℕ]. The ^{∗}-algebra A^{∞} is a tensor product of infinitely many 1-dimensional Weyl algebras. The zero fibre A_{0}^{∞} is the polynomial algebra in the generators N_{j} = a_{j}^{∗}a_{j} for j ∈ ℕ. Hence its spectrum is the infinite product Â_{0}^{∞} = ℝ^{ℕ}, which is not locally compact. The tensor product structure of A^{∞} shows that a character is positive if and only if each component is. That is, (A^{∞} )_{0}^{+} ≅ ℕ^{ℕ} is a product of countably many copies of the discrete space ℕ. Since ℕ is not compact, this is not locally compact either. Hence to build a commutative C^{∗}-hull for A_{0}^{∞} , we must choose some locally compact space X with a continuous, injective map f: X → ℕ^{ℕ}. Here we have simplified notation by assuming that already f(X) ⊆ ℕ^{ℕ}; otherwise, the first step in our construction would replace X by X^{+} := f^{−1} (ℕ^{ℕ}). For X-integrability to be compatible with induction, we also need f(X) to be invariant under the partial action of ℤ[ℕ], and we need the restricted partial action to lift to a continuous partial action on X. The partial action of the group ℤ[ℕ] on ℕ^{ℕ} is the obvious one by translations. It is free, and two points (n_{k}) and (n′_{k}) in ℕ^{ℕ} belong to the same orbit if and only if there is k_{0} such that n_{k} = n′_{k} for all k ≥ k_{0}. Briefly, such points are called tail equivalent. This partial action is minimal, that is, an open, ℤ[ℕ]-invariant subset is either empty or the whole space. Hence ℕ^{ℕ} has no ℤ[ℕ]-invariant, locally closed subsets. Thus ℕ^{ℕ} has no ℤ[ℕ]-invariant, locally compact subspaces. Let K be any compact subset of ℕ^{ℕ}. Then the projection p_{j} : ℕ^{ℕ} → ℕ to the jth coordinate must map K to a compact subset of ℕ. So there is an upper bound M_{j} with p_{j} (K) ⊆ [0, M_{j}]_{ℕ} := [0, M_{j}] ∩ ℕ. Then K ⊆ ∏ _{j∈ℕ} [0, M_{j}]_{ℕ}, and the right hand side is compact. The closure of ∏ _{j∈ℕ} [0, M_{j}]_{ℕ} under tail equivalence is

X_{(M_{k})} := ⋃_{j∈ℕ} ( ℕ^{j} × ∏_{k>j} [0, M_{k}]_{ℕ}) ,

the restricted product of copies of ℕ with respect to the compact-open subsets [0, M_{j}]_{ℕ}. There is a unique topology on X_{(M_{k})} where each subset ℕ^{j} × ∏ _{k>j} [0, M_{k}]_{ℕ} is open and carries the product topology. This topology is locally compact, and the partial action of ℤ[ℕ] on X_{(M_{k})} by translation is continuous.

Lemma 13.4. The Local–Global Principle fails for the X_{(M_{k})} -integrable representations of A^{∞} .

Proof. Since the map X → Â_{0}^{∞} is not a homeomorphism onto its image and Â_{0}^{∞} is metrisable, the Local–Global Principle fails for the X-integrable representations of A_{0}^{∞} by Theorem 8.3. Applying induction from A_{0}^{∞} to A^{∞} to a counterexample for the Local–Global Principle for A_{0}^{∞} produces such a counterexample also for A^{∞} . Explicitly, choose a sequence (n_{k}) such that n_{k} > M_{k} for infinitely many k. Let x_{k} := n_{k}δ_{k} ∈ ℕ^{ℕ}, that is, x_{k} ∈ ℕ^{ℕ} has only one non-zero entry, which is n_{k} in the kth place. This sequence belongs to X_{(M_{k})} and converges to 0 in the product topology on Â_{0}^{∞} , but not in the topology of X_{(M_{k})} . The resulting representation of A_{0}^{∞} on C(ℕ̄) is not X_{(M_{k})} -integrable, but it becomes integrable when we tensor with any Hilbert space representation of C(ℕ̄), see the proof of Theorem 8.3. Now induce this (inducible) representation of A_{0}^{∞} to a representation of A^{∞} on C(ℕ̄). This gives a counterexample for the Local–Global Principle for A^{∞} .

I do not know a class of integrable representations of A^{∞} with a C^{∗}-hull for which the Local–Global Principle holds. Let S be the set of all words in the letters a_{j} , a_{j}^{∗}. If χ ∈ ℕ^{ℕ} is a positive character and x, y ∈ S ∩ A_{k}^{∞} for some k ∈ ℤ[ℕ], then χ(x^{∗}y) ≥ 0. Hence Proposition 11.11 shows that there is no twist in our case, that is, the C^{∗}-hull of the X_{(M_{k})} -integrable representations of A^{∞} is the groupoid C^{∗}-algebra of the transformation groupoid ℤ[ℕ] ⋉ X_{(M_{k})} . This C^{∗}-hull is canonically isomorphic to one of the host algebras for A^{∞} constructed in [11], namely, to the one that is denoted L[n] in [11] with n_{k} = M_{k} + 1 for all k ∈ ℕ. We remark in passing that the construction of a full host algebra from these host algebras in [11] is wrong: the resulting C^{∗}-algebra has too many Hilbert space representations, so it is not a host algebra any more, see the erratum [13].

The compact subset T := ∏ _{k∈ℕ} [0, M_{k}]_{ℕ} that we started with is a complete transversal in ℤ[ℕ] ⋉ X_{(M_{k})} , that is, the range map in ℤ[ℕ] ⋉ X_{(M_{k})} restricted to s^{−1} (T) is an open surjection onto X_{(M_{k})} . Hence the groupoid ℤ[ℕ] ⋉ X_{(M_{k})} is Morita equivalent to its restriction to the compact subset T. This restriction is the tail equivalence relation on T. Its groupoid C^{∗}-algebra is well known: it is the UHF-algebra for ∏ _{k∈ℕ} (M_{k} + 1), that is, the infinite tensor product of the matrix algebras ⨂ _{k∈ℕ} 𝕄_{M_{k} +1} . The C^{∗}-algebra of ℤ[ℕ]⋉X_{(M_{k})} itself is the C^{∗}-stabilisation of this UHF-algebra. Thus the X_{(M_{k})} -integrable representations of A^{∞} are equivalent to the representations of the (stabilisation of the) UHF-algebra of type ∏ _{k∈ℕ} (M_{k}+1). This depends very subtly on the choice of the sequence (M_{k}). There is no canonical ^{∗}-homomorphism between these UHF-algebras if we increase (M_{k}): for some (M_{k}) ≤ (M′_{k}), there is not even a non-zero map between their K-theory groups. Instead, there are canonical morphisms, that is, there is a canonical nondegenerate ^{∗}-homomorphism C^{∗}(ℤ[N] ⋉ X_{(M′_{k})} ) → M(C^{∗}(ℤ[N] ⋉ X_{(M_{k})} )) if (M_{k}) ≤ (M′_{k}). They are constructed as follows. The inclusion map X_{(M_{k})} ↪ X_{(M′_{k})} is continuous with dense range, but not proper. Thus it induces an injective, nondegenerate ^{∗}-homomorphism C_{0}(X_{(M′_{k})} ) → C_{b}(X_{(M_{k})} ). Therefore, if a representation of A_{0}^{∞} is X_{(M_{k})} -integrable, then it is also X_{(M′_{k})} -integrable. If a representation of A^{∞} has X_{(M_{k})} -integrable restriction to A_{0}^{∞} , then its restriction to A_{0}^{∞} is also X_{(M′_{k})} -integrable. When we apply this to the universal representation of A^{∞} on the C^{∗}-hull C^{∗}(ℤ[N] ⋉ X_{(M_{k})} ), this gives the desired canonical morphism C^{∗}(ℤ[N] ⋉ X_{(M′_{k})} ) → C^{∗}(ℤ[N] ⋉ X_{(M_{k})} ) if (M_{k}) ≤ (M′_{k}). It is injective, say, because C^{∗}(ℤ[N] ⋉ X_{(M′_{k})} ) is simple. Now let Θ = (Θ_{jk})_{j,k∈ℕ} be a skew-symmetric matrix. It corresponds first to a matrix λ_{jk} := exp(2πiΘ_{jk}) and then to a 2-cocycle Λ on ℤ[ℕ] as in (13.2). The Rieffel deformation of A^{∞} by Θ is the universal ^{∗}-algebra A^{∞,Θ} with the same generators (a_{j})_{j∈ℕ} and the relations a_{j} a_{j}^{∗} = a_{j}^{∗}a_{j} + 1, a_{j} a_{k} = λ_{jk} a_{k}a_{j}, and a_{j}^{∗}a_{k} = λ_{jk}^{−1} a_{k}a_{j}^{∗} for 0 ≤ j < k. We define X_{(M_{k})} for a sequence (M_{k}) and the X_{(M_{k})} -integrable representations of A^{∞,Θ} as above. By Theorem 12.3, this has a C^{∗}-hull, namely, the Rieffel deformation of the C^{∗}-hull for the X_{(M_{k})} -integrable representations of the undeformed Weyl algebra. The Rieffel deformation gives a twist of the groupoid ℤ[ℕ] ⋉ X_{(M_{k})} , and the C^{∗}-hull is the twisted groupoid C^{∗}-algebra of ℤ[ℕ] ⋉ X_{(M_{k})} for this twist.

Proposition 13.5. Let (M_{k}) ∈ ℕ^{ℕ}. The C^{∗}-hulls for the X_{(M_{k})} -integrable representations of the twisted Weyl algebras A^{∞,Θ} are isomorphic for all Θ.

Proof. The C^{∗}-hull of A^{∞,Θ} is a twisted groupoid algebra of the transformation groupoid ℤ[ℕ] ⋉ X_{(M_{k})} , which is isomorphic to the tail equivalence relation R on X_{(M_{k})} . We are going to prove that any twist X_{(M_{k})} × 𝕋 ↣ Σ ↠ R is trivial. Hence the C^{∗}-hull of A^{∞,Θ} is isomorphic to the untwisted groupoid C^{∗}-algebra of R for all Θ. The arrow space of R is totally disconnected because X_{(M_{k})} is totally disconnected and R is étale. Hence any locally trivial principal bundle over R is 1464 Ralf Meyer

trivial. Thus Σ ≅ R × 𝕋 as a topological space, and the twist is described by a continuous 2-cocycle φ: R^{(2)} := R ×_{s,X,r} R → 𝕋. We must show that φ is a coboundary. Let R_{d} for d ∈ ℕ be the equivalence relation on X_{(M_{k})} defined by (n_{k}) R_{d} (n′_{k}) if and only if n_{k} = n′_{k} for all k ≥ d. This is an increasing sequence of open subsets R_{d} ⊆ R with R = ⋃ R_{d}, and each R_{d} is also an equivalence relation. The equivalence relation R_{d} is isomorphic to the product of the pair groupoid on ℕ^{d} and the space X_{(M_{k+d} )} for the shifted sequence (M_{k+d} )_{k∈ℕ} . Thus the cohomology of R_{d} with coefficients in 𝕋 is isomorphic to the cohomology of the pair groupoid on ℕ^{d} with values in the Abelian group of continuous map X_{(M_{k+d} )} → 𝕋. This cohomology vanishes in positive degrees by Lemma 13.3. Therefore, for each d ∈ ℕ there is ψ_{d}: R_{d} → 𝕋 such that φ|_{R_{d}} : R_{d} ×_{s,r} R_{d} → 𝕋 is the coboundary ∂ψ_{d}. The restriction of ψ_{d+1} to R_{d} and ψ_{d} both have coboundary φ|_{R_{d}} . Hence ψ_{d+1}^{−1} |_{R_{d}} · ψ_{d} is a 1-cocycle on R_{d}. Again by Lemma 13.3, there is χ: X → 𝕋 with ψ_{d+1}^{−1} |_{R_{d}} · ψ_{d} = ∂_{R_{d}} χ. We replace ψ_{d+1} by ψ′_{d+1} := ψ_{d+1} · ∂_{R_{d+1}} χ, where ∂_{R_{d+1}} χ means the coboundary of χ on the groupoid R_{d+1} . This still satisfies ∂ψ′_{d+1} = ∂ψ_{d+1} = φ|_{R_{d+1}} , and ψ′_{d+1} |_{R_{d}} = ψ_{d}. Proceeding like this, we get continuous maps ψ′_{d}: R_{d} → 𝕋 for all d ∈ ℕ that satisfy ψ′_{d+1} |_{R_{d}} = ψ′_{d} and ∂ψ′_{d} = φ|_{R_{d}} for all d ∈ ℕ. These combine to a continuous map ψ′: R → 𝕋 with ∂ψ′ = φ.

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Ralf Meyer

Mathematisches Institut Universität Göttingen Bunsenstraße 3–5 37073 Göttingen Germany rmeyer2@uni-goettingen.de