Subproduct Systems

Orr Moshe Shalit and Baruch Solel ^{1}

Received: July 1, 2009 Revised: August 12, 2009

Communicated by Joachim Cuntz

Abstract. The notion of a subproduct system, a generalization of that of a product system, is introduced. We show that there is an essentially 1 to 1 correspondence between cp-semigroups and pairs (X, T) where X is a subproduct system and T is an injective subproduct system representation. A similar statement holds for subproduct systems and units of subproduct systems. This correspondence is used as a framework for developing a dilation theory for cp-semigroups. Results we obtain:

(i) a ∗-automorphic dilation to semigroups of ∗-endomorphisms over quite general semigroups; (ii) necessary and sufficient conditions for a semigroup of CP maps to have a ∗-endomorphic dilation; (iii) an analogue of Parrot’s example of three contractions with no isometric dilation, that is, an example of three commuting, contractive normal CP maps on B(H) that admit no ∗-endomorphic dilation (thereby solving an open problem raised by Bhat in 1998). Special attention is given to subproduct systems over the semigroup ℕ, which are used as a framework for studying tuples of operators satisfying homogeneous polynomial relations, and the operator algebras they generate. As applications we obtain a noncommutative (projective) Nullstellensatz, a model for tuples of operators subject to homogeneous polynomial relations, a complete description of all representations of Matsumoto’s subshift C^{∗}-algebra when the subshift is of finite type, and a classification of certain operator algebras – including an interesting non-selfadjoint generalization of the noncommutative tori.

2000 Mathematics Subject Classification: 46L55, 46L57, 46L08, 47L30.

1B.S. was supported in part by the US-Israel Binational Foundation and by the Fund for the Promotion of Research at the Technion.

Keywords and Phrases: Product system, subproduct system, semigroups of completely positive maps, dilation, e_{0}-dilation, ∗- automorphic dilation, row contraction, homogeneous polynomial identities, universal operator algebra, q-commuting, subshift C^{∗}- algebra.

Contents

Introduction 802

Part 1. Subproduct systems and cp-semigroups 810 1. Subproduct systems of Hilbert W^{∗}-correspondences 810 2. Subproduct system representations and cp-semigroups 812 3. Subproduct system units and cp-semigroups 819 4. ∗-automorphic dilation of an e_{0}-semigroup 821 5. Dilations and pieces of subproduct system representations 826

Part 2. Subproduct systems over ℕ 834 6. Subproduct systems of Hilbert spaces over ℕ 834 7. Zeros of homogeneous polynomials in noncommutative variables 840 8. Universality of the shift: universal algebras and models 844 9. The operator algebra associated to a subproduct system 848 10. Classification of the universal algebras of q-commuting tuples 853 11. Standard maximal subproduct systems with dim X(1) = 2 and

dim X(2) = 3 857 12. The representation theory of Matsumoto’s subshift C^{∗}-algebras 861 References 866

Introduction

Motivation: dilation theory of CP_{0}-semigroups. We begin by describing the problems that motivated this work. Let H be a separable Hilbert space, and let M ⊆ B(H) be a von Neumann algebra. A CP map on M is a contractive, normal and completely positive map. A CP_{0}-semigroup on M is a family Θ = {Θ_{t}}_{t≥0} of unital CP maps on M satisfying the semigroup property

Θ_{s+t}(a) = Θ_{s}(Θ_{t}(a)) , s, t ≥ 0, a ∈ M,
Θ_{0}(a) = a , a ∈ B(H),

and the continuity condition

lim_{t→t_{0}}⟨Θ_{t}(a)h, g⟩ = ⟨Θ_{t_{0}}(a)h, g⟩ , a ∈ M, h, g ∈ H.

A CP_{0}-semigroup is called an E_{0}-semigroup if each of its elements is a ∗- endomorphism. Let Θ be a CP_{0}-semigroup acting on M, and let α be an E_{0}-semigroup acting on R, where R is a von Neumann subalgebra of B(K) and K ⊇ H. Denote the orthogonal projection of K onto H by p. We say that α is an E_{0}-dilation of Θ if for all t ≥ 0 and b ∈ R

(0.1)  Θ_{t}(pbp) = pα_{t}(b)p.

In the mid 1990’s Bhat proved the following result, known today as “Bhat’s Theorem” (see [9] for the case M = B(H), and also [40, 15, 29, 6] for different proofs and for the general case):

Theorem 0.1. (Bhat). Every CP_{0}-semigroup has a unique minimal E_{0}- dilation.

A natural question is then this: given two commuting CP_{0}-semigroups, can one simultaneously dilate them to a pair of commuting E_{0}-semigroups? In [43] the following partial positive answer was obtained^{2}:

Theorem 0.2. [43, Theorem 6.6] Let {ϕ_{t}}_{t≥0} and {θ_{t}}_{t≥0} be two strongly commuting CP_{0}-semigroups on a von Neumann algebra M ⊆ B(H), where H is a separable Hilbert space. Then there is a separable Hilbert space K containing H and an orthogonal projection p : K → H, a von Neumann algebra R ⊆ B(K) such that M = pRp, and two commuting E_{0}-semigroups α and β on R such that

ϕ_{s} ∘ θ_{t}(pbp) = pα_{s} ∘ β_{t}(b)p

for all s, t ≥ 0 and all b ∈ R.

In other words: every two-parameter CP_{0}-semigroup that satisfies an additional condition of strong commutativity has a two-parameter E_{0}-dilation. The condition of strong commutativity was introduced in [48]. A precise definition will not be given here. The main tools in the proof of Theorem 0.2, and also in some of the proofs of Theorem 0.1, were product systems of W^{∗}-correspondences and their representations. In fact, the only place in the proof of Theorem 0.2 where the assumption of strong commutativity is used, is in the construction of a certain product system. More about that later. In [10], Bhat showed that given a pair of commuting CP maps Θ and Φ on B(H), there is a Hilbert space K ⊇ H and a pair of commuting normal ∗- endomorphisms α and β acting on B(K) such that

Θ^{m} ∘ Φ^{n}(pbp) = pα^{m} ∘ β^{n}(b)p , b ∈ B(K)

for all m, n ∈ ℕ (here p denotes the projection of K onto H). Later on Solel, using a different method (using in fact product systems and their representations), proved this result for commuting CP maps on arbitrary von Neumann algebras [48]. Neither one of the above results requires strong commutativity.

2The same result was obtained in [42] for nonunital semigroups acting on M = B(H).

In light of the above discussion, and inspired by classical dilation theory [47, 49], it is natural to conjecture that every two commuting (not necessarily strongly commuting) CP_{0}-semigroups have an E_{0}-dilation, and in fact that the same is true for any k commuting CP_{0}-semigroups, for any positive integer k. However, the framework given by product systems seems to be too weak to prove this. Trying to bypass this stoppage, we arrived at the notion of a subproduct system.

Background: from product systems to subproduct systems. Product systems of Hilbert spaces over ℝ_{+} were introduced by Arveson some 20 years ago in his study of E_{0}-semigroups [3]. In a few imprecise words, a product system of Hilbert spaces over ℝ_{+} is a bundle {X(t)}_{t∈ℝ_{+}} of Hilbert spaces such that

X(s + t) = X(s) ⊗ X(t) , s, t ∈ ℝ_{+}.

We emphasize immediately that Arveson’s definition of product systems required also that the bundle carry a certain Borel measurable structure, but we do not deal with these matters here. To every E_{0}-semigroup Arveson associated a product system, and it turns out that the product system associated to an E_{0}-semigroup is a complete cocycle conjugacy invariant of the E_{0}-semigroup. Later, product systems of Hilbert C^{∗}-correspondences over ℝ_{+} appeared (see the survey [46] by Skeide). In [15], Bhat and Skeide associate with every semigroup of completely positive maps on a C^{∗}-algebra A a product system of Hilbert A-correspondences. This product system was then used in showing that every semigroup of completely positive maps can be “dilated” to a semigroup of ∗-endomorphisms. Muhly and Solel introduced a different construction [29]: to every CP_{0}-semigroup on a von Neumann algebra M they associated a product system of Hilbert W^{∗}-correspondences over M′, the commutant of M. Again, this product system is then used in constructing an E_{0}-dilation for the original CP_{0}-semigroup. Product systems of C^{∗}-correspondences over semigroups other than ℝ_{+} were first studied by Fowler [21], and they have been studied since then by many authors. In [48], product systems over ℕ^{2} (and their representations) were studied, and the results were used to prove that every pair of commuting CP maps has a ∗-endomorphic dilation. Product systems over ℝ_{+}^{2} were also central to the proof of Theorem 0.2, where every pair of strongly commuting CP_{0}-semigroups is associated with a product system over ℝ_{+}^{2}. However, the construction of the product system is one of the hardest parts in that proof. Furthermore, that construction fails when one drops the assumption of strong commutativity, and it also fails when one tries to repeat it for k strongly commuting semigroups. On the other hand there is another object that may be naturally associated with a semigroup of CP maps over any semigroup: this object is the subproduct system, which, when the CP maps act on B(H), is the bundle of Arveson’s “metric operator spaces” (introduced in [4]). Roughly, a subproduct system of correspondences over a semigroup S is a bundle {X(s)}_{s∈S} of correspondences such that

X(s + t) ⊆ X(s) ⊗ X(t) , s, t ∈ S.

See Definition 1.1 below. Of course, a difficult problem cannot be made easy just by introducing a new notion, and the problem of dilating k-parameter CP_{0}-semigroups remains unsolved. However, subproduct systems did already provide us with an efficient general framework for tackling various problems in operator algebras, and in particular it has led us to a progress toward the solution of the discrete analogue of the above unsolved problem. While preparing this paper we learned that Bhat and Mukherjee have also considered subproduct systems over the semigroup ℝ_{+} ([14]). They called it inclusion systems and used it to compute the index of certain product systems. This paper consists of two parts. In the first part we introduce subproduct systems over general semigroups, show the connection between subproduct systems and cp-semigroups, and use this connection to obtain three main results in dilation theory of cp-semigroups. The first result is that every e_{0}-semigroup over a (certain kind of) semigroup S can be dilated to a semigroup of ∗- automorphisms on some type I factor. The second is some necessary conditions and sufficient conditions for a cp-semigroup to have a (minimal) ∗-endomorphic dilation. The third is an analogue of Parrot’s example of three contractions with no isometric dilation, that is, an example of three commuting, contractive normal CP maps on B(H) that admit no ∗-endomorphic dilation. The CP maps in the stated example can be taken to have arbitrarily small norm, providing the first example of a theorem in the classical theory of isometric dilations that cannot be generalized to the theory of e-dilations of cp-semigroups. Having convinced the reader that subproduct systems are an interesting and important object, we turn in the second part of the paper to take a closer look at the simplest examples of subproduct systems, that is, subproduct systems of Hilbert spaces over ℕ. We study certain tuples of operators and operator algebras that can be naturally associated with every subproduct system, and explore the relationship between these objects and the subproduct systems that give rise to them.

Some preliminaries. M and N will denote von Neumann subalgebras of B(H), where H is some Hilbert space. In Sections 1 through 5, S will denote a sub-semigroup of ℝ_{+}^{k}. In fact, in large parts of the paper S can be taken to be any semigroup with unit, or at least any Ore semigroup (see [25] for a definition), but we prefer to avoid this distraction.

Definition 0.3. A cp-semigroup is a semigroup of CP maps, that is, a family Θ = {Θ_{s}}_{s∈S} of completely positive, contractive and normal maps on M such that

Θ_{s+t}(a) = Θ_{s}(Θ_{t}(a)) , s, t ∈ S, a ∈ M

and

Θ_{0}(a) = a , a ∈ M.

A cp_{0}-semigroup is a semigroup of unital CP maps. An e-semigroup is a semigroup of ∗-endomorphisms. An e_{0}-semigroup is a semigroup of unital ∗- endomorphisms.

For concreteness, one should think of the case S = ℕ^{k}, where a cp-semigroup is a k-tuple of commuting CP maps, or the case S = ℝ_{+}^{k}, where a cp-semigroup is a k-parameter semigroup of CP maps, or k mutually commuting one-parameter cp-semigroups.

Definition 0.4. Let Θ = {Θ_{s}}_{s∈S} be a cp-semigroup acting on a von Neumann algebra M ⊆ B(H). An e-dilation of Θ is a triple (α, K, R) consisting of a Hilbert space K ⊇ H (with orthogonal projection P_{H} : K → H), a von Neumann algebra R ⊆ B(K) that contains M as a corner M = P_{H}RP_{H}, and an e-semigroup α = {α_{s}}_{s∈S} on R such that for all T ∈ R, s ∈ S,

Θ_{s}(P_{H}T P_{H}) = P_{H}α_{s}(T)P_{H}.

Definition 0.5. Let A be a C^{∗}-algebra. A Hilbert C^{∗}-correspondences over A is a (right) Hilbert A-module E which carries a non-degenerate, adjointable, left action of A.

Definition 0.6. Let M be a W^{∗}-algebra. A Hilbert W^{∗}-correspondence over M is a self-adjoint Hilbert C^{∗}-correspondence E over M, such that the map M → L(E) which gives rise to the left action is normal.

Definition 0.7. Let E be a C^{∗}-correspondence over A, and let H be a Hilbert space. A pair (σ, T) is called a completely contractive covariant representation of E on H (or, for brevity, a c.c. representation) if

(1) T : E → B(H) is a completely contractive linear map;

(2) σ : A → B(H) is a nondegenerate ∗-homomorphism; and

(3) T (xa) = T (x)σ(a) and T (a·x) = σ(a)T (x) for all x ∈ E and all a ∈ A. If A is a W^{∗}-algebra and E is W^{∗}-correspondence then we also require that σ be normal.

Given a C^{∗}-correspondence E and a c.c. representation (σ, T) of E on H, one can form the Hilbert space E ⊗_{σ} H, which is defined as the Hausdorff completion of the algebraic tensor product with respect to the inner product

⟨x ⊗ h, y ⊗ g⟩ = ⟨h, σ(⟨x, y⟩)g⟩.

One then defines T̃ : E ⊗_{σ} H → H by

T̃(x ⊗ h) = T (x)h.

Definition 0.8. A c.c. representation (σ, T) is called isometric if for all x, y ∈ E,

T (x)^{∗}T (y) = σ(⟨x, y⟩).

(This is the case if and only if T̃ is an isometry). It is called fully coisometric if T̃ is a coisometry.

Given two Hilbert C^{∗}-correspondences E and F over A, the balanced (or inner) tensor product E ⊗ F is a Hilbert C^{∗}-correspondence over A defined to be the Hausdorff completion of the algebraic tensor product with respect to the inner product

⟨x ⊗ y, w ⊗ z⟩ = ⟨y, ⟨x, w⟩ · z⟩ , x, w ∈ E, y, z ∈ F.

The left and right actions are defined as a · (x ⊗ y) = (a · x) ⊗ y and (x ⊗ y)a = x⊗(ya), respectively, for all a ∈ A, x ∈ E, y ∈ F. When working in the context of W^{∗}-correspondences, that is, if E and F are W *-correspondences and A is a W^{∗}-algebra, then E ⊗ F is understood do be the self-dual extension of the above construction.

Detailed overview of the paper. Subproduct systems, their representations, and their units, are defined in the next section. The following two sections, 2 and 3, can be viewed as a reorganization and sharpening of some known results, including several new observations. Section 2 establishes the correspondence between cp-semigroups and subproduct systems. It is shown that given a subproduct system X of N - correspondences and a subproduct system representation R of X on H, we may construct a cp-semigroup Θ acting on N′. We denote this assignment as Θ = Σ(X, R). Conversely, it is shown that given a cp-semigroup Θ acting on M, there is a subproduct system E (called the Arveson-Stinespring subproduct system of Θ) of M′-correspondences and an injective representation T of E on H such that Θ = Σ(E, T). Denoting this assignment as (E, T) = Ξ(Θ), we have that Σ∘Ξ is the identity. In Theorem 2.6 we show that Ξ∘Σ is also, after restricting to pairs (X, R) with R an injective representation (and up to some “isomorphism”), the identity. This allows us to deduce (Corollary 2.8) that a subproduct system that is not a product system has no isometric representations. We introduce the Fock spaces associated to a subproduct system and the canonical shift representations. These constructs allow us to show that every subproduct system is the Arveson-Stinespring subproduct system of some cp-semigroup. In Section 3 we briefly sketch the picture that is dual to that of Section 2. It is shown that given a subproduct system and a unit of that subproduct system one may construct a cp-semigroup, and that every cp-semigroup arises this way. In Section 4, we construct for every subproduct system X and every fully coisometric subproduct system representation T of X on a Hilbert space, a semigroup T̂ of contractions on a Hilbert space that captures “all the information” about X and T. This construction is a modification of the construction introduced in [41] for the case where X is a product system. It turns out that when X is merely a subproduct system, it is hard to apply T̂ to obtain new results about the representation T. However, when X is a true product system T̂ is very handy, and we use it to prove that every e_{0}-semigroup has a ∗-automorphic dilation (in a certain sense). Section 5 begins with some general remarks regarding dilations and pieces of subproduct system representations, and then the connection between the dilation theories of cp-semigroups and of representations of subproduct systems is made. We define the notion of a subproduct subsystem and then we define dilations and pieces of subproduct system representations. These notions generalize the notions of commuting piece or q-commuting piece of [12] and [19], and also generalizes the definition of dilation of a product system representation of [29]. Proposition 5.8, Theorem 5.12 and Corollary 5.13 show that the 1-1 correspondences Σ and Ξ between cp-semigroups and subproduct systems with representations take isometric dilations of representations to e-dilations and vice-versa. This is used to obtain an example of three commuting, unital and contractive CP maps on B(H) for which there exists no e-dilation acting on a B(K), and no minimal dilation acting on any von Neumann algebra (Theorem 5.14). In Section 5 we also present a reduction of both the problem of constructing an e_{0}-dilation to a cp_{0}-semigroup, and the problem of constructing an e-dilation to a k-tuple of commuting CP maps with small enough norm, to the problem of embedding a subproduct system in a larger product system. We show that not every subproduct system can be embedded in a product system (Proposition 5.15), and we use this to construct an example of three commuting CP maps θ_{1}, θ_{2}, θ_{3} such that for any λ > 0 the three-tuple λθ_{1}, λθ_{2}, λθ_{3} has no e-dilation (Theorem 5.16). This unexpected phenomenon has no counterpart in the classical theory of isometric dilations, and provides the first example of a theorem in classical dilation theory that cannot be generalized to the theory of e-dilations of cp-semigroups. The developments described in the first part of the paper indicate that subproduct systems are worthwhile objects of study, but to make progress we must look at plenty of concrete examples. In the second part of the paper we begin studying subproduct systems of Hilbert spaces over the semigroup ℕ. In Section 6 we show that every subproduct system (of W^{∗}-correspondences) over ℕ is isomorphic to a standard subproduct system, that is, it is a subproduct subsystem of the full product system {E^{⊗n}}_{n∈ℕ} for some W^{∗}-correspondence E. Using the results of the previous section, this gives a new proof to the discrete analogue of Bhat’s Theorem: every cp_{0}-semigroup over ℕ has an e_{0}- dilation. Given a subproduct system we define the standard X-shift, and we show that if X is a subproduct subsystem of Y, then the standard X-shift is the maximal X-piece of the standard Y -shift, generalizing and unifying results from [12, 19, 39]. In Section 7 we explain why subproduct systems are convenient for studying noncommutative projective algebraic geometry. We show that every homogeneous ideal I in the algebra ℂ⟨x_{1}, …, x_{d}⟩ of noncommutative polynomials corresponds to a unique subproduct system X_{I}, and vice-versa. The representations of X_{I} on a Hilbert space H are precisely determined by the d-tuples in the zero set of I,

Z(I) = \{\underline{T} = (T_{1}, …, T_{d}) ∈ B(H)^{d} : ∀p ∈ I.p(\underline{T}) = 0\}.

A noncommutative version of the Nullstellensatz is obtained, stating that

\{p ∈ ℂ⟨x_{1}, …, x_{d}⟩ : ∀\underline{T} ∈ Z(I).p(\underline{T}) = 0\} = I.

Section 8 starts with a review of a powerful tool, Gelu Popescu’s “Poisson Transform” [38]. Using this tool we derive some basic results (obtained previously by Popescu in [39]) which allow us to identify the operator algebra A_{X} generated by the X-shift as the universal unital operator algebra generated by a row contraction subject to homogeneous polynomial identities. We then prove that every completely bounded representation of a subproduct system X is a piece of a scaled inflation of the X-shift, and derive a related “von Neumann inequality”. In Section 9 we discuss the relationship between a subproduct system X and A_{X}, the (non-selfadjoint operator algebra generated by the X-shift). The main result in this section is Theorem 9.7, which says that X ≅ Y if and only if A_{X} is completely isometrically isomorphic to A_{Y} by an isomorphism that preserves the vacuum state. This result is used in Section 10, where we study the universal norm closed unital operator algebra generated by a row contraction (T_{1}, …, T_{d}) satisfying the relations

T_{i}T_{j} = q_{ij}T_{j}T_{i} , 1 ≤ i < j ≤ d,

where q = (q_{i,j})_{i,j=1}^{d} ∈ M_{n}(ℂ) is a matrix such that q_{j,i} = q_{i,j}^{−1}. These nonselfadjoint analogues of the noncommutative tori, are shown to be classified by their subproduct systems when q_{i,j} ≠ 1 for all i, j. In particular, when d = 2, we obtain the universal algebra for the relation

T_{1}T_{2} = qT_{2}T_{1},

which we call A_{q}. It is shown that A_{q} is isomorphically isomorphic to A_{r} if and only if q = r or q = r^{−1}. In Section 11 we describe all standard maximal subproduct systems X with dim X(1) = 2 and dim X(2) = 3, and classify their algebras up to isometric isomorphisms. In the closing section of this paper, Section 12, we find that subproduct systems are also closely related to subshifts and to the subshift C^{∗}-algebras introduced by K. Matsumoto [28]. We show how every subshift gives rise to a subproduct system, and characterize the subproduct systems that come from subshifts. We use this connection together with the results of Section 8 to describe all representations of subshift C^{∗}-algebras that come from a subshift of finite type (Theorem 12.7).

Acknowledgments. The authors owe their thanks to Eliahu Levy for pointing out a mistake in a previous version of the paper, and to Michael Skeide, for several helpful remarks.

Part 1. Subproduct systems and cp-semigroups

1. Subproduct systems of Hilbert W^{∗}-correspondences

Definition 1.1. Let N be a von Neumann algebra. A subproduct system of Hilbert W^{∗}-correspondences over N is a family X = {X(s)}_{s∈S} of Hilbert W^{∗}-correspondences over N such that

(1) X(0) = N,

(2) For every s, t ∈ S there is a coisometric mapping of N -correspondences

U_{s,t} : X(s) ⊗ X(t) → X(s + t),

(3) The maps U_{s,0} and U_{0,s} are given by the left and right actions of N on X(s),

(4) The maps U_{s,t} satisfy the following associativity condition:

(1.1)  U_{s+t,r} (U_{s,t} ⊗ I_{X(r)}) = U_{s,t+r} (I_{X(s)} ⊗ U_{t,r}) .

The difference between a subproduct system and a product system is that in a subproduct system the maps U_{s,t} are only required to be coisometric, while in a product system these maps are required to be unitaries. Thus, given the image U_{s,t}(x ⊗ y) of x ⊗ y in X(s + t), one cannot recover x and y. Thus, subproduct systems may be thought of as irreversible product systems. The terminology is, admittedly, a bit awkward. It may be more sensible – however, impossible at present – to use the term product system for the objects described above and to use the term full product system for product system.

Example 1.2. The simplest example of a subproduct system F = F_{E} = {F (n)}_{n∈ℕ} is given by

F (n) = E^{⊗n},

where E is some W^{∗}-correspondence. F is actually a product system. We shall call this subproduct system the full product system (over E).

Example 1.3. Let E be a fixed Hilbert space. We define a subproduct system (of Hilbert spaces) SSP = SSP_{E} over ℕ using the familiar symmetric tensor products (one can obtain a subproduct system from the anti-symmetric tensor products as well). Define

E^{⊗n} = E ⊗ ⋯ ⊗ E,

(n times). Let p_{n} be the projection of E^{⊗n} onto the symmetric subspace of E^{⊗n}, given by

p_{n}k_{1} ⊗ ⋯ ⊗ k_{n} = \frac{1}{n!} ∑_{σ∈S_{n}} k_{σ^{−1}(1)} ⊗ ⋯ ⊗ k_{σ^{−1}(n)}.

We define

SSP (n) = E^{Ⓢn} := p_{n}E^{⊗n},

the symmetric tensor product of E with itself n times (SSP (0) = ℂ). We define a map U_{m,n} : SSP (m) ⊗ SSP (n) → SSP (m + n) by

U_{m,n}(x ⊗ y) = p_{m+n}(x ⊗ y).

The U’s are coisometric maps because every projection, when considered as a map from its domain onto its range, is coisometric. A straightforward calculation shows that (1.1) holds (see [35, Corollary 17.2]). In these notes we shall refer to SSP (or SSP_{E} to be precise) as the symmetric subproduct system (over E).

Definition 1.4. Let X and Y be two subproduct systems over the same semigroup S (with families of coisometries {U_{s,t}^{X}}_{s,t∈S} and {U_{s,t}^{Y}}_{s,t∈S}). A family V = {V_{s}}_{s∈S} of maps V_{s} : X(s) → Y (s) is called a morphism of subproduct systems if V_{0} : X(0) → Y (0) is the identity, if for all s ∈ S \ {0} the map V_{s} is a coisometric correspondence map, and if for all s, t ∈ S the following identity holds:

(1.2)  V_{s+t} ∘ U_{s,t}^{X} = U_{s,t}^{Y} ∘ (V_{s} ⊗ V_{t}).

V is said to be an isomorphism if V_{s} is a unitary for all s ∈ S \ {0}. X is said to be isomorphic to Y if there exists an isomorphism V : X → Y.

There is an obvious extension of the above definition to the case where X(0) and Y (0) are ∗-isomorphic. The above notion of morphism is not optimized in any way. It is simply precisely what we need in order to develop dilation theory for cp-semigroups.

Definition 1.5. Let N be a von Neumann algebra, let H be a Hilbert space, and let X be a subproduct system of Hilbert N -correspondences over the semigroup S. Assume that T : X → B(H), and write T_{s} for the restriction of T to X(s), s ∈ S, and σ for T_{0}. T (or (σ, T)) is said to be a completely contractive covariant representation of X if

(1) For each s ∈ S, (σ, T_{s}) is a c.c. representation of X(s); and

(2) T_{s+t}(U_{s,t}(x ⊗ y)) = T_{s}(x)T_{t}(y) for all s, t ∈ S and all x ∈ X(s), y ∈

X(t). T is said to be an isometric (fully coisometric) representation if it is an isometric (fully coisometric) representation on every fiber X(s).

Since we shall not be concerned with any other kind of representation, we shall call a completely contractive covariant representation of a subproduct system simply a representation.

Remark 1.6. Item 2 in the above definition of product system can be rewritten as follows:

T̃_{s+t}(U_{s,t} ⊗ I_{H}) = T̃_{s}(I_{X(s)} ⊗ T̃_{t}).

Here T̃_{s} : X(s) ⊗_{σ} H → H is the map given by

T̃_{s}(x ⊗ h) = T_{s}(x)h.

Example 1.7. We now define a representation T of the symmetric subproduct system SSP from Example 1.3 on the symmetric Fock space. Denote by 𝔉_{+} the symmetric Fock space

𝔉_{+} = ⨁_{n∈ℕ} E^{Ⓢn}.

For every n ∈ ℕ, the map T_{n} : SSP (n) = E^{Ⓢn} → B(𝔉_{+}) is defined on the m-particle space E^{Ⓢm} by putting

T_{n}(x)y = p_{n+m}(x ⊗ y)

for all x ∈ X(n), y ∈ E^{Ⓢm}. Then T extends to a representation of the subproduct system SSP on 𝔉_{+} (to see that item 2 of Definition 1.5 is satisfied one may use again [35, Corollary 17.2]).

Definition 1.8. Let X = {X(s)}_{s∈S} be a subproduct system of N - correspondences over S. A family ξ = {ξ_{s}}_{s∈S} is called a unit for X if

(1.3)  ξ_{s} ⊗ ξ_{t} = U_{s,t}^{∗}ξ_{s+t}.

A unit ξ = {ξ_{s}}_{s∈S} is called unital if ⟨ξ_{s}, ξ_{s}⟩ = 1_{N} for all s ∈ S, it is called contractive if ⟨ξ_{s}, ξ_{s}⟩ ≤ 1_{N} for all s ∈ S, and it is called generating if X(s) is spanned by elements of the form (1.4) U_{s_{1}+···+s_{n−1},s_{n}}(⋯ U_{s_{1}+s_{2},s_{3}}(U_{s_{1},s_{2}}(a_{1}ξ_{s_{1}} ⊗ a_{2}ξ_{s_{2}}) ⊗ a_{3}ξ_{s_{3}}) ⊗ ⋯ ⊗ a_{n}ξ_{s_{n}}a_{n+1}),

where s = s_{1} + s_{2} + ⋯ + s_{n}.

From (1.3) follows the perhaps more natural looking

U_{s,t}(ξ_{s} ⊗ ξ_{t}) = ξ_{s+t}.

Example 1.9. A unital unit for the symmetric subproduct system SSP from Example 1.3 is given by defining ξ_{0} = 1 and

ξ_{n} = \underbrace{v ⊗ v ⊗ ⋯ ⊗ v}_{n times}

for n ≥ 1. This unit is generating only if E is one dimensional.

2. Subproduct system representations and cp-semigroups

In this section, following Muhly and Solel’s constructions from [29], we show that subproduct systems and their representations provide a framework for dealing with cp-semigroups, and allow us to obtain a generalization of the classical result of Wigner that any strongly continuous one-parameter group of automorphisms of B(H) is given by X ↦ U_{t}XU_{t}^{∗} for some one-parameter unitary group {U_{t}}_{t∈ℝ}.

2.1. All cp-semigroups come from subproduct system representations.

Proposition 2.1. Let N be a von Neumann algebra and let X be a subproduct system of N -correspondences over S, and let R be completely contractive covariant representation of X on a Hilbert space H, such that R_{0} is unital. Then the family of maps

(2.1)  Θ_{s} : a ↦ R̃_{s}(I_{X(s)} ⊗ a)R̃_{s}^{∗} , a ∈ R_{0}(N)′,

is a semigroup of CP maps on R_{0}(N)′. Moreover, if R is an isometric (a fully coisometric) representation, then Θ_{s} is a ∗-endomorphism (a unital map) for all s ∈ S.

Proof. By Proposition 2.21 in [29], {Θ_{s}}_{s∈S} is a family of contractive, normal, completely positive maps on R_{0}(N)′. Moreover, these maps are unital if R is a fully coisometric representation, and they are ∗-endomorphisms if R is an isometric representation. It remains is to check that Θ = {Θ_{s}}_{s∈S} satisfies the semigroup condition Θ_{s} ∘ Θ_{t} = Θ_{s+t}. Fix a ∈ R_{0}(N)′. For all s, t ∈ S,

Θ_{s}(Θ_{t}(a)) = R̃_{s} (I_{X(s)} ⊗ (R̃_{t}(I_{X(t)} ⊗ a)R̃_{t}^{∗})) R̃_{s}^{∗}
= R̃_{s}(I_{X(s)} ⊗ R̃_{t})(I_{X(s)} ⊗ I_{X(t)} ⊗ a)(I_{X(s)} ⊗ R̃_{t}^{∗})R̃_{s}^{∗}
= R̃_{s+t}(U_{s,t} ⊗ I_{G})(I_{X(s)} ⊗ I_{X(t)} ⊗ a)(U_{s,t}^{∗} ⊗ I_{G})R̃_{s+t}^{∗}
= R̃_{s+t}(I_{X(s·t)} ⊗ a)R̃_{s+t}^{∗}
= Θ_{s+t}(a).

Using the fact that R_{0} is unital, we have

Θ_{0}(a)h = R̃_{0}(I_{N} ⊗ a)R̃_{0}^{∗}h
= R̃_{0}(I_{N} ⊗ a)(1_{N} ⊗ h)
= R_{0}(1_{N})ah
= ah,

thus Θ_{0}(a) = a for all a ∈ N. □

We will now show that every cp-semigroup is given by a subproduct representation as in (2.1) above. We recall some constructions from [29] (building on the foundations set in [4]). Fix a CP map Θ on von Neumann algebra M ⊆ B(H). We define M ⊗_{Θ} H to be the Hausdorff completion of the algebraic tensor product M ⊗ H with respect to the sesquilinear positive semidefinite form

⟨T_{1} ⊗ h_{1}, T_{2} ⊗ h_{2}⟩ = ⟨h_{1}, Θ(T_{1}^{∗}T_{2})h_{2}⟩ .

We define a representation π_{Θ} of M on M ⊗_{Θ} H by

π_{Θ}(S)(T ⊗ h) = ST ⊗ h,

and we define a (contractive) linear map W_{Θ} : H → M ⊗ H by

W_{Θ}(h) = I ⊗ h.

If Θ is unital then W_{Θ} is an isometry, and if Θ is an endomorphism then W_{Θ} is a coisometry. The adjoint of W_{Θ} is given by

W_{Θ}^{∗}(T ⊗ h) = Θ(T)h.

For a given CP semigroup Θ on M, Muhly and Solel defined in [29] a W^{∗}- correspondence E_{Θ} over M′ and a c.c. representation (σ, T_{Θ}) of E_{Θ} on H such that for all a ∈ M

(2.2)  Θ(a) = T̃_{Θ} (I_{E_{Θ}} ⊗ a) T̃_{Θ}^{∗}.

The W^{∗}-correspondence E_{Θ} is defined as the intertwining space

E_{Θ} = L_{M}(H, M ⊗_{Θ} H),

where

L_{M}(H, M ⊗_{Θ} H) := {X ∈ B(H, M ⊗_{Θ} H)| XT = π_{Θ}(T)X, T ∈ M}.

The left and right actions of M′ are given by

S · X = (I ⊗ S)X , X · S = XS

for all X ∈ E_{Θ} and S ∈ M′. The M′-valued inner product on E_{Θ} is defined by ⟨X, Y ⟩ = X^{∗}Y. E_{Θ} is called the Arveson-Stinespring correspondence (associated with Θ). The representation (σ, T_{Θ}) is defined by letting σ = id_{M′}, the identity representation of M′ on H, and by defining

T_{Θ}(X) = W_{Θ}^{∗}X.

(id_{M′}, T_{Θ}) is called the identity representation (associated with Θ). We remark that the paper [29] focused on unital CP maps, but the results we cite are true for nonunital CP maps, with the proofs unchanged. The case where M = B(H) in the following theorem appears, in essence at least, in [4].

Theorem 2.2. Let Θ = {Θ_{s}}_{s∈S} be a cp-semigroup on a von Neumann algebra M ⊆ B(H), and for all s ∈ S let E(s) := E_{Θ_{s}} be the Arveson-Stinespring correspondence associated with Θ_{s}, and let T_{s} := T_{Θ_{s}} denote the identity representation for Θ_{s}. Then E = {E(s)}_{s∈S} is a subproduct system of M′- correspondences, and (id_{M′}, T) is a representation of E on H that satisfies

(2.3)  Θ_{s}(a) = T̃_{s} (I_{E(s)} ⊗ a) T̃_{s}^{∗}

for all a ∈ M and all s ∈ S. T_{s} is injective for all s ∈ S. If Θ is an e-semigroup (cp_{0}-semigroup), then (id_{M′}, T) is isometric (fully coisometric).

Proof. We begin by defining the subproduct system maps U_{s,t} : E(s) ⊗ E(t) → E(s + t). We use the constructions made in [29, Proposition 2.12] and the surrounding discussion. We define

U_{s,t} = V_{s,t}^{∗}Ψ_{s,t} ,

where the map

Ψ_{s,t} : L_{M}(H, M ⊗_{Θ_{s}} H) ⊗ L_{M}(H, M ⊗_{Θ_{t}} H) → L_{M}(H, M ⊗_{Θ_{t}} M ⊗_{Θ_{s}} H)

is given by Ψ_{s,t}(X ⊗ Y) = (I ⊗ X)Y, and

V_{s,t} : L_{M}(H, M ⊗_{Θ_{s+t}} H) → L_{M}(H, M ⊗_{Θ_{t}} M ⊗_{Θ_{s}} H)

is given by V_{s,t}(X) = Γ_{s,t} ∘ X, where Γ_{s,t} : M ⊗_{Θ_{s+t}} H → M ⊗_{Θ_{t}} M ⊗_{Θ_{s}} H is the isometry

Γ_{s,t} : S ⊗_{Θ_{s+t}} h ↦ S ⊗_{Θ_{t}} I ⊗_{Θ_{s}} h.

By [29, Proposition 2.12], U_{s,t} is a coisometric bimodule map for all s, t ∈ S. To see that the U’s compose associatively as in (1.1), take s, t, u ∈ S, X ∈ E(s), Y ∈ E(t), Z ∈ E(u), and compute:

U_{s,t+u}(I_{E(s)} ⊗ U_{t,u})(X ⊗ Y ⊗ Z) = U_{s,t+u}(X ⊗ V_{t,u}^{∗}(I ⊗ Y)Z)
= V_{s,t+u}^{∗} ((I ⊗ X)V_{t,u}^{∗}(I ⊗ Y)Z)
= Γ_{s,t+u}^{∗}(I ⊗ X)Γ_{t,u}^{∗}(I ⊗ Y)Z

and

U_{s+t,u}(U_{s,t} ⊗ I_{E(u)})(X ⊗ Y ⊗ Z) = U_{s+t,u}(V_{s,t}^{∗}(I ⊗ X)Y ⊗ Z)
= V_{s+t,u}^{∗} ((I ⊗ V_{s,t}^{∗}(I ⊗ X)Y)Z)
= Γ_{s+t,u}^{∗}(I ⊗ Γ_{s,t}^{∗})(I ⊗ I ⊗ X)(I ⊗ Y)Z .

So it suffices to show that

Γ_{s,t+u}^{∗}(I ⊗ X)Γ_{t,u}^{∗} = Γ_{s+t,u}^{∗}(I ⊗ Γ_{s,t}^{∗})(I ⊗ I ⊗ X)

It is easier to show that their adjoints are equal. Let a ⊗ h be a typical element of M ⊗_{Θ_{s+t+u}} h.

Γ_{t,u}(I ⊗ X^{∗})Γ_{s,t+u}(a ⊗_{Θ_{s+t+u}} h) = Γ_{t,u}(I ⊗ X^{∗})(a ⊗_{Θ_{t+u}} I ⊗_{Θ_{s}} h)
= Γ_{t,u}(a ⊗_{Θ_{t+u}} X^{∗}(I ⊗_{Θ_{s}} h))
= a ⊗_{Θ_{u}} I ⊗_{Θ_{t}} X^{∗}(I ⊗_{Θ_{s}} h).

On the other hand

(I ⊗ I ⊗ X^{∗})(I ⊗ Γ_{s,t})Γ_{s+t,u}(a ⊗_{Θ_{s+t+u}} h) =
= (I ⊗ I ⊗ X^{∗})(I ⊗ Γ_{s,t})(a ⊗_{Θ_{u}} I ⊗_{Θ_{s+t}} h)
= (I ⊗ I ⊗ X^{∗})(a ⊗_{Θ_{u}} I ⊗_{Θ_{t}} I ⊗_{Θ_{s}} h)
= a ⊗_{Θ_{u}} I ⊗_{Θ_{t}} X^{∗}(I ⊗_{Θ_{s}} h).

This shows that the maps {U_{s,t}} make E into a subproduct system. Let us now verify that T is a representation of subproduct systems. That (id_{M′}, T_{s}) is a c.c. representation of E(s) is explained in [29, page 878]. Let X ∈ E(s), Y ∈ E(t).

T_{s+t}(U_{s,t}(X ⊗ Y)) = W_{Θ_{s+t}}^{∗}Γ_{s,t}^{∗}(I ⊗ X)Y,

while

T_{s}(X)T_{t}(Y) = W_{Θ_{s}}^{∗}XW_{Θ_{t}}^{∗}Y.

But for all h ∈ H,

W_{Θ_{t}}X^{∗}W_{Θ_{s}}h = W_{Θ_{t}}X^{∗}(I ⊗_{Θ_{s}} h)
= I ⊗_{Θ_{t}} X^{∗}(I ⊗_{Θ_{s}} h)
= (I ⊗ X^{∗})(I ⊗_{Θ_{t}} I ⊗_{Θ_{s}} h)
= (I ⊗ X^{∗})Γ_{s,t}(I ⊗_{Θ_{s+t}} h)
= (I ⊗ X^{∗})Γ_{s,t}W_{Θ_{s+t}}h,

which implies W_{Θ_{s}}^{∗}XW_{Θ_{t}}^{∗}Y = W_{Θ_{s+t}}^{∗}Γ_{s,t}^{∗}(I ⊗ X)Y, so we have the desired equality

T_{s+t}(U_{s,t}(X ⊗ Y)) = T_{s}(X)T_{t}(Y).

Equation (2.3) is a consequence of (2.2). The injectivity of the identity representation has already been noted by Solel in [48] (for all h, g ∈ H and a ∈ M, ⟨W_{Θ}^{∗}Xa^{∗}h, g⟩ = ⟨Xa^{∗}h, I ⊗ g⟩ = ⟨(I ⊗ a^{∗})Xh, I ⊗ g⟩ = ⟨Xh, a ⊗ g⟩ ). The final assertion of the theorem is trivial (if Θ_{s} is a ∗-endomorphism, then W_{Θ_{s}} is a coisometry). □

Definition 2.3. Given a cp-semigroup Θ on a W^{∗} algebra M, the pair (E, T) constructed in Theorem 2.2 is called the identity representation of Θ, and E is called the Arveson-Stinespring subproduct system associated with Θ.

Remark 2.4. If follows from [29, Proposition 2.14], if Θ is an e-semigroup, then the identity representation subproduct system is, in fact, a (full) product system.

Remark 2.5. In [27], Daniel Markiewicz has studied the Arveson-Stinespring subproduct system of a CP_{0}-semigroup over ℝ_{+} acting on B(H), and has also shown that it carries a structure of a measurable Hilbert bundle.

2.2. Essentially all injective subproduct system representations come from cp-semigroups. The following generalizes and is motivated by [48, Proposition 5.7]. We shall also repeat some arguments from [32, Theorem 2.1]. By Theorem 2.2, with every cp-semigroup Θ = {Θ_{s}}_{s∈S} on M ⊆ B(H) we can associate a pair (E, T) - the identity representation of Θ - consisting of a subproduct system E (of correspondences over M′) and an injective subproduct system c.c. representation T. Let us write (E, T) = Ξ(Θ). Conversely, given a pair (X, R) consisting of a subproduct system X of correspondences over M′ and a c.c. representation R of X such that R_{0} = id, one may define by equation (2.1) a cp-semigroup Θ acting on M, which we denote as Θ = Σ(X, R). The meaning of equation (2.3) is that Σ ∘ Ξ is the identity map on the set of cpsemigroups of M. We will show below that Ξ ∘ Σ, when restricted to pairs (X, R) such that R is injective, is also, essentially, the identity. When (X, R) is not injective, we will show that Ξ ∘ Σ(X, R) “sits inside” (X, R).

Theorem 2.6. Let N be a W^{∗}-algebra, let X = {X(s)}_{s∈S} be a subproduct system of N -correspondences, and let R be a c.c. representation of X on H, such that σ := R_{0} is faithful and nondegenerate. Let M ⊆ B(H) be the commutant σ(N)′ of σ(N). Let Θ = Σ(X, R), and let (E, T) = Ξ(Θ). Then there is a morphism of subproduct systems W : X → E such that

(2.4)  R_{s} = T_{s} ∘ W_{s} , s ∈ S.

W_{s}^{∗}W_{s} = I_{X(s)} − q_{s}, where q_{s} is the orthogonal projection of X(s) onto KerR_{s}. In particular, W is an isomorphism if and only if R_{s} is injective for all s ∈ S.

Remark 2.7. The construction of the morphism W below basically comes from [48, 32], and it remains only to show that it respects the subproduct system structure. Some of the details in the proof will be left out.

Proof. We may construct a subproduct system X′ of M′-correspondences (recall that M′ = σ(N)), and a representation R′ of X′ on H such that R′_{0} is the identity, in such a way that (X, R) may be naturally identified with (X′, R′). Indeed, put

X′(0) = M′ , X′(s) = X(s) for s ≠ 0,

where the inner product is defined by

⟨x, y⟩_{X′} = σ(⟨x, y⟩_{X}),

and the left and right actions are defined by

a · x · b := σ^{−1}(a)xσ^{−1}(b),

for a, b ∈ M′ and x, y ∈ X′(s), s ∈ S \ {0}. Defining R′_{0} = id and W_{0} = σ; and R′_{s} = R_{s} for and W_{s} = id for s ∈ S \ {0}, we have that W is a morphism X → X′ that sends R to R′. Assume, therefore, that N = M′ and that σ is the identity representation. We begin by defining for every s ≠ 0

v_{s} : M ⊗_{Θ_{s}} H → X(s) ⊗ H

by

v_{s}(a ⊗ h) = (I_{X(s)} ⊗ a)R̃_{s}^{∗}h.

It is straightforward to show that for all s ∈ S the map v_{s} is a well-defined isometry. [(I_{X(s)} ⊗ M)R̃_{s}^{∗}H] is invariant under I_{X(s)} ⊗ M, thus the projection onto the orthocomplement of this subspace is in (I_{X(s)} ⊗ M)′ = L(X(s)) ⊗ I_{H}, so it has the form q_{s} ⊗ I_{H} for some projection q_{s} ∈ L(X(s)). In fact, it is easy to check that q_{s} is the orthogonal projection of X(s) onto KerR_{s}. By the definition of v_{s} and by the covariance properties of T, we have for all a ∈ M and b ∈ M′,

v_{s}(a ⊗ I) = (I ⊗ a)v_{s} , v_{s}(I ⊗ b) = (b ⊗ I)v_{s}.

Fix s ∈ S and x ∈ E(s). For all ξ ∈ X(s), h ∈ H, write

ψ(ξ)h = x^{∗}v_{s}^{∗}(ξ ⊗ h).

It is easy to verify that the linear map ξ ↦ ψ(ξ) maps X(s) into M′ and is a bounded right module map. From the self duality of X(s) it follows that there is a unique element in X(s), which we denote by V_{s}(x), such that for all ξ ∈ X(s), h ∈ H,

(2.5)  ⟨V_{s}(x), ξ⟩h = x^{∗}v_{s}^{∗}(ξ ⊗ h).

The map V_{s} is then a linear bimodule map. To show that V_{s} preserves inner products, write L_{ξ}, ξ ∈ X(s), for the operator L_{ξ} : H → X(s) ⊗ H that maps h to ξ ⊗ h and note that equation (2.5) becomes

L_{V_{s}(x)}^{∗}L_{ξ} = x^{∗}v_{s}^{∗}L_{ξ} , ξ ∈ X(s),

or L_{V_{s}(x)} = v_{s}x, for all x ∈ E(s). But since v_{s} preserves inner products, we have for all x, y ∈ E(s):

⟨x, y⟩ = x^{∗}y = x^{∗}v_{s}^{∗}v_{s}y = L_{V_{s}(x)}^{∗}L_{V_{s}(y)} = ⟨V_{s}(x), V_{s}(y)⟩.

We now prove that V_{s}V_{s}^{∗} = I_{X(s)}−q_{s}. ξ ∈ ImV_{s}^{⊥} if and only if L_{ξ}^{∗}v_{s}E(s)H = 0. But by [29, Lemma 2.10], E(s)H = M⊗_{Θ_{s}} H, thus L_{ξ}^{∗}v_{s}E(s)H = 0 if and only if ⟨ξ, η⟩ = 0 for all η ∈ (I_{X(s)} − q_{s})X(s), which is the same as ξ ∈ q_{s}X(s). Fix h, k ∈ H. For x ∈ E(s), we compute:

⟨T_{s}(x)h, k⟩ = ⟨W_{Θ_{s}}^{∗}xh, k⟩
= ⟨xh, I ⊗_{Θ_{s}} k⟩
= ⟨v_{s}xh, v_{s}(I ⊗_{Θ_{s}} k)⟩
= ⟨V_{s}(x) ⊗ h, R̃_{s}^{∗}k⟩
= ⟨R_{s}(V_{s}(x))h, k⟩,

thus T_{s} = R_{s} ∘ V_{s} for all s ∈ S. Define W_{s} = V_{s}^{∗}. Then T_{s} = R_{s} ∘ W_{s}^{∗}. Multiplying both sides by W_{s} we obtain T_{s} ∘ W_{s} = R_{s} ∘ W_{s}^{∗}W_{s}. But W_{s}^{∗}W_{s} = I − q_{s} is the orthogonal projection onto (KerR_{s})^{⊥}, thus we obtain (2.4). Finally, we need to show that W = {W_{s}} respects the subproduct system structure: for all s, t ∈ S, x ∈ X(s) and y ∈ X(t), we must show that

W_{s+t}(U_{s,t}^{X}(x ⊗ y)) = U_{s,t}^{E}(W_{s}(x) ⊗ W_{t}(y)).

Since T_{s+t} is injective, it is enough to show that after applying T_{s+t} to both sides of the above equation we get the same thing. But T_{s+t} applied to the left hand side gives

T_{s+t}W_{s+t}(U_{s,t}^{X}(x ⊗ y)) = R_{s+t}(U_{s,t}^{X}(x ⊗ y)) = R_{s}(x)R_{t}(y),

and T_{s+t} applied to the right hand side gives

T_{s+t}(U_{s,t}^{E}(W_{s}(x) ⊗ W_{t}(y))) = T_{s}(W_{s}(x))T_{t}(W_{t}(y)) = R_{s}(x)R_{t}(y).
□

Corollary 2.8. Let X be a subproduct system that has an isometric representation V such that V_{0} is faithful and nondegenerate. Then X is a (full) product system.

Proof. Let Θ = Σ(X, V). Then Θ is an e-semigroup. Thus, if (E, T) = Ξ(Θ) is the identity representation of Θ, then, by Remark 2.4, E is a (full) product system. But if V_{0} is faithful and V is isometric then V is injective. By the above theorem, X is isomorphic to E, so it is a product system. □

2.3. Subproduct systems arise from cp-semigroups. The shift representation. A question arises: does every subproduct system arise as the Arveson-Stinespring subproduct system associated with a cp-semigroup? By Theorem 2.6, this is equivalent to the question does every subproduct system have an injective representation? We shall answer this question in the affirmative by constructing for every such subproduct system a canonical injective representation. The following constructs will be of most interest when S is a countable semigroup, such as ℕ^{k}.

Definition 2.9. Let X = {X(s)}_{s∈S} be a subproduct system. The X-Fock space 𝔉_{X} is defined as

𝔉_{X} = ⨁_{s∈S} X(s).

The vector Ω := 1 ∈ N = X(0) is called the vacuum vector of 𝔉_{X}. The X-shift representation of X on 𝔉_{X} is the representation

S^{X} : X → B(𝔉_{X}),

given by S^{X}(x)y = U_{s,t}^{X}(x ⊗ y), for all x ∈ X(s), y ∈ X(t) and all s, t ∈ S.

Strictly speaking, S^{X} as defined above is not a representation because it represents X on a C^{∗}-correspondence rather than on a Hilbert space. However, since for any C^{∗}-correspondence E, L(E) is a C^{∗}-algebra, one can compose a faithful representation π : L(E) → B(H) with S^{X} to obtain a representation on a Hilbert space. A direct computation shows that S̃_{s}^{X} : X(s) ⊗ 𝔉_{X} → 𝔉_{X} is a contraction, and also that S^{X}(x)S^{X}(y) = S^{X}(U_{s,t}^{X}(x ⊗ y)) so S^{X} is a completely contractive representation of X. S^{X} is also injective because S^{X}(x)Ω = x for all x ∈ X. Thus,

Corollary 2.10. Every subproduct system is the Arveson-Stinespring subproduct system of a cp-semigroup.

3. Subproduct system units and cp-semigroups

In this section, following Bhat and Skeide’s constructions from [15], we show that subproduct systems and their units may also serve as a tool for studying cp-semigroups.

Proposition 3.1. Let N be a von Neumann algebra and let X be a subproduct system of N -correspondences over S, and let ξ = {ξ_{s}}_{s∈S} be a contractive unit of X, such that ξ_{0} = 1_{N}. Then the family of maps

(3.1)  Θ_{s} : a ↦ ⟨ξ_{s}, aξ_{s}⟩ ,

is a semigroup of CP maps on N. Moreover, if ξ is unital, then Θ_{s} is a unital map for all s ∈ S.

Proof. It is standard that Θ_{s} given by (3.1) is a contractive completely positive map on N, which is unital if and only if ξ is unital. The fact that Θ_{s} is normal goes a little bit deeper, but is also known (one may use [29, Remark 2.4(i)]).

We show that {Θ_{s}}_{s∈S} is a semigroup. It is clear that Θ_{0}(a) = a for all a ∈ N. For all s, t ∈ S,

Θ_{s}(Θ_{t}(a)) = ⟨ξ_{s}, ⟨ξ_{t}, aξ_{t}⟩ ξ_{s}⟩
= ⟨ξ_{t} ⊗ ξ_{s}, , aξ_{t} ⊗ ξ_{s}⟩
= ⟨U_{t,s}^{∗}ξ_{s+t}, , aU_{t,s}^{∗}ξ_{s+t}⟩
= ⟨ξ_{s+t}, aξ_{s+t}⟩
= Θ_{s+t}(a).
□

We recall a central construction in Bhat and Skeide’s approach to dilation of cp-semigroup [15], that goes back to Paschke [36]. Let M be a W^{∗}-algebra, and let Θ be a normal completely positive map on M ^{3}. The GNS representation of Θ is a pair (F_{Θ}, ξ_{Θ}) consisting of a Hilbert W^{∗}-correspondence F_{Θ} and a vector ξ_{Θ} ∈ F_{Θ} such that

Θ(a) = ⟨ξ_{Θ}, aξ_{Θ}⟩ for all a ∈ M.

F_{Θ} is defined to be the correspondence M ⊗_{Θ} M - which is the self-dual extension of the Hausdorff completion of the algebraic tensor product M ⊗ M with respect to the inner product

⟨a ⊗ b, c ⊗ d⟩ = b^{∗}Θ(a^{∗}c)d.

ξ_{Θ} is defined to be ξ_{Θ} = 1 ⊗ 1. Note that ξ_{Θ} is a unit vector, that is - ⟨ξ_{Θ}, ξ_{Θ}⟩ = 1, if and only if Θ is unital.

Theorem 3.2. Let Θ = {Θ_{s}}_{s∈S} be a cp-semigroup on a W^{∗}-algebra M. For every s ∈ S let (F (s), ξ_{s}) be the GNS representation of Θ_{s}. Then F = {F (s)}_{s∈S} is a subproduct system of M-correspondences, and ξ = {ξ_{s}}_{s∈S} is a generating contractive unit for F that gives back Θ by the formula

(3.2)  Θ_{s}(a) = ⟨ξ_{s}, aξ_{s}⟩ for all a ∈ M.

Θ is a cp_{0}-semigroup if and only if ξ is a unital unit.

Proof. For all s, t ∈ S define a map V_{s,t} : F (s + t) → F (s) ⊗ F (t) by sending ξ_{s+t} to ξ_{s} ⊗ ξ_{t} and extending to a bimodule map. Because

⟨aξ_{s} ⊗ ξ_{t}b, cξ_{s} ⊗ ξ_{t}d⟩ = ⟨ξ_{t}b, ⟨aξ_{s}, cξ_{s}⟩ ξ_{t}d⟩
= ⟨ξ_{t}b, Θ_{s}(a^{∗}c)ξ_{t}d⟩
= b^{∗} ⟨ξ_{t}, Θ_{s}(a^{∗}c)ξ_{t}⟩ d
= b^{∗}Θ_{t+s}(a^{∗}c)d
= ⟨aξ_{t+s}b, cξ_{t+s}d⟩ ,

V_{s,t} extends to a well defined isometric bimodule map from F (s+t) into F (s)⊗ F (t). We define the map U_{s,t} to be the adjoint of V_{s,t} (here it is important that we are working with W^{∗} algebras - in general, an isometry from one Hilbert C^{∗}-module into another need not be adjointable, but bounded module maps between self-dual Hilbert modules are always adjointable, [36, Proposition 3.4]). The collection {U_{s,t}}_{s,t∈S} makes F into a subproduct system. Indeed, these maps are coisometric by definition, and they compose in an associative manner. To see the latter, we check that (I_{F (r)} ⊗ V_{s,t})V_{r,s+t} = (V_{r,s} ⊗ I_{F (t)})V_{r+s,t} and take adjoints.

3The construction works also for completely positive maps on unital C^{∗}-algebras, but in Theorem 3.2 below we will need to work with normal maps on W^{∗}-algebras.

(I_{F (r)} ⊗ V_{s,t})V_{r,s+t}(aξ_{r+s+t}b) = (I_{F (r)} ⊗ V_{s,t})(aξ_{r} ⊗ ξ_{s+t}b)
= aξ_{r} ⊗ ξ_{s} ⊗ ξ_{t}b.

Similarly, (V_{r,s} ⊗ I_{F (t)})V_{r+s,t}(aξ_{r+s+t}b) = aξ_{r} ⊗ ξ_{s} ⊗ ξ_{t}b. Since F (r + s + t) is spanned by linear combinations of elements of the form aξ_{r+s+t}b, the U’s make F into a subproduct system, and ξ is certainly a unit for F. Equation (3.2) follows by definition of the GNS representation. Now,

⟨ξ_{s}, ξ_{s}⟩ = Θ_{s}(1) , s ∈ S,

so ξ is a contractive unit because Θ_{s}(1) ≤ 1, and ξ it is unital if and only if Θ_{s} is unital for all s. ξ is in fact more then just a generating unit, as F (s) is spanned by elements with the form described in equation (1.4) with (s_{1}, …, s_{n}) a fixed n-tuple such that s_{1} + ⋯ + s_{n} = s. □

Definition 3.3. Given a cp-semigroup Θ on a W^{∗} algebra M, the subproduct system F and the unit ξ constructed in Theorem 3.2 are called, respectively, the GNS subproduct system and the GNS unit of Θ. The pair (F, ξ) is called the GNS representation of Θ.

Remark 3.4. There is a precise relationship between the identity representation (Definition 2.3) and the GNS representation of a cp-semigroup. The GNS representation of a CP map is the dual of the identity representation in a sense that is briefly described in [31]. This notion of duality has been used to move from the product-system-and-representation picture to the productsystem-with-unit picture, and vice versa. See for example [44] and the references therein. It is more-or-less straightforward to use this duality to get Theorem 3.2 from Theorem 2.2 (or the other way around).

4. ∗-automorphic dilation of an e_{0}-semigroup

We now apply some of the tools developed above to dilate an e_{0}-semigroup to a semigroup of ∗-automorphisms. We shall need the following proposition, which is a modification (suited for subproduct systems) of the method introduced in [41] for representing a product system representation as a semigroup of contractive operators on a Hilbert space.

Proposition 4.1. Let N be a von Neumann algebra and let X be a subproduct system of W^{∗}-correspondences over S. Let (σ, T) be a fully coisometric covariant representation of X on the Hilbert space H, and assume that σ is unital.

Denote

H_{s} := (X(s) ⊗_{σ} H)/KerT̃_{s}

and

H = ⨁_{s∈S} H_{s}.

Then there exists a semigroup of coisometries T̂ = {T̂_{s}}_{s∈S} on H such that for all s ∈ S, x ∈ X(s) and h ∈ H,

T̂_{s} (δ_{s} · x ⊗ h) = T_{s}(x)h.

T̂ also has the property that for all s ∈ S and all t ≥ s

(4.1)  T̂_{s}^{∗}T̂_{s}|_{H_{t}} = I_{H_{t}} , (t ≥ s).

Proof. First, we note that the assumptions on σ and on the left action of N imply that H_{0} ≅ H via the identification a ⊗ h ↔ σ(a)h. This identification will be made repeatedly below. Define H_{0} to be the space of all finitely supported functions f on S such that for all s ∈ S,

f(s) ∈ H_{s}.

H_{0} is generated by the functions δ_{s} ·ξ, where δ_{s} is the function taking the value 1 at s and 0 elsewhere, and ξ ∈ H_{s}. We equip H_{0} with the inner product

⟨δ_{s} · ξ, δ_{t} · η⟩ = δ_{s,t}⟨ξ, η⟩,

for all s, t ∈ S, ξ ∈ H_{s}, η ∈ H_{t} (here δ_{s,t} is Kronecker’s delta function). Let H be the completion of H_{0} with respect to this inner product. We have

H ≅ ⨁_{s∈S} H_{s}.

It will sometimes be convenient to identify the subspace δ_{s} · H_{s} ⊆ H with H_{s}, and for s = 0 this gives us an inclusion H ⊆ H. We define a family T̂ = {T̂_{s}}_{s∈S} of operators on H_{0} as follows. First, we define T̂_{0} to be the identity. Now assume that s > 0. If t ∈ S and t ≱ s, then we define T̂_{s}(δ_{t} · ξ) = 0 for all ξ ∈ H_{t}. If t ≥ s > 0 we would like to define (as we did in [41])

(4.2)  T̂_{s} (δ_{t} · (x_{t−s} ⊗ x_{s} ⊗ h)) = δ_{t−s} · (x_{t−s} ⊗ T̃_{s}(x_{s} ⊗ h)) ,

but since X is not a true product system, we cannot identify X(t − s) ⊗ X(s) with X(t). For a fixed t > 0, we define for all s ≤ t, ξ ∈ X(t) and h ∈ H

Ť_{s} (δ_{t} · (ξ ⊗ h)) = δ_{t−s} · ((I_{X(t−s)} ⊗ T̃_{s})(U_{t−s,s}^{∗}ξ ⊗ h)) .

Ť_{s} can be extended to a well defined contraction from X(t)⊗H to X(t−s)⊗H, for all t ≥ s, and has an adjoint given by

(4.3)  Ť_{s}^{∗}δ_{t−s} · η ⊗ h = δ_{t} · ((U_{t−s,s} ⊗ I_{H})(η ⊗ T̃_{s}^{∗}h)) .

We are going to obtain T̂_{s} as the map H_{t} → H_{t−s} induced by Ť_{s}. Let Y ∈ H_{t} satisfy T̃_{t}(Y) = 0. We shall show that Ť_{s}δ_{t} · Y = 0 in δ_{t−s} · H_{t−s}. But

Ť_{s}δ_{t} · Y = δ_{t−s} · ((I_{X(t−s)} ⊗ T̃_{s})(U_{t−s,s}^{∗} ⊗ I_{H})Y ) ,

and

T̃_{t−s} ((I_{X(t−s)} ⊗ T̃_{s})(U_{t−s,s}^{∗} ⊗ I_{H})Y ) (∗) = T̃_{t}(U_{t−s,s} ⊗ I_{H})(U_{t−s,s}^{∗} ⊗ I_{H})Y
(∗∗) = T̃_{t}(Y) = 0,

where the equation marked by (*) follows from the fact that T is a representation of subproduct systems, and the one marked by (**) follows from the fact that U_{t−s,s} is a coisometry. Thus, for all s, t ∈ S,

Ť_{s} (δ_{t} · KerT̃_{t}) ⊆ δ_{t−s} · KerT̃_{t−s},

thus Ť_{s} induces a well defined contraction T̂_{s} on H given by

(4.4)  T̂_{s} (δ_{t} · (ξ ⊗ h)) = δ_{t−s} · ((I_{X(t−s)} ⊗ T̃_{s})(U_{t−s,s}^{∗}ξ ⊗ h)) ,

where ξ⊗h and (I_{X(t−s)}⊗T̃_{s})(U_{t−s,s}^{∗}ξ⊗h) stand for these elements’ equivalence classes in (X(t) ⊗ H)/KerT̃_{t} and (X(t − s) ⊗ H)/KerT̃_{t−s}, respectively. It follows that we have the following, more precise, variant of (4.2):

T̂_{s} (δ_{t} · (U_{t−s,s}(x_{t−s} ⊗ x_{s}) ⊗ h)) = δ_{t−s} · (x_{t−s} ⊗ T̃_{s}(x_{s} ⊗ h)) .

In particular,

T̂_{s} (δ_{s} · x_{s} ⊗ h) = T_{s}(x_{s})h,

for all s ∈ S, x_{s} ∈ X(s), h ∈ H. It will be very helpful to have a formula for T̂_{s}^{∗} as well. Assume that ∑_{i} ξ_{i}⊗h_{i} ∈ KerT̃_{t}.

Ť_{s}^{∗} (δ_{t} · ∑_{i} ξ_{i} ⊗ h_{i}) = δ_{s+t} · ((U_{t,s} ⊗ I_{H})(∑_{i} ξ_{i} ⊗ T̃_{s}^{∗}h_{i})) ,

and applying T̃_{s+t} to the right hand side (without the δ) we get

T̃_{s+t} ((U_{t,s} ⊗ I_{H})(∑_{i} ξ_{i} ⊗ T̃_{s}^{∗}h_{i})) = T̃_{t}(I_{X(t)} ⊗ T̃_{s})(∑_{i} ξ_{i} ⊗ T̃_{s}^{∗}h_{i})

= T̃_{t}(∑_{i} ξ_{i} ⊗ T̃_{s}T̃_{s}^{∗}h_{i})

= T̃_{t}(∑_{i} ξ_{i} ⊗ h_{i}) = 0,

because T is a fully coisometric representation. So

Ť_{s}^{∗} (δ_{t} · KerT̃_{t}) ⊆ δ_{s+t} · KerT̃_{s+t},

and this means that Ť_{s}^{∗} induces on H a well defined contraction which is equal to T̂_{s}^{∗}, and is given by the formula (4.3). We now show that T̂ is a semigroup. Let s, t, u ∈ S. If either s = 0 or t = 0 then it is clear that the semigroup property T̂_{s}T̂_{t} = T̂_{s+t} holds. Assume that s, t > 0. If u ≱ s + t, then both T̂_{s}T̂_{t} and T̂_{s+t} annihilate δ_{u} · ξ, for all ξ ∈ H_{u}. Assuming u ≥ s + t, we shall show that T̂_{s}T̂_{t} and T̂_{s+t} agree on elements of the form

Z = δ_{u} · (U_{u−t,t}(U_{u−t−s,s} ⊗ I)(x_{u−s−t} ⊗ x_{s} ⊗ x_{t})) ⊗ h,

and since the set of all such elements is total in H_{u}, this will establish the semigroup property.

T̂_{s}T̂_{t}Z = T̂_{s} (δ_{u−t} (U_{u−t−s,s}(x_{u−s−t} ⊗ x_{s}) ⊗ T̃_{t}(x_{t} ⊗ h)))
= δ_{u−s−t} (x_{u−s−t} ⊗ T̃_{s}(x_{s} ⊗ T̃_{t}(x_{t} ⊗ h)))
= δ_{u−s−t} (x_{u−s−t} ⊗ T̃_{s}(I ⊗ T̃_{t})(x_{s} ⊗ x_{t} ⊗ h))
= δ_{u−s−t} (x_{u−s−t} ⊗ T̃_{s+t} (U_{s,t}(x_{s} ⊗ x_{t}) ⊗ h))
= T̂_{t+s}δ_{u} · (U_{u−t−s,t+s} (x_{u−s−t} ⊗ U_{s,t}(x_{s} ⊗ x_{t})) ⊗ h)
= T̂_{t+s}Z.

The final equality follows from the associativity condition (1.1). To see that T̂ is a semigroup of coisometries, we take ξ ∈ X(t), h ∈ H, and compute

T̃_{t} (T̂_{s}T̂_{s}^{∗}δ_{t} · (ξ ⊗ h)) =
= T̃_{t} ((I_{X(t)} ⊗ T̃_{s})(U_{t,s}^{∗} ⊗ I_{H})(U_{t,s} ⊗ I_{H})(I_{X(t)} ⊗ T̃_{s}^{∗})(ξ ⊗ h))
= T̃_{s+t}(U_{t,s} ⊗ I_{H})(I_{X(t)} ⊗ T̃_{s}^{∗})(ξ ⊗ h)
= T̃_{t}(ξ ⊗ T̃_{s}T̃_{s}^{∗}h) = T̃_{t}(ξ ⊗ h),

so T̂_{s}T̂_{s}^{∗} is the identity on H_{t} for all t ∈ S, thus T̂_{s}T̂_{s}^{∗} = I_{H}. Equation (4.1) follows by a similar computation, which is omitted. □

We can now obtain a ∗-automorphic dilation for any e_{0}-semigroup over any subsemigroup of ℝ_{+}^{k}. The following result should be compared with similarlooking results of Arveson-Kishimoto [8], Laca [25], Skeide [45], and Arveson-Courtney [7] (none of these cited results is strictly stronger or weaker than the result we obtain for the case of e_{0}-semigroups).

Theorem 4.2. Let Θ be a e_{0}-semigroup acting on a von Neumann algebra M. Then Θ can be dilated to a semigroup of ∗-automorphisms in the following sense: there is a Hilbert space K, an orthogonal projection p of K onto a subspace H of K, a normal, faithful representation φ : M → B(K) such that φ(1) = p, and a semigroup α = {α_{s}}_{s∈S} of ∗-automorphisms on B(K) such that for all a ∈ M and all s ∈ S

(4.5)  α_{s}(φ(a))|_{H} = φ(Θ_{s}(a)),

so, in particular,

(4.6)  pα_{s}(φ(a))p = φ(Θ_{s}(a)).

The projection p is increasing for α, in the sense that for all s ∈ S,

(4.7)  α_{s}(p) ≥ p.

Remark 4.3. Another way of phrasing the above theorem is by using the terminology of “weak Markov flows”, as used in [15]. Denoting φ by j_{0}, and defining j_{s} := α_{s} ∘ j_{0}, we have that (B(K), j) is a weak Markov flow for Θ on K, which just means that for all t ≤ s ∈ S and all a ∈ M,

(4.8)  j_{t}(1)j_{s}(a)j_{t}(1) = j_{t}(Θ_{s−t}(a)).

Equation (4.8) for t = 0 is just (4.6), and the case t ≥ 0 follows from the case t = 0.

Remark 4.4. The assumption that Θ is a unital semigroup is essential, since (4.6) and (4.7) imply that Θ(1) = 1.

Remark 4.5. It is impossible, in the generality we are working in, to hope for a semigroup of automorphisms that extends Θ in the sense that

(4.9)  α_{s}(φ(a)) = φ(Θ_{s}(a)),

because that would imply that Θ is injective.

Proof. Let (E, T) be the identity representation of Θ. Since Θ preserves the unit, T is a fully coisometric representation. Let T̂ and H be the semigroup and Hilbert space representing T as described in Proposition 4.1. {T̂_{s}^{∗}}_{s∈S} is a commutative semigroup of isometries. By a theorem of Douglas [20], {T̂_{s}^{∗}}_{s∈S} can be extended to a semigroup {V̂_{s}^{∗}}_{s∈S} of unitaries acting on a space K ⊇ H. We obtain a semigroup of unitaries V = {V̂_{s}}_{s∈S} that is a dilation of T̂, that is

P_{H}V̂_{s}|_{H} = T̂_{s} , s ∈ S.

For any b ∈ B(K), and any s ∈ S, we define

α_{s}(b) = V̂_{s}bV̂_{s}^{∗}.

Clearly, α = {α_{s}}_{s∈S} is a semigroup of ∗-automorphisms. Put p = P_{H}, the orthogonal projection of K onto H. Define φ : M → B(K) by φ(a) = p(I ⊗ a)p, where I ⊗ a : H → H is given by

(I ⊗ a)δ_{t} · x ⊗ h = δ_{t} · x ⊗ ah , x ⊗ h ∈ E(t) ⊗ H.

φ is well defined because T is an isometric representation (so KerT̃_{t} is always zero). We have that φ is a faithful, normal ∗-representation (the fact that T_{0} is the identity representation ensures that φ is faithful). It is clear that φ(1) = p.

To see (4.7), we note that since V̂_{s}^{∗} is an extension of T̂_{s}^{∗}, we have T̂_{s}^{∗} = V̂_{s}^{∗}p = pV̂_{s}^{∗}p, thus

pα_{s}(p)p = pV̂_{s}pV̂_{s}^{∗}p
= pV̂_{s}V̂_{s}^{∗}p
= p,

that is, pα_{s}(p)p = p, which implies that α_{s}(p) ≥ p. We now prove (4.6). Let δ_{t} · x ⊗ h be a typical element of H. We compute

pα_{s}(φ(a))pδ_{t} · x ⊗ h = pV̂_{s}p(I ⊗ a)pV̂_{s}^{∗}pδ_{t} · x ⊗ h
= T̂_{s}(I ⊗ a)T̂_{s}^{∗}δ_{t} · x ⊗ h
= T̂_{s}(I ⊗ a)δ_{s+t} · (U_{t,s} ⊗ I_{H})(x ⊗ T̃_{s}^{∗}h)
= T̂_{s}δ_{s+t} · (U_{t,s} ⊗ I_{H})(x ⊗ (I ⊗ a)T̃_{s}^{∗}h)
= δ_{t} · x ⊗ (T̃_{s}(I ⊗ a)T̃_{s}^{∗}h)
= δ_{t} · x ⊗ (Θ_{s}(a)h)
= φ(Θ_{s}(a))δ_{t} · x ⊗ h.

Since both pα_{s}(φ(a))p and φ(Θ_{s}(a)) annihilate K ⊖ H, we have (4.6). To prove (4.5), it just remains to show that

pα_{s}(φ(a))|_{H} = α_{s}(φ(a))|_{H},

that is, that α_{s}(φ(a))H ⊆ H. Now, V̂_{s}^{∗} is an extension of T̂_{s}^{∗}. Moreover (4.1) shows that if ξ ∈ H_{u} with u ≥ s, then ∥T̂_{s}(ξ)∥ = ∥ξ∥. Thus

∥ξ∥^{2} = ∥V̂_{s}ξ∥^{2} = ∥P_{H}V̂_{s}ξ∥^{2} + ∥(I_{K} − P_{H})V̂_{s}ξ∥^{2} = ∥T̂_{s}ξ∥^{2} + ∥(I_{K} − P_{H})V̂_{s}ξ∥^{2}.

So V̂_{s}ξ = T̂_{s}ξ for ξ ∈ H_{u} with u ≥ s. Now, for a typical element δ_{t} · x ⊗ h in H_{t}, t ∈ S, we have

α_{s}(φ(a))δ_{t} · x ⊗ h = V̂_{s}(I ⊗ a)V̂_{s}^{∗}δ_{t} · x ⊗ h
= V̂_{s}(I ⊗ a)T̂_{s}^{∗}δ_{t} · x ⊗ h
= V̂_{s}δ_{s+t} · (U_{s,t} ⊗ I_{H})(x ⊗ (I ⊗ a)T̃_{s}^{∗}h)
= T̂_{s}δ_{s+t} · (U_{s,t} ⊗ I_{H})(x ⊗ (I ⊗ a)T̃_{s}^{∗}h) ∈ H,

because δ_{s+t} · x ⊗ (I ⊗ a)T̃_{s}^{∗}h ∈ H_{s+t}, and s + t ≥ s. □

5. Dilations and pieces of subproduct system representations

5.1. Dilations and pieces of subproduct system representations.

Definition 5.1. Let X and Y be subproduct systems of M correspondences (M a W^{∗}-algebra) over the same semigroup S. Denote by U_{s,t}^{X} and U_{s,t}^{Y} the coisometric maps that make X and Y, respectively, into subproduct systems.

X is said to be a subproduct subsystem of Y (or simply a subsystem of Y for short) if for all s ∈ S the space X(s) is a closed subspace of Y (s), and if the orthogonal projections p_{s} : Y (s) → X(s) are bimodule maps that satisfy

(5.1)  p_{s+t} ∘ U_{s,t}^{Y} = U_{s,t}^{X} ∘ (p_{s} ⊗ p_{t}) , s, t ∈ S.

One checks that if X is a subproduct subsystem of Y then

(5.2) p_{s+t+u} ∘ U_{s,t+u}^{Y}(I ⊗ (p_{t+u} ∘ U_{t,u}^{Y})) = p_{s+t+u} ∘ U_{s+t,u}^{Y}((p_{s+t} ∘ U_{s,t}^{Y}) ⊗ I),

for all s, t, u ∈ S. Conversely, given a subproduct system Y and a family of orthogonal projections {p_{s}}_{s∈S} that are bimodule maps satisfying (5.2), then by defining X(s) = p_{s}Y (s) and U_{s,t}^{X} = p_{s+t} ∘ U_{s,t}^{Y} one obtains a subproduct subsystem X of Y (with (5.1) satisfied). The following proposition is a consequence of the definitions.

Proposition 5.2. There exists a morphism X → Y if and only if Y is isomorphic to a subproduct subsystem of X.

Remark 5.3. In the notation of Theorem 2.6, we may now say that given a subproduct system X and a representation R of X, then the Arveson-Stinespring subproduct system E of Θ = Σ(X, R) is isomorphic to a subproduct subsystem of X.

The following definitions are inspired by the work of Bhat, Bhattacharyya and Dey [12].

Definition 5.4. Let X and Y be subproduct systems of W^{∗}-correspondences (over the same W^{∗}-algebra M) over S, and let T be a representation of Y on a Hilbert space K. Let H be some fixed Hilbert space, and let S = {S_{s}}_{s∈S} be a family of maps S_{s} : X(s) → B(H). (Y, T, K) is called a dilation of (X, S, H) if

(1) X is a subsystem of Y,

(2) H is a subspace of K, and

(3) for all s ∈ S, T̃_{s}^{∗}H ⊆ X(s) ⊗ H and T̃_{s}^{∗}|_{H} = S̃_{s}^{∗}. In this case we say that S is an X-piece of T, or simply a piece of T. T is said to be an isometric dilation of S if T is an isometric representation.

The third item can be replaced by the three conditions 1’ T_{0}(·)P_{H} = P_{H}T_{0}(·)P_{H} = S_{0}(·),

2’ P_{H} T̃_{s}|_{X(s)⊗H} = S̃_{s} for all s ∈ S, and
3’ P_{H} T̃_{s}|_{Y (s)⊗K⊖X(s)⊗H} = 0.

So our definition of dilation is identical to Muhly and Solel’s definition of dilation of representations when X = Y is a product system [29, Theorem and Definition 3.7].

Proposition 5.5. Let T be a representation of Y, let X be a subproduct subsystem of Y, and let S an X-piece of T. Then S is a representation of X.

Proof. S is a completely contractive linear map as the compression of a completely contractive linear map. Item 1’ above together with the coinvariance of T imply that S is coinvariant: if a, b ∈ M and x ∈ X(s), then

S_{s}(axb) = P_{H}T_{s}(axb)P_{H} = P_{H}T_{0}(a)T_{s}(x)T_{0}(b)P_{H}
= P_{H}T_{0}(a)P_{H}T_{s}(x)P_{H}T_{0}(b)P_{H}
= S_{0}(a)S_{s}(x)S_{0}(b).

Finally, (using Item 3’ above),

S_{s+t}(U_{s,t}^{X}(x ⊗ y))h = S_{s+t}(p_{s+t}U_{s,t}^{Y}(x ⊗ y))h
= S̃_{s+t}(p_{s+t}U_{s,t}^{Y}(x ⊗ y) ⊗ h)
= P_{H} T̃_{s+t}(U_{s,t}^{Y}(x ⊗ y) ⊗ h)
= P_{H}T_{s}(x)T_{t}(y)h
= P_{H}T_{s}(x)P_{H}T_{t}(y)h
= S_{s}(x)S_{t}(y)h.
□

Example 5.6. Let E be a Hilbert space of dimension d, and let X be the symmetric subproduct system constructed in Example 1.3. Fix an orthonormal basis {e_{1}, …, e_{n}} of E. There is a one-to-one correspondence between c.c. representations S of X (on some H) and commuting row contractions (S_{1}, …, S_{d}) (of operators on H), given by

S ↔ \underline{S} = (S(e_{1}), …, S(e_{d})).

If Y is the full product system over E, then any dilation (Y, T, K) gives rise to a tuple \underline{T} = (T (e_{1}), …, T (e_{d})) that is a dilation of \underline{S} in the sense of [12], and vice versa. Moreover, \underline{S} is then a commuting piece of \underline{T} in the sense of [12].

Consider a subproduct system Y and a representation T of Y on K. Let X be some subproduct subsystem of Y. Define the following set of subspaces of K:

(5.3)  P(X, T) = {H ⊆ K : T̃_{s}^{∗}H ⊆ X(s) ⊗ H for all s ∈ S}.

As in [12], we observe that P(X, T) is closed under closed linear spans (and intersections), thus we may define

K^{X}(T) = ⋁_{H∈P(X,T)} H.

K^{X}(T) is the maximal element of P(X, T).

Definition 5.7. The representation T^{X} of X on K^{X}(T) given by

T^{X}(x)h = P_{K^{X}(T)}T (x)h,

for x ∈ X(s) and h ∈ K^{X}(T), is called the maximal X-piece of T.

By Proposition 5.5, T^{X} is indeed a representation of X.

5.2. Consequences in dilation theory of cp-semigroups.

Proposition 5.8. Let X and Y be subproduct systems of W^{∗}-correspondences (over the same W^{∗}-algebra M) over S, and let S and T be representations of X on H and of Y on K, respectively. Assume that (Y, T, K) is a dilation of (X, S, H). Then the cp-semigroup Θ acting on V_{0}(M)′, given by

Θ_{s}(a) = T̃_{s}(I_{Y (s)} ⊗ a)T̃_{s}^{∗} , a ∈ V_{0}(M)′,

is a dilation of the cp-semigroup Φ acting on T_{0}(M)′ given by

Φ_{s}(a) = S̃_{s}(I_{X(s)} ⊗ a)S̃_{s}^{∗} , a ∈ T_{0}(M)′,

in the sense that for all b ∈ V_{0}(M)′ and all s ∈ S,

Φ_{s}(P_{H}bP_{H}) = P_{H}Θ_{s}(b)P_{H}.

Proof. This follows from the definitions. □

Although the above proposition follows immediately from the definitions, we hope that it will prove to be important in the theory of dilations of cpsemigroups, because it points to a conceptually new way of constructing dilations of cp-semigroups, as the following proposition and corollary illustrate.

Proposition 5.9. Let X = {X(s)}_{s∈S} be a subproduct system, and let S be a fully coisometric representation of X on H such that S_{0} is unital. If there exists a (full) product system Y = {Y (s)}_{s∈S} such that X is a subproduct subsystem of Y, then S has an isometric and fully coisometric dilation.

Proof. Define a representation T of Y on H by

(5.4)  T_{s} = S_{s} ∘ p_{s},

where, as above, p_{s} is the orthogonal projection Y (s) → X(s). A straightforward verification shows that T is indeed a fully coisometric representation of Y on H. By [43, Theorem 5.2], (Y, T, H) has a minimal isometric and fully coisometric dilation (Y, V, K). (Y, V, K) is also clearly a dilation of (X, S, H). □

Corollary 5.10. Let Θ = {Θ_{s}}_{s∈S} be a cp_{0}-semigroup and let (E, T) = Ξ(Θ) be the Arveson-Stinespring representation of Θ. If there is a (full) product system Y such that E is a subproduct subsystem of Y, then Θ has an e_{0}- dilation.

Proof. Combine Propositions 2.1, 5.8 and 5.9. □

Thus, the problem of constructing e_{0}-dilations to cp_{0}-semigroups is reduced to the problem of embedding a subproduct system into a full product system. In the next subsection we give an example of a subproduct system that cannot be embedded into full product system. When this can be done in general is a challenging open question.

Corollary 5.11. Let Θ = {Θ_{s}}_{s∈ℕ^{k}} be a cp-semigroup generated by k commuting CP maps θ_{1}, …, θ_{k}, and let (E, T) = Ξ(Θ) be the Arveson representation of Θ. Assume, in addition, that

∑_{i=1}^{k} ∥θ_{i}∥ ≤ 1.

If there is a (full) product system Y such that E is a subproduct subsystem of Y, then Θ has an e-dilation.

Proof. As in (5.4), we may extend T to a product system representation of Y on H, which we also denote by T. Denote by e_{i} the element of ℕ^{k} with 1 in the ith element and zeros elsewhere. Then

∑_{i=1}^{k} ∥T̃_{e_{i}} T̃_{e_{i}}^{∗}∥ = ∑_{i=1}^{k} ∥θ_{i}∥ ≤ 1.

By the methods of [41], one may show that S has a minimal (regular) isometric dilation. This isometric dilation provides the required e-dilation of Θ. □

Theorem 5.12. Let M ⊆ B(H) be a von Neumann algebra, let X be a subproduct system of M′-correspondences, and let R be an injective representation of X on a Hilbert space H. Let Θ = Σ(X, R) be the cp-semigroup acting on R_{0}(M′)′ given by (2.1). Assume that (α, K, R) is an e-dilation of Θ, and let (Y, V) = Ξ(α) be the Arveson-Stinespring subproduct system of α together with the identity representation. Assume, in addition, that the map R′ ∋ b ↦ P_{H}bP_{H} is a ∗-isomorphism of R′ onto R_{0}(M′). Then (Y, V, K) is a dilation of (X, R, H).

Proof. For every s ∈ S, define W_{s} : Y (s) → B(H) by W_{s}(y) = P_{H}V_{s}(y)P_{H}. We claim that W = {W_{s}}_{s∈S} is a representation of Y on H. First, note that P_{H}α_{s}(I − P_{H})P_{H} = Θ_{s}(P_{H}(I − P_{H})P_{H}) = 0, thus P_{H} Ṽ_{s}(I ⊗ (I − P_{H}))Ṽ_{s}^{∗}P_{H} = 0, and consequently P_{H} Ṽ_{s}(I ⊗ P_{H}) = P_{H} Ṽ_{s}. It follows that W_{s}(y) = P_{H}V_{s}(y)P_{H} = P_{H}V_{s}(y). From this it follows that

W_{s}(y_{1})W_{t}(y_{2}) = P_{H}V_{s}(y_{1})P_{H}V_{t}(y_{2}) = P_{H}V_{s}(y_{1})V_{t}(y_{2})
= P_{H}V_{s+t}(U_{s,t}^{Y}(y_{1} ⊗ y_{2})) = W_{s+t}(U_{s,t}^{Y}(y_{1} ⊗ y_{2})).

By Theorem 2.6, we may assume that (X, R) = (E, T) = Ξ(Θ) is the Arveson-Stinespring representation of Θ. Because α is a dilation of Θ, we have

W̃_{s}(I ⊗ a)W̃_{s}^{∗} = P_{H} Ṽ_{s}(I ⊗ a)Ṽ_{s}^{∗}P_{H} = Θ_{s}(a),

That is, Θ = Σ(Y, W). Thus, by Theorem 2.6 and Remark 5.3, we may assume that E is a subproduct subsystem of Y, and that T_{s} ∘ p_{s} = W_{s}, p_{s} being the projection of Y (s) onto E(s). In other words, for all y ∈ Y,

T̃_{s}(p_{s} ⊗ I_{H}) = P_{H} W̃_{s}.

Therefore, W̃_{s}^{∗}H ⊆ E(s) ⊗ H, and W̃_{s}^{∗}|_{H} = T̃_{s}^{∗}. That is, (Y, W, H) is a dilation of (E, T, H). But (Y, V, K) is a dilation of (Y, W, H), so it is also a dilation of (E, T, H). □

The assumption that R′ ∋ b ↦ P_{H}bP_{H} ∈ M′ is a ∗-isomorphism is satisfied when M = B(H) and R = B(K). More generally, it is satisfied whenever the central projection of P_{H} in R is I_{K} (see Propositions 5.5.5 and 5.5.6 in [22]). Let (α, K, R) be an e-dilation of a semigroup Θ on M ⊆ B(H). (α, K, R) is called a minimal dilation if the central support of P_{H} in R is I_{K} and if

R = W^{∗} ( ⋃_{s∈S} α_{s}(M)) .

Corollary 5.13. Let Θ be cp-semigroup on M ⊆ B(H), and let (α, K, R) be a minimal dilation of Θ. Then Ξ(α) is an isometric dilation of Ξ(Θ).

5.3. cp-semigroups with no e-dilations. Obstructions of a new nature. By Parrot’s famous example [34], there exist 3 commuting contractions that do not have a commuting isometric dilation. In 1998 Bhat asked whether 3 commuting CP maps necessarily have a commuting ∗-endomorphic dilation [10]. Note that it is not obvious that the non-existence of an isometric dilation for three commuting contractions would imply the non-existence of a ∗-endomorphic dilation for 3 commuting CP maps. However, it turns out that this is the case.

Theorem 5.14. There exists a cp-semigroup Θ = {Θ_{n}}_{n∈ℕ^{3}} acting on a B(H) for which there is no e-dilation (α, K, B(K)). In fact, Θ has no minimal edilation (α, K, R) on any von Neumann algebra R.

Proof. Let T_{1}, T_{2}, T_{3} ∈ B(H) be three commuting contractions that have no isometric dilation and such that T_{1}^{n_{1}}T_{2}^{n_{2}}T_{3}^{n_{3}} ≠ 0 for all n = (n_{1}, n_{2}, n_{3}) ∈ ℕ^{3} (one may take commuting contractions R_{1}, R_{2}, R_{3} with no isometric dilation as in Parrot’s example [34], and define T_{i} = R_{i} ⊕ 1). For all n = (n_{1}, n_{2}, n_{3}) ∈ ℕ^{3}, define

Θ_{n}(a) = T_{1}^{n_{1}}T_{2}^{n_{2}}T_{3}^{n_{3}}a(T_{3}^{n_{3}})^{∗}(T_{2}^{n_{2}})^{∗}(T_{1}^{n_{1}})^{∗} , a ∈ B(H).

Note that Θ = Σ(X, R), where X = {X(n)}_{n∈ℕ^{3}} is the subproduct system given by X(n) = ℂ for all n ∈ ℕ^{3}, and R is the (injective) representation that sends 1 ∈ X(n) to T_{1}^{n_{1}}T_{2}^{n_{2}}T_{3}^{n_{3}} (the product in X is simply multiplication of scalars). Assume, for the sake of obtaining a contradiction, that Θ = {Θ_{n}}_{n∈ℕ^{3}} has an e-dilation (α, K, B(K)). Let (Y, V) = Ξ(α) be the Arveson-Stinespring subproduct system of α together with the identity representation. By Theorem 5.12, (Y, V, K) is a dilation of (X, R, H). It follows that V_{1}, V_{2}, V_{3} are a commuting isometric dilation of T_{1}, T_{2}, T_{3} where V_{1} := V (1) with 1 ∈ X(1, 0, 0), V_{2} := V (1) with 1 ∈ X(0, 1, 0), and V_{3} := V (1) with 1 ∈ X(0, 0, 1). This is a contradiction. Finally, a standard argument shows that if (α, K, R) is a minimal dilation of Θ, then R = B(K). □

Until this point, all the results that we have seen in the dilation theory of cpsemigroups have been anticipated by the classical theory of isometric dilations. We shall now encounter a phenomena that has no counterpart in the classical theory. By [49, Proposition 9.2], if T_{1}, …, T_{k} is a commuting k-tuple of contractions such that

(5.5)  ∑_{i=1}^{k} ∥T_{i}∥^{2} ≤ 1,

then T_{1}, …, T_{k} has a commuting regular unitary dilation (and, in particular, an isometric dilation). One is tempted to conjecture that if θ_{1}, …, θ_{k} is a commuting k-tuple of CP maps such that

(5.6)  ∑_{i=1}^{k} ∥θ_{i}∥ ≤ 1,

then the tuple θ_{1}, …, θ_{k} has an e-dilation. Indeed, if θ_{i}(a) = T_{i}aT_{i}^{∗}, where T_{1}, …, T_{k} is a commuting k-tuple satisfying (5.5), then it is easy to construct an e-dilation of θ_{1}, …, θ_{k} from the isometric dilation of T_{1}, …, T_{k}. However, it turns out that (5.6) is far from being sufficient for an e-dilation to exist. We need some preparations before exhibiting an example.

Proposition 5.15. There exists a subproduct system that is not a subsystem of any product system.

Proof. We construct a counter example over ℕ^{3}. Let e_{1}, e_{2}, e_{3} be the standard basis of ℕ^{3}. We let X(e_{1}) = X(e_{2}) = X(e_{3}) = ℂ^{2}. Let X(e_{i} + e_{j}) = ℂ^{2} ⊗ ℂ^{2} for all i, j = 1, 2, 3. Put X(n) = {0} for all n ∈ ℕ^{k} such that |n| > 2. To complete the construction of X we need to define the product maps U_{m,n}^{X}. Let U_{e_{i},e_{j}}^{X} be the identity on ℂ^{2} ⊗ ℂ^{2} for all i, j except for i = 3, j = 2, and let U_{e_{3},e_{2}}^{X} be the flip. Define the rest of the products to be zero maps (except the maps U_{0,n}^{X}, U_{m,0}^{X} which are identities). This product is evidently coisometric, and it is also associative, because the product of any three nontrivial elements vanishes. Let Y be a product system “dilating” X. Then for all k, m, n ∈ ℕ^{k} we have

U_{k+m,n}^{Y}(U_{k,m}^{Y} ⊗ I) = U_{k,m+n}^{Y}(I ⊗ U_{m,n}^{Y}),

or

U_{k+m,n}^{Y} = U_{k,m+n}^{Y}(I ⊗ U_{m,n}^{Y})(U_{k,m}^{Y} ⊗ I)^{∗},

and

U_{k,m+n}^{Y} = U_{k+m,n}^{Y}(U_{k,m}^{Y} ⊗ I)(I ⊗ U_{m,n}^{Y})^{∗}.

Iterating these identities, we have, on the one hand,

U_{e_{3},e_{1}+e_{2}} = U_{e_{3}+e_{2},e_{1}}^{Y}(U_{e_{3},e_{2}}^{Y} ⊗ I)(I ⊗ U_{e_{2},e_{1}}^{Y})^{∗}
= U_{e_{2},e_{3}+e_{1}}^{Y}(I ⊗ U_{e_{3},e_{1}}^{Y})(U_{e_{2},e_{3}}^{Y} ⊗ I)^{∗}(U_{e_{3},e_{2}}^{Y} ⊗ I)(I ⊗ U_{e_{2},e_{1}}^{Y})^{∗}
= U_{e_{1}+e_{2},e_{3}}^{Y}(U_{e_{2},e_{1}}^{Y} ⊗ I)(I ⊗ U_{e_{1},e_{3}}^{Y})^{∗}
(I ⊗ U_{e_{3},e_{1}}^{Y})(U_{e_{2},e_{3}}^{Y} ⊗ I)^{∗}(U_{e_{3},e_{2}}^{Y} ⊗ I)(I ⊗ U_{e_{2},e_{1}}^{Y})^{∗},

and on the other hand

U_{e_{3},e_{1}+e_{2}} = U_{e_{3}+e_{1},e_{2}}^{Y}(U_{e_{3},e_{1}}^{Y} ⊗ I)(I ⊗ U_{e_{1},e_{2}}^{Y})^{∗}
= U_{e_{1},e_{3}+e_{2}}^{Y}(I ⊗ U_{e_{3},e_{2}}^{Y})(U_{e_{1},e_{3}}^{Y} ⊗ I)^{∗}(U_{e_{3},e_{1}}^{Y} ⊗ I)(I ⊗ U_{e_{1},e_{2}}^{Y})^{∗}
= U_{e_{1}+e_{2},e_{3}}^{Y}(U_{e_{1},e_{2}}^{Y} ⊗ I)(I ⊗ U_{e_{2},e_{3}}^{Y})^{∗}
(I ⊗ U_{e_{3},e_{2}}^{Y})(U_{e_{1},e_{3}}^{Y} ⊗ I)^{∗}(U_{e_{3},e_{1}}^{Y} ⊗ I)(I ⊗ U_{e_{1},e_{2}}^{Y})^{∗}.

Canceling U_{e_{1}+e_{2},e_{3}}^{Y}, we must have

(U_{e_{1},e_{2}}^{Y} ⊗ I)(I ⊗ U_{e_{2},e_{3}}^{Y})^{∗}(I ⊗ U_{e_{3},e_{2}}^{Y})(U_{e_{1},e_{3}}^{Y} ⊗ I)^{∗}(U_{e_{3},e_{1}}^{Y} ⊗ I)(I ⊗ U_{e_{1},e_{2}}^{Y})^{∗}

= (U_{e_{2},e_{1}}^{Y} ⊗ I)(I ⊗ U_{e_{1},e_{3}}^{Y})^{∗}(I ⊗ U_{e_{3},e_{1}}^{Y})(U_{e_{2},e_{3}}^{Y} ⊗ I)^{∗}(U_{e_{3},e_{2}}^{Y} ⊗ I)(I ⊗ U_{e_{2},e_{1}}^{Y})^{∗}.

Now, U_{e_{i},e_{j}}^{X} were unitary to begin with, so the above identity implies

(U_{e_{1},e_{2}}^{X} ⊗ I)(I ⊗ U_{e_{2},e_{3}}^{X})^{∗}(I ⊗ U_{e_{3},e_{2}}^{X})(U_{e_{1},e_{3}}^{X} ⊗ I)^{∗}(U_{e_{3},e_{1}}^{X} ⊗ I)(I ⊗ U_{e_{1},e_{2}}^{X})^{∗}

= (U_{e_{2},e_{1}}^{X} ⊗ I)(I ⊗ U_{e_{1},e_{3}}^{X})^{∗}(I ⊗ U_{e_{3},e_{1}}^{X})(U_{e_{2},e_{3}}^{X} ⊗ I)^{∗}(U_{e_{3},e_{2}}^{X} ⊗ I)(I ⊗ U_{e_{2},e_{1}}^{X})^{∗}.

Recalling the definition of the product in X (the product is usually the identity), this reduces to

I ⊗ U_{e_{3},e_{2}}^{X} = U_{e_{3},e_{2}}^{X} ⊗ I.

This is absurd. Thus, X cannot be dilated to a product system. □

We can now strengthen Theorem 5.14:

Theorem 5.16. There exists a cp-semigroup Θ = {Θ_{n}}_{n∈ℕ^{3}} acting on a B(H), such that for all λ > 0, λΘ has no e-dilation (α, K, B(K)), and no minimal e-dilation (α, K, R) on any von Neumann algebra R.

Proof. Let X be as in Proposition 5.15. Let Θ be the cp-semigroup generated by the X-shift, as in Section 2.3 of the paper. Of course, Θ, as a semigroup over ℕ^{3}, can be generated by three commuting CP maps θ_{1}, θ_{2}, θ_{3}. X cannot be embedded into a full product system, so by Theorem 5.12, Θ has no minimal e-dilation, nor does it have an e-dilation acting on a B(K). Note that if Θ is scaled its product system is left unchanged (this follows from Theorem 2.6: if you take X and scale the representation S^{X} you get a scaled version of Θ). So no matter how small you take λ > 0, λθ_{1}, λθ_{2}, λθ_{3} cannot be dilated to three commuting ∗-endomorphisms on B(K), nor to a minimal three-tuple on any von Neumann algebra. □

Note that the obstruction here seems to be of a completely different nature from the one in the example given in Theorem 5.14. The subproduct system arising there is already a product system, and, indeed, the cp-semigroup arising there can be dilated once it is multiplied by a small enough scalar.

Part 2. Subproduct systems over ℕ

6. Subproduct systems of Hilbert spaces over ℕ

We now specialize to subproduct systems of Hilbert W^{∗}-correspondences over the semigroup ℕ, so from now on any subproduct system is to be understood as such (soon we will specialize even further to subproduct systems of Hilbert spaces).

6.1. Standard and maximal subproduct systems. If X is a subproduct system over ℕ, then X(0) = M (some von Neumann algebra), X(1) equals some W^{∗}-correspondence E, and X(n) can be regarded as a subspace of E^{⊗n}. The following lemma allows us to consider X(m + n) as a subspace of X(m) ⊗ X(n).

Lemma 6.1. Let X = {X(n)}_{n∈ℕ} be a subproduct system. X is isomorphic to a subproduct system Y = {Y (n)}_{n∈ℕ} with coisometries {U_{m,n}^{Y}}_{m,n∈ℕ} that satisfies

Y (1) = X(1)

and

(6.1)  Y (m + n) ⊆ Y (m) ⊗ Y (n).

Moreover, if p_{m+n} is the orthogonal projection of Y (1)^{⊗(m+n)} onto Y (m + n), then

(6.2)  U_{m,n}^{Y} = p_{m+n}|_{Y (m)⊗Y (n)}

and the projections {p_{n}}_{n∈ℕ} satisfy

(6.3)  p_{k+m+n} = p_{k+m+n}(I_{E^{⊗k}} ⊗ p_{m+n}) = p_{k+m+n}(p_{k+m} ⊗ I_{E^{⊗n}}).

Proof. Denote by U_{m,n}^{X} the subproduct system maps X(s) ⊗ X(t) → X(s + t). Denote E = X(1). We first note that for every n there is a well defined coisometry U_{n} : E^{⊗n} → X(n) given by composing in any way a sequence of maps U_{k,m}^{X} (for example, one can take U_{3} = U_{2,1}^{X}(U_{1,1}^{X} ⊗ I_{E}) and so on). We define Y (n) = Ker(U_{n})^{⊥}, and we let p_{n} be the orthogonal projection from E^{⊗n} onto Y (n). p_{n} = U_{n}^{∗}U_{n}, so, in particular, p_{n} is a bimodule map. For all m, n ∈ ℕ we have that

E^{⊗m} ⊗ Ker(U_{n}) ⊆ Ker(U_{m+n}).

Thus E^{⊗m}⊗Ker(U_{n})^{⊥} ⊇ Ker(U_{m+n})^{⊥}, so p_{m+n} ≤ I_{E^{⊗m}} ⊗p_{n}. This means that (6.3) holds. In addition, defining U_{m,n}^{Y} to be p_{m+n} restricted to Y (m)⊗Y (n) ⊆ E^{⊗(m+n)} gives Y the associative multiplication of a subproduct system.

It remains to show that X is isomorphic to Y. For all n, X(n) is spanned by elements of the form U_{n}(x_{1} ⊗ ⋯ ⊗ x_{n}), with x_{1}, …, x_{n} ∈ E. We define a map V_{n} : X(n) → Y (n) by

V_{n}(U_{n}(x_{1} ⊗ ⋯ ⊗ x_{n})) = p_{n}(x_{1} ⊗ ⋯ ⊗ x_{n}).

It is immediate that V_{n} preserves inner products (thus it is well defined) and that it maps X(n) onto Y (n). Finally, for all m, n ∈ ℕ and x ∈ E^{⊗m}, y ∈ E^{⊗n},

V_{m+n}(U_{m,n}^{X}(U_{m}(x) ⊗ U_{n}(y))) = V_{m+n}(U_{m+n}(x ⊗ y))
= p_{m+n}(x ⊗ y)
= p_{m+n}(p_{m}x ⊗ p_{n}y)
= p_{m+n}((V_{m}U_{m}(x)) ⊗ (V_{n}U_{n}(y)))
= U_{m+n}^{Y}((V_{m}U_{m}(x)) ⊗ (V_{n}U_{n}(y))),

and (1.2) holds. □

Definition 6.2. A subproduct system Y satisfying (6.1), (6.2) and (6.3) above will be called a standard subproduct system.

Note that a standard subproduct system is a subproduct subsystem of the full product system {E^{⊗n}}_{n∈ℕ}.

Corollary 6.3. Every cp-semigroup over ℕ has an e-dilation.

Proof. The unital case follows from Corollary 5.10 together with the above lemma. The nonunital case follows from a similar construction (where the dilation of a non-fully-coisometric representation is obtained by adapting [41, Theorem 4.2] instead of [43, Theorem 5.2]). □

Let k ∈ ℕ, and let E = X(1), X(2), …, X(k) be subspaces of E, E^{⊗2}, …, E^{⊗k}, respectively, such that the orthogonal projections p_{n} : E^{⊗n} → X(n) satisfy

p_{n} ≤ I_{E^{⊗i}} ⊗ p_{j}

and

p_{n} ≤ p_{i} ⊗ I_{E^{⊗j}}

for all i, j, n ∈ ℕ_{+} satisfying i+j = n ≤ k. In this case one can define a maximal standard subproduct system X with the prescribed fibers X(1), …, X(k) by defining inductively for n > k

X(n) = ( ⋂_{i+j=n} E^{⊗i} ⊗ X(j)) ⋂ ( ⋂_{i+j=n} X(i) ⊗ E^{⊗j}) .

It is easy to see that

X(n) = ⋂_{n_{1}+...+n_{m}=n} X(n_{1}) ⊗ ⋯ ⊗ X(n_{m}) = ⋂_{i+j=n} X(i) ⊗ X(j).

We then have obvious formulas for the projections {p_{n}}_{n∈ℕ} as well, for example

p_{n} = ⋀_{i+j=n} p_{i} ⊗ p_{j} , (n > k).

6.2. Examples.

Example 6.4. In the case k = 1, the maximal standard subproduct system with prescribed fiber X(1) = E, with E a Hilbert space, is the full product system F_{E} of Example 1.2. If dim E = d, we think of this subproduct system as the product system representing a (row-contractive) d-tuple (T_{1}, …, T_{d}) of non commuting operators, that is, d operators that are not assumed to satisfy any relations (the idea behind this last remark must be rather vague at this point, but it shall become clearer as we proceed). In the case k = 2, if X(2) is the symmetric tensor product E with itself then the maximal standard subproduct system with prescribed fibers X(1), X(2) is the symmetric subproduct system SSP_{E} of Example 1.3. We think of SSP as the subproduct system representing a commuting d-tuple.

Example 6.5. Let E be a two dimensional Hilbert space with basis {e_{1}, e_{2}}. Let X(2) be the space spanned by e_{1} ⊗ e_{1}, e_{1} ⊗ e_{2}, and e_{2} ⊗ e_{1}. In other words, X(2) is what remains of E^{⊗2} after we declare that e_{2} ⊗e_{2} = 0. We think of the maximal standard subproduct system X with prescribed fibers X(1) = E, X(2) as the subproduct system representing pairs (T_{1}, T_{2}) of operators subject only to the condition T_{2}^{2} = 0. E^{⊗n} has a basis consisting of all vectors of the form e_{α} = e_{α_{1}} ⊗ ⋯ ⊗ e_{α_{n}} where α = α_{1} ⋯ α_{n} is a word of length n in “1” and “2”. X(n) then has a basis consisting of all vectors e_{α} where α is a word of length n not containing “22” as a subword. Let us compute dim X(n), that is, the number of such words. Let A_{n} denote the number of words not containing “22” that have leftmost letter “1”, and let B_{n} denote the number of words not containing “22” that have leftmost letter “2”. Then we have the recursive relation A_{n} = A_{n−1}+B_{n−1} and B_{n} = A_{n−1}. The solution of this recursion gives

dim X(n) = A_{n} + B_{n} ≈ (\frac{1 + \sqrt{5}}{2})^{n} .

As one might expect, the dimension of X(n) grows exponentially fast.

Example 6.6. Suppose that we want a “subproduct system that will represent a pair of operators (T_{1}, T_{2}) such that T_{i}T_{2} = 0 for i = 1, 2”. Although we have not yet made clear what we mean by this, let us proceed heuristically along the lines of the preceding examples. We let E be as above, but now we declare e_{1} ⊗e_{2} = e_{2} ⊗e_{2} = 0. In other words, we define X(2) = {e_{1} ⊗e_{2}, e_{2} ⊗e_{2}}^{⊥}. One checks that the maximal standard subproduct system X with prescribed fibers X(1) = E, X(2) is given by X(n) = span{e_{1} ⊗ e_{1} ⊗ ⋯ ⊗ e_{1}, e_{2} ⊗ e_{1} ⊗ ⋯ ⊗ e_{1}}. This is an example of a subproduct system with two dimensional fibers.

At this point two natural questions might come to mind. First, is every standard subproduct system X the maximal subproduct system with prescribed fibers X(1), …, X(k) for some k ∈ ℕ? Second, does dim X(n) grow exponentially fast (or remain a constant) for every subproduct system X? The next example answers both questions negatively.

Example 6.7. Let E be as in the preceding examples, and let X(n) be a subspace of E^{⊗n} having basis the set

{e_{α} : |α| = n, α does not contain the words 22, 212, 2112, 21112, …}.

Then X = {X(n)}_{n∈ℕ} is a standard subproduct system, but it is smaller than the maximal subproduct system defined by any initial k fibers. Also, X(n) is the span of e_{α} with α = 11 ⋯ 11, 21 ⋯ 11, 121 ⋯ 11, …, 11 ⋯ 12, thus

dim X(n) = n + 1,

so this is an example of a subproduct system with fibers that have a linearly growing dimension.

Of course, one did not have to go far to find an example of a subproduct system with linearly growing dimension: indeed, the dimension of the fibers of the symmetric subproduct system SSP_{ℂ^{d}} is known to be

dim SSP_{ℂ^{d}}(n) = ( \begin{matrix}n + d − 1 \\ n \end{matrix} ) .

Taking d = 2 we get the same dimension as in Example 6.7. However, SSP := SSP_{ℂ^{2}} and the subproduct system X of Example 6.7 are not isomorphic: for any nonzero x ∈ SSP (1), the “square” U_{1,1}^{SSP} (x ⊗ x) ∈ SSP (2) is never zero, while U_{1,1}^{X}(e_{2} ⊗ e_{2}) = 0. Here is an interesting question that we do not know the answer to: given a solution f : ℕ → ℕ to the functional inequality

f(m + n) ≤ f(m)f(n) , m, n ∈ ℕ,

does there exists a subproduct system X such that dim X(n) = f(n) for all n ∈ ℕ?

Remark 6.8. One can cook up simple examples of subproduct systems that are not standard. We will not write these examples down, as we already know that such a subproduct system is isomorphic to a standard one.

6.3. Representations of subproduct systems. Fix a W^{∗}-correspondence E. Every completely contractive linear map T_{1} : E → B(H) gives rise to a c.c. representation T^{n} of the full product system F_{E} = {E^{⊗n}}_{n∈ℕ} by defining for all x ∈ E^{⊗n} and h ∈ H

(6.4)  T^{n}(x)h = T̃_{1}(I_{E} ⊗ T̃_{1}) ⋯ (I_{E^{⊗(n−1)}} ⊗ T̃_{1})(x ⊗ h),

where T̃_{1} : E ⊗ H → H is given by T̃_{1}(e ⊗ h) = T_{1}(e)h. We will denote the operator acting on x ⊗ h in the right hand side of (6.4) as T̃^{n}, so as not to confuse with T̃_{n}, which sometimes has a different meaning (namely: if T denotes a c.c. representation of a subproduct system X then

T̃_{n} : X(n) ⊗ H → H

is given by

T̃_{n}(x ⊗ h) = T (x)h

for all x ∈ X(n), h ∈ H. Of course, when X = F_{E}, T is a representation of F_{E} and T_{1} is the restriction of T to E, then T̃^{n} = T̃_{n} for all n). If X is a standard subproduct system and X(1) = E, we obtain a completely contractive representation of X(n) by restricting T^{n} to X(n). Let us denote this restriction by T_{n}, and denote the family {T_{n}}_{n∈ℕ} by T.

Proposition 6.9. Let X be a standard subproduct system with projections {p_{n}}_{n∈ℕ}, and let T_{1} : E → B(H) be a completely contractive map. Construct the family of maps T = {T_{n}}_{n∈ℕ}, with T_{n} : X(n) → B(H) as in the preceding paragraph. Then the following are equivalent:

(1) T is a representation of X.

(2) For all m, n ∈ ℕ,

(6.5)  T̃_{m}(I_{X(m)} ⊗ T̃_{n})(p_{m} ⊗ p_{n} ⊗ I_{H})(p_{m+n}^{⊥} ⊗ I_{H}) = 0.

(3) For all n ∈ ℕ,

(6.6)  T̃^{n}(p_{n}^{⊥} ⊗ I_{H}) = 0.

Proof. If T is a representation, then

T̃_{m}(I_{X(m)}⊗T̃_{n})(p_{m}⊗p_{n}⊗I_{H})(p_{m+n}^{⊥}⊗I_{H}) = T̃_{m+n}(p_{m+n}⊗I_{H})(p_{m+n}^{⊥}⊗I_{H}) = 0,

so 1 ⇒ 2. To prove 2 ⇒ 3 note first that (6.6) is clear for n = 1. Assuming that (6.6) holds for n = 1, 2, …, k − 1, we will show that it holds for n = k.

T̃^{k}(p_{k}^{⊥} ⊗ I_{H}) = T̃^{1}(I ⊗ T̃^{k−1})(p_{k}^{⊥} ⊗ I_{H})

= T̃^{1}(I ⊗ T̃^{k−1})(I_{E} ⊗ p_{k−1}^{⊥} ⊗ I_{H} + I_{E} ⊗ p_{k−1} ⊗ I_{H})(p_{k}^{⊥} ⊗ I_{H})

(∗) = T̃^{1}(I ⊗ T̃^{k−1}(p_{k−1} ⊗ I_{H}))(p_{k}^{⊥} ⊗ I_{H})
= T̃_{1}(I ⊗ T̃_{k−1}(p_{k−1} ⊗ I_{H}))(p_{k}^{⊥} ⊗ I_{H})
(∗∗) = 0.

The equality marked by (*) is true by the inductive hypothesis, and the one marked by (**) follows from (6.5). Finally, 3 ⇒ 1: by (6.6) we have T̃^{n}(p_{n} ⊗ I_{H}) = T̃^{n}. On the other hand, T̃^{n}(p_{n} ⊗ I_{H}) = T̃_{n}(p_{n} ⊗ I_{H}). Thus

T̃_{m+n}(p_{m+n} ⊗ I_{H}) = T̃^{m+n}(p_{m+n} ⊗ I_{H})
= T̃^{m+n}
= T̃^{m}(I_{X(m)} ⊗ T̃^{n})
= T̃_{m}(I_{X(m)} ⊗ T̃_{n})(p_{m} ⊗ p_{n} ⊗ I_{H}),

which shows that T is a representation. □

Proposition 6.10. Let X be the maximal standard subproduct system with prescribed fibers X(1), …, X(k), and let T_{1} : E → B(H) be a completely contractive map. Construct T as in Proposition 6.9. Then T is a representation of X if and only if

(6.7)  T̃^{n}(p_{n}^{⊥} ⊗ I_{H}) = 0 for all n = 1, 2, …, k.

Proof. The necessity of (6.7) follows from Proposition 6.9. By the same proposition, to show that the condition is sufficient it is enough to show that (6.7) holds for all n ∈ ℕ. Given m ∈ ℕ, we have p_{m} = ⋀_{q} q, where q runs over all projections of the form q = I_{X(i)} ⊗ p_{j} or q = p_{i} ⊗ I_{X(j)}, with i, j ∈ ℕ_{+} and i + j = m. But then p_{m}^{⊥} = ⋁_{q} q^{⊥}, thus if (6.7) holds for all n < m then it also holds for n = m. □

6.4. Fock spaces and standard shifts.

Definition 6.11. Let X be a subproduct system of Hilbert spaces. Fix an orthonormal basis {e_{i}}_{i∈I} of E = X(1). X(n), when considered as a subspace of 𝔉_{X}, is called the n particle space. The standard X-shift (related to {e_{i}}_{i∈I}) on 𝔉_{X} is the tuple of operators \underline{S}^{X} = (S_{i}^{X})_{i∈I} in B(𝔉_{X}) given by

S_{i}^{X}(x) = U_{1,n}(e_{i} ⊗ x),

for all i ∈ I, n ∈ ℕ and x ∈ X(n).

It is clear that S_{i}^{X} = S^{X}(e_{i}), where S^{X} is the shift representation given by Definition 2.9. If F denotes the usual full product system (Example 1.2) then 𝔉_{F} is the usual Fock space and the tuple (S_{i}^{F})_{i∈I} is the standard shift (the I orthogonal shift of [37]). We shall denote 𝔉_{F} as 𝔉 and (S_{i}^{F})_{i∈I} as (S_{i})_{i∈I}. It is then obvious that the tuple (S_{i}^{X})_{i∈I} is a row contraction, as it is the compression of the row contraction (S_{i})_{i∈I}. Indeed, assuming (as we may, thanks to Lemma 6.1) that U_{m,n} is an orthogonal projection p_{m+n} : X(m) ⊗ X(n) → X(m + n), and denoting p = ⊕_{n}p_{n}, we have for all i that S_{i}^{X} = pS_{i}|𝔉_{X}.

Example 6.12. The q-commuting Fock space of [19] also fits into this framework. Indeed, let (as in [19]) Γ(ℂ^{d}) be the full Fock space, let Γ_{q}(ℂ^{d}) denote the q-commuting Fock space, and let Y (n) be the “n particle q-commuting space” with orthogonal projection p_{n} : (ℂ^{d})^{n} → Y (n). Then a straightforward calculation shows that the projections {p_{n}}_{n∈ℕ} satisfy equation (6.3) of Lemma 6.1, thus Y = {Y (n)}_{n∈ℕ} is a subproduct system (satisfying (6.1) and (6.2)). With our notation from above we have that 𝔉_{Y} = Γ_{q}(ℂ^{d}) and that the tuple (S_{i}^{Y} , …, S_{d}^{Y}) is the standard q-commuting shift.

S^{F}, the standard shift of the full product system on the full Fock space, will be denoted by S, and will be called simply the standard shift. By the notation introduced in Definition 5.7, the symbol S^{X} is also used to denote the maximal X-piece of the standard shift S. The following proposition – which is a generalization of [12, Proposition 6], [19, Proposition 11] and [39, Proposition 2.9] – shows that this is consistent.

Proposition 6.13. Let X subproduct subsystem of a subproduct system Y. Then the maximal X-piece of the standard Y -shift is the standard X-shift.

Proof. Let E = Y (1), and let F = F_{E} be the full product system. Viewing F (n) ⊗ 𝔉_{F} as direct sum of |I|^{n} copies of 𝔉_{F}, (S̃)_{n} is just the row isometry (S_{i_{1}} ∘ ⋯ ∘ S_{i_{n}})_{i_{1},...,i_{n}∈I} from the space of columns 𝔉_{F} ⊕ 𝔉_{F} ⊕ ⋯ into 𝔉_{F}. In other words, for h ∈ 𝔉_{F} and i_{1}, …, i_{n} ∈ I,

(S̃)_{n}((e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}}) ⊗ h) = S_{i_{1}} ∘ ⋯ ∘ S_{i_{n}}h = (e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}}) ⊗ h.

This is an isometry, and the adjoint works by sending (e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}}) ⊗ h ∈ 𝔉_{F} back to (e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{n}}) ⊗ h ∈ F (n) ⊗ 𝔉_{F}, and by sending the 0, 1, …, n − 1 particle spaces to 0. Now, if Z is any standard subproduct subsystem of F, then

\binom{}{S^{Z}}_{n} = P_{𝔉_{Z}} \binom{}{S}_{n} |_{Z(n)⊗𝔉_{Z}} ,

thus

(6.8)  \binom{}{S^{Z}}_{n}^{∗} = P_{Z(n)⊗𝔉_{Z}} \binom{}{S}_{n}^{∗} |_{𝔉_{Z}} .

Now if h is in the k particle space of 𝔉_{F} with k < n, then (S̃^{Z})_{n}^{∗}h = 0. If k ≥ n, then since Z(k) ⊆ Z(n) ⊗ Z(k − n) we may write h = ∑ ξ_{i} ⊗ η_{i}, where ξ_{i} ∈ Z(n) and η_{i} ∈ Z(k − n). Thus by (6.8) we find that

(6.9)  (S̃^{Z})_{n}^{∗} (∑ ξ_{i} ⊗ η_{i}) = ∑ p_{n}^{Z}ξ_{i} ⊗ p_{k−n}^{Z}η_{i} = ∑ ξ_{i} ⊗ η_{i}.

From these considerations it follows that the standard X-shift is in fact an X-piece of the standard Y shift, as (S̃^{Y})_{n}^{∗}|_{𝔉_{X}} = (S̃^{X})_{n}^{∗}. It remains to show that the X-shift is maximal. Assume that there is a Hilbert space H, 𝔉_{X} ⊆ H ⊆ 𝔉_{Y}, such that the compression of S^{Y} to H is an X-piece of Y, that is, H ∈ P(X, S^{Y}) (see equation (5.3)). Let h ∈ H ⊖ 𝔉_{X}. We shall prove that h = 0. Being orthogonal to all of 𝔉_{X}, p_{n}^{Y} h must be orthogonal to X(n) for all n. Thus, we may assume that h ∈ Y (n) ⊖ X(n) for some n. But then by (6.9)

(S̃^{Y})_{n}^{∗}h = h ⊗ Ω.

But since H ∈ P(X, S^{Y}), we must have h ⊗ Ω ∈ X(n) ⊗ H, and this, together with h ∈ Y (n) ⊖ X(n), forces h = 0. □

7. Zeros of homogeneous polynomials in noncommutative variables

In the next section we will describe a model theory for representations of subproduct systems. But before that we dedicate this section to build a precise connection between subproduct systems together with their representations and tuples of operators that are the zeros of homogeneous polynomials in non commuting variables.

Remark 7.1. The notions that we are developing give a framework for studying tuples of operators satisfying relations given by homogeneous polynomials. One can go much further by considering subspaces of Fock spaces and “representations”, i.e., maps of the Fock space into B(H), that give a framework for studying tuples of operators satisfying arbitrary (not-necessarily homogeneous) polynomial and even analytic identities. Gelu Popescu [39] has already begun developing such a theory.

We begin by setting up the usual notation. Let I be a fixed set of indices, and let ℂ⟨(x_{i})_{i∈I}⟩ be the algebra of complex polynomials in the non commuting variables (x_{i})_{i∈I}. We denote \underline{x} = (x_{i})_{i∈I}, and we consider \underline{x} as a “tuple variable”. We shall sometimes write ℂ⟨\underline{x}⟩ for ℂ⟨(x_{i})_{i∈I}⟩. The set of all words in I is denoted by 𝔽_{I}^{+}. For a word α ∈ 𝔽_{I}^{+}, let |α| denote the length of α, i.e., the number of letters in α. For every word α = α_{1} ⋯ α_{k} in I denote \underline{x}^{α} = x_{α_{1}} ⋯ x_{α_{k}}. If α = 0 is the empty word, then this is to be understood as 1. k is also referred to in this context as the degree of the monomial x^{α}. ℂ⟨\underline{x}⟩ is by definition the linear span over ℂ of all such monomials, and every element in ℂ⟨\underline{x}⟩ is called a polynomial. A polynomial is called homogeneous if it is the sum of monomials of equal degree. A homogeneous ideal is a two-sided ideal that is generated by homogeneous polynomials. If \underline{T} = (T_{i})_{i∈I} is a tuple of operators on a Hilbert space H and α = α_{1} ⋯ α_{k} is a word with letters in I, we define

\underline{T}^{α} = T_{α_{1}}T_{α_{2}} ⋯ T_{α_{k}}.

We define \underline{T}^{0} = I_{H}. If p(x) = ∑_{α} c_{α}\underline{x}^{α} ∈ ℂ⟨\underline{x}⟩, we define p(\underline{T}) = ∑_{α} c_{α}\underline{T}^{α}. If E is a Hilbert space with orthonormal basis {e_{i}}_{i∈I}, An element e_{α_{1}} ⊗ ⋯ ⊗ e_{α_{k}} ∈ E^{⊗k} will be written in short form as e_{α}, where α = α_{1} ⋯ α_{k}. If p(x) = ∑_{α} c_{α}\underline{x}^{α} ∈ ℂ⟨\underline{x}⟩, we define p(e) = ∑_{α} c_{α}e_{α}. Here e_{0} (0 the empty word) is understood as the vacuum vector Ω.

Proposition 7.2. Let E be a Hilbert space with orthonormal basis {e_{i}}_{i∈I}. There is an inclusion reversing correspondence between proper homogeneous ideals I ◁ ℂ⟨\underline{x}⟩ and standard subproduct systems X = \{X(n)\}_{n∈ℕ} with X(1) ⊆ E. When |I| < ∞ this correspondence is bijective.

Proof. Let X be such a subproduct system. We define an ideal

(7.1)  I^{X} := span\{p ∈ ℂ⟨\underline{x}⟩ : ∃n > 0, p(e) ∈ E^{⊗n} ⊖ X(n)\}.

Once it is established that I^{X} is a two-sided ideal the fact that it is homogeneous will follow from the definition. Let p ∈ ℂ⟨\underline{x}⟩ be such that p(e) ∈ E^{⊗n} ⊖ X(n) for some n > 0. It suffices to show that for every monomial \underline{x}^{α} we have that \underline{x}^{α}p(\underline{x}) ∈ I^{X}, that is,

e_{α} ⊗ p(e) ∈ E^{⊗|α|+n} ⊖ X(|α| + n).

But since X is standard, X(|α| + n) ⊆ X(|α|) ⊗ X(n), thus

E^{⊗|α|} ⊗ (E^{⊗n} ⊖ X(n)) ⊆ E^{⊗|α|+n} ⊖ X(|α| + n).

It follows that I^{X} is a homogeneous ideal. Conversely, let I be a homogeneous ideal. We construct a subproduct system X_{I} as follows. Let I^{(n)} be the set of all homogeneous polynomials of degree n in I. Define

(7.2)  X_{I}(n) = E^{⊗n} ⊖ {p(e) : p ∈ I^{(n)}}.

Denote by p_{n} the orthogonal projection of E^{⊗n} onto X_{I}(n). To show that X_{I} is a subproduct system it is enough (by symmetry) to prove that for all m, n ∈ ℕ

p_{m+n} ≤ I_{E^{⊗m}} ⊗ p_{n},

or, in other words, that

(7.3)  X_{I}(m + n) ⊆ E^{⊗m} ⊗ X_{I}(n).

Let x ∈ X_{I}(m + n), let α ∈ I^{m}, and let q ∈ I^{(n)}. Since I is an ideal, \underline{x}^{α}q(\underline{x}) is in I^{(m+n)}, thus ⟨x, e_{α} ⊗ q(e)⟩ = 0. This proves (7.3). Assume now that |I| < ∞. We will show that the maps X ↦ I^{X} and I ↦ X_{I} are inverses of each other. Let J be a homogeneous ideal in ℂ⟨\underline{x}⟩. Then

I^{X_{J}} = span\{p ∈ ℂ⟨\underline{x}⟩ : ∃n > 0, p(e) ∈ E^{⊗n} ⊖ X_{J}(n)\}
(∗) = span\{p ∈ ℂ⟨\underline{x}⟩ : ∃n > 0, p(e) ∈ \{q(e) : q ∈ J^{(n)}\}\}
= span{p : ∃n > 0, p ∈ J^{(n)}}
(∗∗) = J,

where (*) follows from the definition of X_{J}, and (**) from the fact that J is a homogeneous ideal. For the other direction, let Y be a standard subproduct subsystem of F_{E} = \{E^{⊗n}\}_{n∈ℕ}. Clearly, (I^{Y})^{(n)} = \{p ∈ ℂ⟨\underline{x}⟩ : p(e) ∈ E^{⊗n} ⊖ Y (n)\}. Thus

X_{I^{Y}} (n) = E^{⊗n} ⊖ {p(e) : p ∈ (I^{Y})^{(n)}}
= E^{⊗n} ⊖ \{p(e) : p ∈ \{q ∈ ℂ⟨\underline{x}⟩ : q(e) ∈ E^{⊗n} ⊖ Y (n)\}
= E^{⊗n} ⊖ (E^{⊗n} ⊖ Y (n))
= Y (n).
□

We record the definitions of I^{X} and X_{I} from the above theorem for later use:

Definition 7.3. Let E be a Hilbert space with orthonormal basis {e_{i}}_{i∈I} (|I| is not assumed finite). Given a homogeneous ideal I ◁ℂ⟨\underline{x}⟩, the subproduct system X_{I} defined by (7.2) will be called the subproduct system associated with I. If X is a given subproduct subsystem of F_{E}, then the ideal I^{X} of ℂ⟨\underline{x}⟩ defined by (7.1) will be called the ideal associated with X.

We note that X_{I} depends on the choice of the space E and basis {e_{i}}_{i∈I}, but different choices will give rise to isomorphic subproduct systems.

Proposition 7.4. Let X and Y be standard subproduct systems with dim X(1) = dim Y (1) = d < ∞. Then X is isomorphic to Y if and only if there is a unitary linear change of variables in ℂ⟨x_{1}, …, x_{d}⟩ that sends I^{X} onto I^{Y}.

Fix some infinite dimensional separable Hilbert space H. As in classical algebraic geometry, given a homogeneous ideal I ◁ ℂ⟨\underline{x}⟩, it is natural to introduce and to study the zero set of I

Z(I) := \{\underline{T} = (T_{i})_{i∈I} ∈ B(H)^{I} : ∀p ∈ I, p(\underline{T}) = 0\}.

Also, given a set Z ⊆ B(H)^{I}, one may form the following two-sided ideal in ℂ⟨\underline{x}⟩

I(Z) := \{p ∈ ℂ⟨\underline{x}⟩ : ∀\underline{T} ∈ Z, p(\underline{T}) = 0\}.

In the following theorem we shall use the notation of 6.3. This simple result is the justification for viewing subproduct systems as a framework for studying tuples of operators satisfying certain homogeneous polynomial relations.

Theorem 7.5. Let E be a Hilbert space with orthonormal basis {e_{i}}_{i∈I} (not necessarily with |I| < ∞), and let I be a proper homogeneous ideal in ℂ⟨(x_{i})_{i∈I}⟩. Let X_{I} be the associated subproduct system. Let T_{1} : E → B(H) be a given representation of E. Define a tuple \underline{T} = (T (e_{i}))_{i∈I}. Construct the family of maps T = {T_{n}}_{n∈ℕ}, with T_{n} : X(n) → B(H) as in the paragraphs before Proposition 6.9. Then T is a representation of X if and only if \underline{T} ∈ Z(I).

Proof. On the one hand, E^{⊗n} ⊖ X_{I}(n) = \overline{span}\{p(e) : p ∈ I^{(n)}\}. On the other hand, for every p ∈ I^{(n)} and every h ∈ H,

T̃^{n}(p(e) ⊗ h) = p(\underline{T})h.

Hence, the Theorem follows from Proposition 6.9. □

Lemma 7.6. Let J ◁ ℂ⟨(x_{i})_{i∈I}⟩, |I| < ∞, be a proper homogeneous ideal. Let S^{X_{J}} be the X_{J}-shift representation, and define \underline{T} = (T_{i})_{i∈I} by T_{i} = S^{X_{J}} (e_{i}), i ∈ I. If p ∈ ℂ⟨\underline{x}⟩ is a homogeneous polynomial, then p(\underline{T}) = 0 if and only if p ∈ J.

Proof. The “if” part follows from Theorem 7.5. For the “only if” part, let p ∉ J be a homogeneous polynomial of degree n. Applying p(\underline{T}) to the vacuum vector Ω, we have

p(\underline{T})Ω = P p(e),

where P is the orthogonal projection of E^{⊗n} onto X_{J}(n). But as p ∉ J, p(e) is not in E^{⊗n} ⊖ X_{J}(n) = ker P, thus P p(e) ≠ 0. In particular, p(\underline{T}) ≠ 0. □

We have the following noncommutative projective Nullstellensatz.

Theorem 7.7. Let H be a fixed infinite dimensional separable Hilbert space. Let J be a homogeneous ideal in ℂ⟨(x_{i})_{i∈I}⟩, with |I| < ∞. Then

I(Z(J)) = J.

In particular, Z(J) = \{\underline{0} = (0, 0, …)\} if and only if J is the ideal generated by all the x_{i}, i ∈ I.

Proof. I(Z(J)) ⊇ J is immediate. To see the converse, first note that equality is obvious when J = ℂ⟨\underline{x}⟩, so we may assume that J is proper. Also note that since J is homogeneous Z(J) is scale invariant. From this it follows that I(Z(J)) is also a homogeneous ideal. Indeed, if h, g ∈ H, and p(\underline{x}) = ∑_{α} c_{α}\underline{x}^{α} ∈ I(Z(J)), then for all λ ∈ ℂ one has for every tuple \underline{T} = (T_{i})_{i∈I} ∈ Z(I),

0 = ⟨p(λ\underline{T})h, g⟩ = ∑_{k} ( ∑_{|α|=k} c_{α}⟨\underline{T}^{α}h, g⟩) λ^{k},

and since a nonzero univariate polynomial has only finitely many zeros, it follows the homogeneous components of p are all in I(Z(J)). Assume now that p is a homogeneous polynomial not in J. Let S^{X_{J}} be the X_{J}-shift representation, and define \underline{T} = (T_{i})_{i∈I} by T_{i} = S^{X_{J}} (e_{i}), i ∈ I. It is clear that B(H)^{I} contains some unitarily equivalent copy of \underline{T}, which we also denote by \underline{T}. By Theorem 7.5, \underline{T} ∈ Z(J). But by Lemma 7.6, p(\underline{T}) ≠ 0, so p ∉ I(Z(J)). This completes the proof. □

8. Universality of the shift: universal algebras and models

In [5], Arveson established a model for commuting, row-contractive tuples. Using an idea from that paper that appeared also in [12] and [19] – an idea that rests upon Popescu’s “Poisson Transform” introduced in [38] (and pushed forward in [33] and [39]) – we construct below a model for representations of subproduct systems. Roughly speaking, we will show that every representation of a subproduct system X is a piece of a scaled inflation of the shift. Our model should be compared with a similar model obtained by Popescu in [39]. We will also see below that the operator algebra generated by the shift S^{X} is the universal operator algebra generated by a representation of X.

8.1. Notation for this section. We continue to use the notation set in the previous section. Let X be a standard subproduct system of Hilbert spaces over ℕ, to be fixed throughout this section. Let p_{n} : E^{⊗n} → X(n) be the projections. Denote E = X(1). Let {e_{i}}_{i∈I} be an orthonormal basis for E, fixed once and for all. We denote the standard X-shift tuple by \underline{S^{X}} = (S_{i}^{X})_{i∈I} , and we denote the standard X-shift representation of X on 𝔉_{X} by S^{X}. We consider 𝔉_{X} to be a subspace of the full Fock space 𝔉, we denote the full shift by \underline{S} = (S_{i})_{i∈I}, and we denote the full shift representation of F on 𝔉 := 𝔉_{F} by S. Given a representation T : X → B(H), we will write \underline{T} = (T_{i})_{i∈I} for the tuple (T (e_{i}))_{i∈I}. We denote by A_{X} the unital algebra

A_{X} := \overline{span}\{\underline{S^{X}}^{α} : α ∈ 𝔽_{I}^{+}\}.

We denote by E_{X} the operator system

E_{X} := \overline{span}A_{X}A_{X}^{∗},

and by T_{X} = C^{∗}(\underline{S^{X}}) the C^{∗}-algebra generated by S_{i}^{X}, i ∈ I and I_{𝔉_{X}}. We denote by K(𝔉_{X}) the algebra of compact operators on 𝔉_{X} If T and U are two representations of X on Hilbert spaces H and K, respectively, then we define

T ⊕ U

to be the representation of X on H ⊕ K given by (T ⊕ U)(x) = T (x) ⊕ U(x). We also define

T ⊗ I_{K}

to be the representation of X on H ⊗ K given by (T ⊗ I_{K})(x) = T (x) ⊗ I_{K}.

8.2. Popescu’s “Poisson Transform”. After obtaining the results of this section, we discovered that they were obtained earlier by Popescu [39]. We are presenting them here since they are important for the rest of this paper but we leave out some of the arguments.

Proposition 8.1. K(𝔉_{X}) ⊆ E_{X}.

Proof. The result follows from the equations

(8.1)  I − ∑_{i∈I} S_{i}^{X} (S^{X})_{i}^{∗} = P_{ℂ}.

and

(\underline{S^{X}})^{β} (I − ∑_{i∈I} S_{i}^{X} (S^{X})_{i}^{∗}) \underline{S^{X}}^{α∗}x = p_{|β|}⟨e_{α}, x⟩e_{β}.

As the elements p_{|β|}e_{β} span 𝔉_{X}, it follows that K(𝔉_{X}) ⊆ E_{X}. Full details can be found in [39, Theorem 1.3] □

Given a representation T of X on a Hilbert space H and given an integer m ∈ ℕ, we denote by m · T the representation

m · T : X → B(\underbrace{H ⊕ H ⊕ ⋯ ⊕ H}_{m times})

given by m · T (x) = \underbrace{T (x) ⊕ T (x) ⊕ ⋯ ⊕ T (x)}_{m times}. \underline{T} is a row contraction (i.e.,

∑_{i∈I} T_{i}T_{i}^{∗} ≤ I_{H}) if and only if T is completely contractive. When \underline{T} is a row contraction the defect operator Δ(\underline{T}) is defined as

Δ(\underline{T}) = I − ∑_{i∈I} T_{i}T_{i}^{∗},

and the Poisson Kernel [38] associated with \underline{T} is the family of isometries \{K_{r} (\underline{T})\}_{0≤r<1}

K_{r} (\underline{T}) : H → 𝔉 ⊗ H,

given by

K_{r} (\underline{T}) h = ∑_{α∈𝔽_{I}^{+}} e_{α} ⊗ (r^{|α|}Δ(r\underline{T})^{1/2}\underline{T}^{α∗}h) .

(See the beginning of [38, Section 8] for the remark that \underline{T} has “property (P)”, and [38, Lemma 3.2] for the fact that these are isometries). When it makes sense, we also define K_{1} (\underline{T}) by the same formula with r = 1. The Poisson transform is then defined as a map

Φ = Φ\underline{T} : C^{∗}(\underline{S}) → B(H)

Φ(a) = Φ\underline{T} (a) = lim_{r↗1} K_{r} (\underline{T})^{∗} (a ⊗ I)K_{r} (\underline{T}) .

By [38, Theorem 3.8], Φ is a unital, completely positive, completely contractive, satisfies

Φ(\underline{S}^{α}\underline{S}^{β∗}) = \underline{T}^{α}\underline{T}^{β∗},

and is multiplicative on Alg(\underline{S}, I_{𝔉}), the algebra generated by \underline{S} and I_{𝔉} (Φ is in fact an Alg(\underline{S}, I_{𝔉})-morphism).

Theorem 8.2. Let T be a c.c. representation of X on H. There exists a unital, completely positive, completely contractive map

Ψ : E_{X} → B(H)

that satisfies

Ψ ((\underline{S^{X}})^{α}(\underline{S^{X}})^{β∗}) = \underline{T}^{α}\underline{T}^{β∗} , α, β ∈ 𝔽_{I}^{+}

and

(8.2)  Ψ(ab) = Ψ(a)Ψ(b) , a ∈ A_{X}, b ∈ E_{X}.

Proof. By the lemma below, the range of K_{r} (\underline{T}) is contained in 𝔉_{X} ⊗ H for all 0 ≤ r < 1, thus

(P_{𝔉_{X}} ⊗ I_{H})K_{r} (\underline{T}) = K_{r} (\underline{T}) .

We may then define

Ψ(\underline{T})(((\underline{S^{X}})^{α}(\underline{S^{X}})^{β∗})) = lim_{r↗1} K_{r} (\underline{T})^{∗} (((\underline{S^{X}})^{α}(\underline{S^{X}})^{β∗}) ⊗ I) K_{r} (\underline{T})
(∗) = lim_{r↗1} K_{r} (\underline{T})^{∗} ((\underline{S}^{α}\underline{S}^{β∗}) ⊗ I) K_{r} (\underline{T})
= \underline{T}^{α}\underline{T}^{β∗},

where in (*) we have made use of the coinvariance of 𝔉_{X} under \underline{S}. This obviously extends to the desired map on E_{X}. □

Lemma 8.3. K_{r} (\underline{T}) H ⊆ 𝔉_{X} ⊗ H.

Proof. This was proved in [39, Equation (2.5)] for r = 1. The same argument (using the fact that p, there, can be chosen homogeneous) works also for r < 1. □

8.3. The universal algebra generated by a tuple subject to homogeneous polynomial identities.

Theorem 8.4. J ◁ ℂ⟨(x_{i})_{i∈I}⟩, be a homogeneous ideal. Then A_{X_{J}} is the universal unital operator algebra generated by a row contraction in Z(J), that is: A_{X_{J}} is a norm closed unital operator algebra generated by a tuple in Z(J), (namely, (S_{i}^{X_{J}})_{i∈I}), and if B ⊆ B(H) is another norm closed unital operator algebra generated by a row contraction (T_{i})_{i∈I} ∈ Z(J), then there is a unique unital and completely contractive homomorphism φ of A_{X_{J}} onto B, such that φ(S_{i}^{X_{J}}) = T_{i} for all i ∈ I.

Proof. This follows from Theorems 7.5 and 8.2. □

8.4. A model for representations: every completely bounded representation of X is a piece of an inflation of S^{X}. We will now construct a model for representations of subproduct systems. In [39, Section 2], a similar but different model – that includes also a fully coisometric part and not only the shift – has been obtained.

Theorem 8.5. Let \underline{T} be a completely bounded representation of the subproduct system X on a separable Hilbert space H, and let K be an infinite dimensional, separable Hilbert space. Then for all r > ∥T ∥_{cb}, T is unitarily equivalent to a piece of

(8.3)  S^{X} ⊗ rI_{K}.

Moreover, ∥T ∥_{cb} is equal the infimum of r such that T is a piece of an operator as in (8.3).

Proof. It is known that ∥T ∥_{cb} = ∥(T_{i})_{i∈I}∥_{row}, where T_{i} = T (e_{i}). Thus if r > r_{0} = ∥T ∥_{cb}, then ∑_{i∈I} T_{i}T_{i}^{∗} ≤ r_{0}^{2}I < r^{2}I. Put W_{i} = r^{−1}T_{i}, so ∑_{i∈I} W_{i}W_{i}^{∗} ≤ r_{0}^{2}/r^{2}I. Then K_{1} (\underline{W}) is an isometry (it is equal to K_{r_{0}/r}(r/r_{0}\underline{W}), and r/r_{0}\underline{W} is a row contraction). Thus we may define a map (as in the proof of Theorem 8.2)

Ψ : B(𝔉_{X}) → B(H)

by

Ψ(a) = K_{1} (\underline{W})^{∗} (a ⊗ I) K_{1} (\underline{W}) .

Ψ is a normal completely positive unital map that satisfies

Ψ ((\underline{S^{X}})^{α}(\underline{S^{X}})^{β∗}) = \underline{W}^{α}\underline{W}^{β∗} , α, β ∈ 𝔽_{I}^{+}.

Since Ψ is normal it has a normal minimal Stinespring dilation Ψ(a) = V^{∗}π(a)V, with π : B(𝔉_{X}) → B(L) a normal ∗-homomorphism and V : H → L an isometry. It is well known that π is equivalent to a multiple of the identity representation. Thus we obtain, up to unitary equivalence and after identifying H with V H, that r^{−1}T_{i} = P_{H}π(S_{i}^{X})P_{H} = P_{H}(S_{i}^{X} ⊗ I_{G})P_{H}, for some Hilbert space G. To see that T is a piece of S^{X} ⊗ I_{G} we need to show that (S_{i}^{X} ⊗ I_{G})^{∗}|_{H} = T_{i}^{∗} for all i ∈ I. In other words, we need to show that P_{H}π(S_{i}^{X}) = P_{H}π(S_{i}^{X})P_{H}. But, for all b ∈ E_{X},

P_{H}π(S_{i}^{X})π(b)P_{H} = P_{H}π(S_{i}^{X}b)P_{H}
= Ψ(S_{i}^{X}b)
(∗) = Ψ(S_{i}^{X})Ψ(b)
= P_{H}π(S_{i}^{X})P_{H}π(b)P_{H},

where (*) follows from (8.2). By Proposition 8.1, the strong operator closure of E_{X} is B(𝔉_{X}). P_{H}π(S_{i}^{X}) = P_{H}π(S_{i}^{X})P_{H} now follows from the minimality and normality of the dilation. It is clear that r^{−1}T is a also piece of S^{X} ⊗I_{K} for every K with dim K ≥ dim G, so we may choose K to be infinite dimensional. We want to show that necessarily dim K ≥ dim H. Since S^{X} ⊗ I_{K} is a dilation of r^{−1}T, I_{L} −∑_{i∈I} S_{i}^{X}(S_{i}^{X})^{∗} ⊗I_{K} is a dilation of I_{H} −∑_{i∈I} r^{−2}T_{i}T_{i}^{∗}. But the latter operator is invertible so it has rank dim H. Thus the rank of P_{ℂ} ⊗ I_{K} = I_{L} − ∑_{i∈I} S_{i}^{X}(S_{i}^{X})^{∗} ⊗ I_{K}, which is dim K, must be greater. Now the final assertion is clear. □

We can now obtain a general von Neumann inequality.

Theorem 8.6. Let X be a subproduct system, and let T be a c.c. representation of X on a Hilbert space H. Let {e_{1}, …, e_{d}} be an orthonormal set in X(1), and define T_{i} = T (e_{i}) and S_{i}^{X} = S^{X}(e_{i}) for i = 1, …, d. Then for every polynomials p and q in d non commuting variables,

∥p(T_{1}, …, T_{d})q(T_{1}, …, T_{d})^{∗}∥ ≤ ∥p(S_{1}^{X}, …, S_{d}^{X})q(S_{1}^{X}, …, S_{d}^{X})^{∗}∥.

Proof. Since T is a piece of S^{X} ⊗ rI_{K} for all r > 1, we have

p(T_{1}, …, T_{d})q(T_{1}, …, T_{d})^{∗} = P (p(rS_{1}, …, rS_{d})q(rS_{1}, …, rS_{d})^{∗} ⊗ I_{K})P

for some projection P, and the result follows by taking r ↘ 1. □

9. The operator algebra associated to a subproduct system

9.1. Let X be a subproduct system. Recall the definitions of A_{X} and E_{X} from 8.1. If {e_{i}}_{i∈I} is an orthonormal basis for X(1), then A_{X} is the unital operator algebra generated by (S_{i}^{X})_{i∈I} with S_{i}^{X} = S^{X}(e_{i}). If {f_{i}}_{i∈I} is another orthonormal basis then the tuple (S^{X}(f_{i}))_{i∈I} is not necessarily unitarily equivalent to (S_{i}^{X})_{i∈I}. For instance (with the above notation), if X and {e_{1}, e_{2}} are as in Example 6.7, and

f_{1} = \frac{1}{\sqrt{2}}(e_{1} + e_{2}) , f_{2} = \frac{1}{\sqrt{2}}(e_{1} − e_{2}),

then S_{1}^{X}, S_{2}^{X} are partial isometries, whereas T_{1} = S^{X}(f_{1}) and T_{2} = S^{X}(f_{2}) are not. Thus, the unitary equivalence of the row (\underline{S}_{i}^{X}) does not determine the isomorphism class of the subproduct system X.

Proposition 9.1. Let X and Y be two subproduct systems with X(1) = E and Y (1) = F. Assume that {e_{i}}_{i∈I} is an orthonormal basis for E and that {f_{i}}_{i∈I} is an orthonormal basis for F. Then the shifts (S_{i}^{X})_{i∈I} and (S_{i}^{Y})_{i∈I} are unitarily equivalent as rows (i.e., there exists a unitary V : 𝔉_{X} → 𝔉_{Y} such that V S_{i}^{X} = S_{i}^{Y} V for all i ∈ I), if and only if there is an isomorphism of subproduct systems W : X → Y such that W e_{i} = f_{i} for all i ∈ I.

Proof. If X and Y are isomorphic with the isomorphism W sending e_{i} to f_{i}, then define a unitary V : 𝔉_{X} → 𝔉_{Y} by

V = ⨁_{n∈ℕ} W |_{X(n)}.

V S_{i}^{X} = S_{i}^{Y} V follows from the properties of W. Conversely, a unitary V intertwining \underline{S}^{X} and \underline{S}^{Y} must send Ω_{X} to Ω_{Y}. Indeed, such a unitary must send {Ω_{X}}^{⊥} (which is equal to ∨_{i}ImS_{i}^{X}) onto a subspace of {Ω_{Y} }^{⊥} that has codimension 1 in 𝔉_{Y}, thus it must send {Ω_{X}}^{⊥} onto {Ω_{Y} }^{⊥}. It follows that V Ω_{X} = Ω_{Y}. Thus, given a unitary V intertwining \underline{S}^{X} and \underline{S}^{Y}, we may define W |X(n) : X(n) → Y (n) by

W S_{α}^{X}Ω = V S_{α}^{X}Ω = S_{α}^{Y} Ω,

for all |α| = n, and it is easy to see that the maps W |_{X(n)} assemble to form an isomorphism of subproduct systems. □

In the example preceding the proposition, we saw how the shift “tuple” (S_{1}^{X}, S_{2}^{X}) depends essentially on the choice of basis in E. However, the closed unital algebra generated by (S_{1}^{X}, S_{2}^{X}) is isomorphic to the one generated by (T_{1}, T_{2}). Similar remarks hold for E_{X} and T_{X}.

Example 9.2. Let X be the subproduct system given by X(0) = ℂ, X(1) = ℂ^{2} and X(n) = 0 for all n ≥ 2. Let Y be the subproduct system given by Y (0) = Y (1) = Y (2) = ℂ and Y (n) = 0 for all n ≥ 3. Then since E_{X} and E_{Y} contain the compact operators on F_{X} and F_{Y} (the Fock spaces), we have E_{X} = T_{X} ≅ M_{3}(ℂ) ≅ T_{Y} = E_{Y}. On the other hand, let {e_{1}, e_{2}} be an orthonormal basis for X(1). Then if Ω is the vacuum vector, then A_{X} is generated by S^{X}(Ω) = I, S^{X}(e_{1}), S^{X}(e_{2}). In the base {Ω, e_{1}, e_{2}} for F_{X}, these operators have the form

(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , (\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}) .

Thus,

A_{X} ≅ \{(\begin{matrix}a & 0 & 0 \\ b & a & 0 \\ c & 0 & a \end{matrix}) |a, b, c ∈ ℂ\} .

On the other hand, A_{Y} is generated by

I = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , S^{Y} (f_{1}) = (\begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) , (S^{Y} (f_{1}))^{2} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}) ,

where {f_{1}} is an orthonormal basis for Y (1). Thus

A_{Y} ≅ \{(\begin{matrix}a & 0 & 0 \\ b & a & 0 \\ c & b & a \end{matrix}) |a, b, c ∈ ℂ\} .

So A_{X} ≇ A_{Y} (in A_{X} the solutions of T^{2} = 0 form a two dimensional subspace, and in A_{Y} they form a one dimensional subspace).

9.2. A_{X} as a graded algebra. For every subproduct system X there exists a unique completely contractive multiplicative linear functional ρ_{0} : E_{X} → ℂ that sends λI to λ and S_{α}^{X} to 0 when |α| > 0. The existence of ρ_{0} follows from Theorem 8.2 (using the Poisson Transform), but it is also clear that ρ_{0} is just the vector state associated with the vacuum vector Ω_{X}:

ρ_{0}(T) = ⟨T Ω_{X}, Ω_{X}⟩ , T ∈ A_{X}.

ρ_{0} can be considered also as a conditional expectation ρ_{0} : A_{X} → ℂ · Ω_{X}, inducing a direct sum

(9.1)  A_{X} = ρ_{0}A_{X} ⊕ ker ρ_{0} = ℂ · I ⊕ ∑_{i} S_{i}^{X}A_{X}.

A_{X} contains a dense graded subalgebra, with the homogeneous elements of degree n being S^{X}(ξ), where ξ ∈ X(n). To be precise, we have the following proposition.

Proposition 9.3. Every T ∈ A_{X} can be written in a unique way as

T = ∑_{n=0}^{∞} T_{n},

where T_{n} ∈ \overline{span}\{S^{X}(ξ) : ξ ∈ X(n)\} and the sum is Cesaro convergent in the norm topology.

Proof. The proof uses a familiar gadget in operator algebra theory, the gauge action of the torus. For every t ∈ [−π, π], let W_{t} : X → X be the subproduct system automorphism given by

X(n) ∋ ξ ↦ e^{int}ξ ∈ X(n).

The gauge action on A_{X} is given by

γ_{t}(T) = W_{t}T W_{t}^{∗} , T ∈ A_{X}.

Note that if α ∈ I^{n}, then

γ_{t}(S_{α}^{X}) = e^{int}S_{α}^{X},

and it follows also that for all T ∈ \overline{span}\{S^{X}(ξ) : ξ ∈ X(n)\},

γ_{t}(T) = e^{int}T.

Moreover, for all T ∈ A_{X}, the path t ↦ γ_{t}(T) is strong operator continuous. Given T ∈ A_{X}, we define

Φ_{n}(T) = \frac{1}{2π} ∫_{−π}^{π} γ_{t}(T)e^{−int}dt,

where this is interpreted in the strong operator sense. It is easy to show that Φ_{n} is an idempotent whose range is \overline{span}\{S^{X}(ξ) : ξ ∈ X(n)}. Define linear maps on A_{X} by

Ψ_{N}(T) = ∑_{n=0}^{N} (1 − \frac{n}{N} ) Φ_{n}(T).

It is then a standard argument (using the Fejer kernel) to prove that ∑_{n} Φ_{n}(T) is Cesaro convergent to T in the norm topology, that is, to show that for all T ∈ A_{X},

∥Ψ_{N}(T) − T ∥ ⟶^{N→∞} 0.

It remains to prove the uniqueness assertion. Assume that T = ∑_{n} T_{n}, where the sum is Cesaro convergent to T, and T_{n} ∈ \overline{span}\{S^{X}(ξ) : ξ ∈ X(n)\}. Then for all N > n,

Φ_{n} ( ∑_{m=0}^{N} (1 − \frac{m}{N} ) T_{m}) = (1 − \frac{n}{N} ) T_{n} ⟶^{N→∞} T_{n}.

On the other hand,

Φ_{n} ( ∑_{m=0}^{N} (1 − \frac{m}{N} ) T_{m}) ⟶^{N→∞} Φ_{n}(T),

whence T_{n} = Φ_{n}(T). □

9.3. Vacuum state preserving isometric isomorphisms of A_{X}.

Lemma 9.4. Let φ : A_{X} → A_{Y} be an isometric isomorphism. Then φ is unital.

Proof. A theorem of Arazy and Solel [1] implies that an isometric map between A_{X} and A_{Y} must send I ∈ A_{X} to an isometry in A_{X} ∩ A_{X}^{∗}. It follows that φ(I) = cI, |c| = 1. But since φ is a homomorphism, then c = 1. □

Lemma 9.5. For all n ∈ ℕ, ξ ∈ X(n)

∥S^{X}(ξ)∥ = ∥S^{X}(ξ)Ω_{X}∥ = ∥ξ∥.

Proof. Because S^{X}(ξ) maps the orthogonal summands X(k) of 𝔉_{X} into the orthogonal summands X(k + n), it suffices to show that for all η ∈ X(k), ∥S^{X}(ξ)η∥ ≤ ∥ξ∥∥η∥ (because S^{X}(ξ)Ω_{X} = ξ). Now, S^{X}(ξ)η = p_{n+k}^{X}(ξ ⊗ η), thus

∥S^{X}(ξ)η∥^{2} ≤ ∥ξ ⊗ η∥^{2} = ∥ξ∥^{2}∥η∥^{2}.

□

Lemma 9.6. Let φ : A_{X} → A_{Y} be an isometric isomorphism that preserves the direct sum decomposition (9.1). Then φ preserves the grading: if ξ ∈ X(n) then φ(S^{X}(ξ)) is in the norm closure of span{S^{Y} (η) : η ∈ Y (n)}.

Proof. Since φ is a homomorphism, it suffices to show, say, that φ(S_{1}^{X}) has “degree one”, that is, it is in the norm closure of span{S^{Y} (η) : η ∈ Y (1)}. By assumption, we may write φ(S_{1}^{X}) = ∑_{i} a_{i}S_{i}^{Y} + T, with T in the closure of span{S^{Y} (η) : η ∈ Y (n), n ≥ 2}. But φ^{−1}(∑_{i} a_{i}S_{i}^{Y} + T) = S_{1}^{X}, and φ^{−1}(T) is in the norm closure of span{S^{X}(ξ) : η ∈ X(n), n ≥ 2}, so φ^{−1}(∑_{i} a_{i}S_{i}^{Y}) = S_{1}^{X} + B, with B = −φ^{−1}(T) (note that φ^{−1} also preserves the direct sum decomposition (9.1)). If T = 0 then we are done, so assume T ≠ 0. Then B ≠ 0, also. But

1 = ∥S_{1}^{X}∥ = ∥S_{1}^{X}Ω_{X}∥ < ∥(S_{1}^{X} + B)Ω_{X}∥ ≤ ∥S_{1}^{X} + B∥ = ∥ ∑_{i} a_{i}S_{i}^{Y} ∥,

and at the same time

∥ ∑_{i} a_{i}S_{i}^{Y} ∥ = ∥ ∑_{i} a_{i}S_{i}^{Y} Ω_{Y} ∥ < ∥(∑_{i} a_{i}S_{i}^{Y} + T)Ω_{Y} ∥ ≤

≤ ∥ ∑_{i} a_{i}S_{i}^{Y} + T ∥ = ∥S_{1}^{X}∥ = 1.

From T ≠ 0 we arrived at 1 < 1, thus T = 0. □

Theorem 9.7. X ≅ Y if and only if A_{X} and A_{Y} are isometrically isomorphic with an isomorphism that preserves the direct sum decomposition (9.1), and this happens if and only if A_{X} and A_{Y} are isometrically isomorphic with a grading preserving isomorphism. In fact, if φ : A_{X} → A_{Y} is a grading preserving isometric isomorphism then there is an isomorphism V : X → Y such that for all T ∈ A_{X}, φ(T) = V T V^{∗}.

Proof. X ≅ Y implies A_{X} ≅ A_{Y} because these algebras are then generated by unitarily equivalent tuples. For the converse, we will assume that X and Y are standard subproduct systems. The isomorphism V : X → Y is defined on the fiber X(n) by

V (ξ) = V (S^{X}(ξ)Ω_{X}) = φ(S^{X}(ξ))Ω_{Y} , ξ ∈ X(n).

If it is well defined, then it is onto. Lemma 9.5 shows that V is an isometry on the fibers:

∥S^{X}(ξ)Ω_{X}∥ = ∥S^{X}(ξ)∥ = ∥φ(S^{X}(ξ))∥ = ∥φ(S^{X}(ξ))Ω_{Y} ∥.

Lemma 9.6 implies that V (ξ) sits in Y (n). V respects the subproduct structure: if m, n ∈ ℕ, ξ ∈ X(n), η ∈ X(m), then

V p_{m,n}^{X}(ξ ⊗ η) = V S^{X}(p_{m,n}^{X}(ξ ⊗ η))Ω_{X}
= φ(S^{X}(p_{m,n}^{X}(ξ ⊗ η)))Ω_{Y}
= φ(S^{X}(ξ)S^{X}(η))Ω_{Y}
= φ(S^{X}(ξ))φ(S^{X}(η))Ω_{Y}
(∗) = p_{m,n}^{Y} (φ(S^{X}(ξ))Ω_{Y} ⊗ φ(S^{X}(η))Ω_{Y} )
= p_{m,n}^{Y}(V (ξ) ⊗ V (η)).

(*) follows from the facts S^{Y} (y)Ω_{Y} = y and S^{Y} (y_{1})S^{Y} (y_{2})Ω_{Y} = S^{Y} (p_{m,n}^{Y}(y_{1} ⊗ y_{2}))Ω_{Y} = p_{m,n}^{Y}(y_{1} ⊗ y_{2}) = p_{m,n}^{Y}(S^{Y} (y_{1})Ω_{Y} ⊗ S^{Y} (y_{2})Ω_{Y}). Finally, let us show that for all T ∈ A_{X}, φ(T) = V T V^{∗}. What we mean by this is that for all ξ ∈ X, φ(S^{X}(ξ)) = V S^{X}(ξ)V^{∗}. Let φ(S^{X}(η))Ω_{Y} = V (η) be a typical element in 𝔉_{Y}.

V S^{X}(ξ)V^{∗}φ(S^{X}(η))Ω_{Y} = V S^{X}(ξ)η
= V p^{X}(ξ ⊗ η)
= φ(S^{X}(p^{X}(ξ ⊗ η)))Ω_{Y}
= φ(S^{X}(ξ)S^{X}(η))Ω_{Y}
= φ(S^{X}(ξ))φ(S^{X}(η))Ω_{Y} ,

This completes the proof. □

10. Classification of the universal algebras of q-commuting tuples

Definition 10.1. A matrix q is called admissible if q_{ii} = 0 and 0 ≠ q_{ij} = q_{ji}^{−1} for all i ≠ j.

10.1. The q-commuting algebras A_{q} and their universality. Let {e_{1}, …, e_{d}} be an orthonormal basis for E := ℂ^{d}, to be fixed (together with d) throughout this section. Let q ∈ M_{d}(ℂ) be an admissible matrix, and let X_{q} be the maximal standard subproduct system with fibers

X_{q}(1) = E , X_{q}(2) = E ⊗ E ⊖ span{e_{i} ⊗ e_{j} − q_{ij}e_{j} ⊗ e_{i} : 1 ≤ i, j ≤ d, i ≠ j}.

When q_{ij} = 1 for all i < j, then X_{q} is the symmetric subproduct system SSP. The Fock spaces 𝔉_{X_{q}} have been studied in [19]. For brevity, we shall write S_{i}^{q} instead of S_{i}^{X_{q}}. We denote by A_{q} the algebra A_{X_{q}}. By Theorem 8.4, the algebra A_{q} is the universal norm closed unital operator algebra generated by a row contraction (T_{1}, …, T_{d}) satisfying the relations

T_{i}T_{j} = q_{ij}T_{j}T_{i} , 1 ≤ i < j ≤ d.

10.2. The character space of A_{q}. Let M_{q} be the space of all (contractive) multiplicative and unital linear functionals on A_{q}, endowed with the weak-∗ topology. We shall call M_{q} the character space of A_{q}. Every ρ ∈ M_{q} is uniquely determined by the d-tuple of complex numbers (x_{1}, …, x_{d}), where x_{i} = ρ(S_{i}^{q}) for i = 1, …, d. Since a contractive linear functional is completely contractive, (x_{1}, …, x_{d}) must be a row contraction, that is, |x_{1}|^{2} +…+|x_{d}|^{2} ≤ 1. In other words, (x_{1}, …, x_{d}) is in the unit ball B_{d} of ℂ^{d}. The multiplicativity of ρ implies that (x_{1}, …, x_{d}) must lie inside the set

Z_{q} := {(z_{1}, …, z_{d}) ∈ B_{d} : (1 − q_{ij})z_{i}z_{j} = 0, 1 ≤ i < j ≤ d}.

Conversely, Theorem 8.4 implies that every (x_{1}, …, x_{d}) ∈ Z_{q} gives rise to a character ρ ∈ M_{q} that sends S_{i}^{q} to x_{i}. Thus the map

M_{q} ∋ ρ ↦ (ρ(S_{1}^{q}), …, ρ(S_{d}^{q})) ∈ Z_{q}

is injective and surjective. It is also obviously continuous (with respect to the weak-∗ and standard topologies). Since M_{q} is compact, we have the homeomorphism

(10.1)  M_{q} ≅ Z_{q}.

Note that the vacuum state ρ_{0} corresponds to the point 0 ∈ Z_{q} ⊂ ℂ^{d}. When q_{ij} = 1, the condition (1 − q_{ij})z_{i}z_{j} = 0 is trivially satisfied, so when q_{i,j} = 1 for all i, j, then Z_{q} is the unit ball B_{d}. When q_{ij} ≠ 1, the condition is that either z_{i} = 0 or z_{j} = 0. Thus, if for all i, j, q_{ij} ≠ 1, then Z_{q} is the union of d discs glued together at their origins.

10.3. Classification of the A_{q}, q_{ij} ≠ 1. Given a permutation σ (on a set with d elements), let U_{σ} be the matrix that induces the same permutation on the standard basis of ℂ^{d}.

Proposition 10.2. Let q and r be two admissible d × d matrices. Assume that there is a permutation σ ∈ S_{d} such that r = U_{σ}qU_{σ}^{−1}, and let λ_{1}, …, λ_{d} be any complex numbers on the unit circle. Then the map

(10.2)  e_{i} ↦ λ_{i}e_{σ(i)}

extends to an isomorphism of X_{q} onto X_{r}, and thus the map

S_{i}^{q} ↦ λ_{i}S_{σ(i)}^{r}

extends to a completely isometric isomorphism between A_{q} and A_{r}.

Proof. For all n, the map (10.2) extends to a unitary V_{n} of E^{⊗n}. For n = 2, this unitary sends e_{i} ⊗ e_{j} − q_{ij}e_{j} ⊗ e_{i} to λ_{i}λ_{j}e_{σ(i)} ⊗ e_{σ(j)} − λ_{i}λ_{j}q_{ij}e_{σ(j)} ⊗ e_{σ(i)}. But r = U_{σ}qU_{σ}^{−1} implies r_{σ(i)σ(j)} = q_{ij}, thus

V_{2} : e_{i} ⊗ e_{j} − q_{ij}e_{j} ⊗ e_{i} ↦ λ_{i}λ_{j}e_{σ(i)} ⊗ e_{σ(j)} − λ_{i}λ_{j}r_{σ(i)σ(j)}e_{σ(j)} ⊗ e_{σ(i)},

so V_{2} is a unitary between X_{q}(2) and X_{r}(2) that respects the product. By induction, it follows that V = {V_{n}|_{X_{q}(n)}}_{n} is an isomorphism of subproduct systems. The final assertion follows from Proposition 9.1. □

Theorem 10.3. Let q and r be two admissible d×d matrices such that q_{ij}, r_{ij} ≠ 1 for all i, j. Then X_{q} is isomorphic to X_{r} if and only if there is a permutation σ ∈ S_{d} such that r = U_{σ}qU_{σ}^{−1}. In this case the isomorphisms are precisely those of the form

e_{i} ↦ λ_{i}e_{σ(i)},

where λ_{1}, …, λ_{d} are any complex numbers on the unit circle, and σ is such that r = U_{σ}qU_{σ}^{−1}.

Proof. One direction is Proposition 10.2, so assume that there is an isomorphism of subproduct systems V : X_{q} → X_{r}. Let f_{i} := V^{−1}e_{i}. There is a d × d unitary matrix U = (u_{ij}) such that f_{i} = ∑_{j} u_{ij}e_{j}. As V is an isomorphism of subproduct systems, we have for all i ≠ j

V p_{2}^{X_{q}}(f_{i} ⊗ f_{j} − r_{ij}f_{j} ⊗ f_{i}) = p_{2}^{X_{r}}(e_{i} ⊗ e_{j} − r_{ij}e_{j} ⊗ e_{i}) = 0,

thus

(∑_{k} u_{ik}e_{k}) ⊗ (∑_{l} u_{jl}e_{l}) − r_{ij}(∑_{k} u_{jk}e_{k}) ⊗ (∑_{l} u_{il}e_{l}) ∈

∈ span{e_{m} ⊗ e_{n} − q_{mn}e_{n} ⊗ e_{m} : m ≠ n},

or

(10.3) ∑_{k,l}(u_{ik}u_{jl} − r_{ij}u_{jk}u_{il})e_{k} ⊗ e_{l} ∈ span{e_{m} ⊗ e_{n} − q_{mn}e_{n} ⊗ e_{m} : m ≠ n}.

The coefficients of the vectors e_{k} ⊗ e_{k} in the sum above must vanish, thus u_{ik}u_{jk} − r_{ij}u_{jk}u_{ik} = 0 for all i ≠ j. Since r_{ij} ≠ 1, we must have u_{jk}u_{ik} = 0 for all k and all i ≠ j. Thus the unitary matrix U has precisely one nonzero element in each column, and it therefore must be of the form U_{σ}^{−1}D, where D is a diagonal unitary matrix. Equation (10.3) becomes

u_{iσ(i)}u_{jσ(j)}e_{σ(i)} ⊗ e_{σ(j)} − r_{ij}u_{jσ(j)}u_{iσ(i)}e_{σ(j)} ⊗ e_{σ(i)} ∈ ∈ span{e_{m} ⊗ e_{n} − q_{mn}e_{n} ⊗ e_{m} : m ≠ n},

but this can only happen if

u_{iσ(i)}u_{jσ(j)}e_{σ(i)} ⊗ e_{σ(j)} − r_{ij}u_{jσ(j)}u_{iσ(i)}e_{σ(j)} ⊗ e_{σ(i)}

is proportional to

e_{σ(i)} ⊗ e_{σ(j)} − q_{σ(i)σ(j)}e_{σ(j)} ⊗ e_{σ(i)},

that is u_{iσ(i)}u_{jσ(j)}q_{σ(i)σ(j)} = u_{jσ(j)}u_{iσ(i)}r_{ij}, or r_{ij} = q_{σ(i)σ(j)}. Replacing σ with σ^{−1}, the proof is complete. □

Corollary 10.4. Let q be an admissible d × d matrix such that there is no permutation σ ∈ S_{d} such that q = U_{σ}qU_{σ}^{−1}. Assume that q_{ij} ≠ 1 for all i, j. Then the only automorphisms of X_{q} are unitary scalings of the basis {e_{1}, …, e_{d}}.

Theorem 10.5. Let q and r be two admissible d×d matrices such that q_{ij}, r_{ij} ≠ 1 for all i, j. Then A_{q} is isometrically isomorphic to A_{r} if and only if there is a permutation σ ∈ S_{d} such that r = U_{σ}qU_{σ}^{−1}. In this case the isometric isomorphisms between A_{q} and A_{r} are precisely those of the form

S_{i}^{q} ↦ λ_{i}S_{σ(i)}^{r},

where λ_{1}, …, λ_{d} are any complex numbers on the unit circle.

Proof. If r = U_{σ}qU_{σ}^{−1}, then by Proposition 10.2 and Theorem 9.7 A_{q} and A_{r} are isomorphic (with an isomorphism that preserves the direct sum decomposition (9.1)). Conversely, assume that φ : A_{q} → A_{r} is a completely isometric isomorphism. Then φ induces a homeomorphism between M_{r} and M_{q} by ρ ↦ ρ ∘ φ. Recall that M_{q} and M_{r} are both homeomorphic to d discs glued together at the origin. Thus the homeomorphism ρ ↦ ρ ∘ φ must take ρ_{0} of X_{r} to ρ_{0} of X_{q}, because these are the unique points in M_{r} and M_{q}, respectively, that when removed from M_{r} and M_{q} leave d disconnected punctured discs. Thus φ sends the vacuum state of A_{r} to the vacuum state of A_{q}, and must therefore preserve the direct sum decomposition (9.1). By Theorem 9.7, there is an isomorphism of subproduct systems V : X_{q} → X_{r} such that φ(•) = V • V^{∗}. By Theorem 10.3 we conclude that there is a permutation σ ∈ S_{d} such that r = U_{σ}qU_{σ}^{−1}. It also follows that φ(S_{i}^{q}) = λ_{i}S_{σ(i)}^{r}. □

Corollary 10.6. Let q be an admissible d × d matrix such that there is no permutation σ ∈ S_{d} such that q = U_{σ}qU_{σ}^{−1}. Then the only isometric automorphisms of A_{q} are unitary scalings of the shift {S_{1}^{q}, …, S_{d}^{q}}.

As a corollary of the above discussion we have:

Corollary 10.7. Let q and r be two admissible d × d matrices such that q_{ij}, r_{ij} ≠ 1 for all i, j. Then A_{q} is isometrically isomorphic to A_{r} if and only if X_{q} ≅ X_{r}.

10.4. X_{q} and A_{q}, d = 2. In the particular case d = 2, we let a complex number q parameterize the spaces X_{q} (we may allow also q = 0) defined to be the maximal standard subproduct system with fibers

X_{q}(1) = ℂ^{2} , X_{q}(2) = ℂ^{2} ⊗ ℂ^{2} ⊖ span{e_{1} ⊗ e_{2} − qe_{2} ⊗ e_{1}}.

Since M_{1} ≅ B_{2}, A_{1} is not isomorphic to any A_{q} with q ≠ 1 (recall that when q ≠ 1, M_{q} is homeomorphic to two discs glued together at the origin). Thus Theorem 10.5 gives:

Corollary 10.8. Assume that d = 2. Then X_{q} ≅ X_{r} if and only if A_{q} is isometrically isomorphic to A_{r}, and either one of these happens if and only if either r = q or r = q^{−1}.

Elias Katsoulis has pointed out to us that the above corollary also follows from the techniques of [18].

The above result is reminiscent to the fact that two rotation algebras A_{θ} and A_{θ′} are isomorphic if and only if either e^{2πiθ} = e^{2πiθ′} or (e^{2πiθ})^{−1} = e^{2πiθ′}. One cannot help but wonder whether one can draw a deeper connection between these results then the superficial one, in particular, can the classification of rotation algebras be deduced from the classification of the algebras A_{q}? By Corollaries 10.4 and 10.6 we have the following.

Corollary 10.9. Let d = 2 and let q ≠ 1. Then subproduct system X_{q} has no automorphisms aside form the unitary scalings of the basis. The algebra A_{q} has no isometric automorphisms other than unitary scalings of the generators.

On the other hand, a direct calculation shows that every unitary on ℂ^{2} extends to an automorphism of X_{1}, and thus induces a non-obvious automorphism of A_{1}.

11. Standard maximal subproduct systems with dim X(1) = 2 and dim X(2) = 3

Again, let {e_{1}, …, e_{d}} be an orthonormal basis for E := ℂ^{d}. We will soon turn attention to the case d = 2. For a matrix A ∈ M_{d}(ℂ), we define the symmetric part of A to be A^{s} := (A + A^{t})/2 and the antisymmetric part of A to be A^{a} := (A − A^{t})/2. Denote by X_{A} the maximal standard subproduct system with fibers

X_{A}(1) = E , X_{A}(2) = E ⊗ E ⊖ span { ∑_{i,j=1}^{d} a_{ij}e_{i} ⊗ e_{j}} .

We will write S^{A} for the shift S^{X_{A}}. We will also write A_{A} for A_{X_{A}}.

Proposition 11.1. Let A, B ∈ M_{d}(ℂ). Then there is an isomorphism V : X_{A} → X_{B} if and only if there exists λ ∈ ℂ and a unitary d × d matrix U such that B = λU^{t}AU. In this case, U extends to the isomorphism V between X_{A} and X_{B} by V_{1} = U.

Proof. Let V : X_{A} → X_{B} be an isomorphism of subproduct systems. There is a d × d unitary matrix U = (u_{ij}) such that

f_{i} := V_{1}(e_{i}) = ∑_{j=1}^{d} u_{ij}e_{j}.

Then

0 = V_{1}(p_{2}^{X}(∑_{i,j} a_{ij}e_{i} ⊗ e_{j}))

= p_{2}^{Y} (∑_{i,j} a_{ij}f_{i} ⊗ f_{j}),

so ∑_{i,j} a_{ij}f_{i} ⊗f_{j} must be a spanning vector of span {∑_{i,j} b_{ij}e_{i} ⊗ e_{j}}. Writing out fully what this means,

λ ∑_{i,j} a_{ij} ∑_{k,l} u_{ik}u_{jl}e_{k} ⊗ e_{l} = ∑_{k,l} b_{kl}e_{k} ⊗ e_{l}

for some λ ∈ ℂ, so

b_{kl} = λ ∑_{i,j} a_{ij}u_{ik}u_{jl}.

But the right hand side is precisely the kl-th element of λU^{t}AU. Conversely, assuming B = λU^{t}AU, one can read the above argument from finish to start to obtain an isomorphism V : X_{A} → X_{B}. □

We see that for X_{A} and X_{B} to be isomorphic the ranks of A and B must be the same, as well as the ranks of their symmetric and anti-symmetric parts. For example, if A is symmetric and B is not then X_{A} ≇ X_{B}, a result which may not seem obvious at first glance.

Theorem 11.2. Assume that d = 2. Let A, B ∈ M_{2}(ℂ) be any two matrices. Then A_{A} is isometrically isomorphic to A_{B} if and only if X_{A} ≅ X_{B}, and this happens if and only if there exists λ ∈ ℂ and a unitary 2 × 2 matrix U such that B = λU^{t}AU.

The proof of Theorem 11.2 will occupy the rest of this section. Denote by M_{A} the character space of A_{A}, that is, the topological space of contractive multiplicative and unital linear functionals on A_{A}, endowed with the weak-∗ topology.

Lemma 11.3. The topology of M_{A} depends on the rank r(A^{s}) of the symmetric part A^{s} of A:

(1) If r(A^{s}) = 0 then M_{A} ≅ B_{2}, the unit ball in ℂ^{2}.

(2) If r(A^{s}) = 1 then M_{A} ≅ D, the unit disc in ℂ.

(3) If r(A^{s}) = 2 then M_{A} is homeomorphic to two discs pasted together at the origin.

Proof. We proceed similarly to the lines of 10.2. Every character ρ ∈ M_{A} is uniquely determined by λ_{1} = ρ(S_{1}^{A}) and λ_{2} = ρ(S_{2}^{A}), which lie in B_{2}. Conversely, every (λ_{1}, λ_{2}) ∈ B_{2} that satisfies

∑_{i,j} a_{ij}λ_{i}λ_{j} = 0

gives rise to a character ρ by defining λ_{1} = ρ(S_{1}^{A}) and λ_{2} = ρ(S_{2}^{A}). Thus,

M_{A} ≅ V_{A} := {(λ_{i}, λ_{j}) ∈ B_{2} : ∑_{i,j} a_{ij}λ_{i}λ_{j} = 0} .

Clearly, V_{A} = V_{A^{s}}. However, every symmetric 2×2 matrix is complex congruent to one of the following:

D_{0} = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) , D_{1} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) or D_{2} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ,

i.e., there exists a nonsingular matrix T such that A^{s} = T^{t}D_{i}T, for i = r(A^{s}). But then V_{A^{s}} = T^{−1}V_{D_{i}} ≅ V_{D_{i}}, so it remains to verify that V_{D_{i}} is homeomorphic to the spaces listed in the statement of the lemma. □

Corollary 11.4. If r(A^{s}) ≠ r(B^{s}) then A_{A} ≇ A_{B}.

We can use this corollary to break down the classification of the algebras A_{A} to the classification of the algebras A_{A} with fixed r(A^{s}). The easiest case is

r(A^{s}) = 0, because then A is either the zero matrix or a multiple of (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}),

and these two matrices give rise to non isomorphic algebras (these are the algebras generated by the full and symmetric shifts, respectively). The next easiest case is r(A^{s}) = 2.

Lemma 11.5. If A, B ∈ M_{2}(ℂ) and r(A^{s}) = r(B^{s}) = 2, then A_{A} is isometrically isomorphic to A_{B} if and only if X_{A} ≅ X_{B}, and this happens if and only if there exists λ ∈ ℂ and a unitary 2 × 2 matrix U such that B = λU^{t}AU. Any isometric isomorphism between A_{A} and A_{B} arises as conjugation by the subproduct system isomorphism arising from U.

Proof. In light of Theorem 9.7 and Proposition 11.1, it suffices to show that any isometric isomorphism φ : A_{A} → A_{B} sends the vacuum state to the vacuum state. But the vacuum state in M_{A} and in M_{B} corresponds to the point where the two discs are glued together. Since φ induces a homeomorphism between M_{B} and M_{A}, it must send the vacuum state to the vacuum state. □

Remark 11.6. In the previous section we have seen already that there is a continuum of non-(completely isometrically)-isomorphic algebras A_{A} and subproduct systems X_{A} with r(A^{s}) = 2, namely the algebras A_{q}. One can see that these algebras A_{A} are not exhausted by the algebras A_{q} of the previous

section. For example, all the algebras A_{A} with A = (\begin{matrix}1 & 0 \\ 0 & q \end{matrix}), with q > 0, are

non-isomorphic, and only for q = 1 is this algebra isomorphic to an A_{q} (in this case q = −1).

We now come to the trickiest case, r(A^{s}) = 1.

Lemma 11.7. If A, B ∈ M_{2}(ℂ) are two symmetric matrices of rank 1, then there exists λ ∈ ℂ and a unitary 2 × 2 matrix U such that B = λU^{t}AU, and consequently X_{A} ≅ X_{B} and A_{A} is isometrically isomorphic to A_{B}.

Proof. We only have to prove the first assertion, and we may assume that

B = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}). We may also assume that there is a unit vector v = (v_{1}, v_{2})^{t} such that A = vv^{t}. Now let

U = (\begin{matrix}\overline{v_{1}} & \overline{v_{2}} \\ \overline{v_{2}} & −\overline{v_{1}} \end{matrix}) .

Then

U^{t}AU = (\begin{matrix}\overline{v_{1}} & \overline{v_{2}} \\ \overline{v_{2}} & −\overline{v_{1}} \end{matrix})^{t} vv^{t} (\begin{matrix}\overline{v_{1}} & \overline{v_{2}} \\ \overline{v_{2}} & −\overline{v_{1}} \end{matrix}) = (\begin{matrix}\overline{v_{1}} & \overline{v_{2}} \\ \overline{v_{2}} & −\overline{v_{1}} \end{matrix})^{t} (\begin{matrix}v_{1} & 0 \\ v_{2} & 0 \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) .

□

Below we will also need the following lemma.

Lemma 11.8. Let A be a 2 × 2 matrix for which r(A^{s}) = 1. Then there exists one and only one q ≥ 0 for which there is a λ ∈ ℂ and a unitary U such that

(\begin{matrix}1 & q \\ −q & 0 \end{matrix}) = λU^{t}AU.

Furthermore, if A is non-symmetric then A is congruent to the matrix

(\begin{matrix}1 & 1 \\ −1 & 0 \end{matrix}) .

Proof. Direct verification, using Lemma 11.7 and the fact that congruations preserves, up to a scalar, the anti-symmetric part. □

Let us write A_{q} for the matrix

A_{q} = (\begin{matrix}1 & q \\ −q & 0 \end{matrix}) .

By the above lemma, we may restrict attention only to the algebras A_{A_{q}} with q ≥ 0. Recall that the character space M_{A_{q}} of A_{A_{q}} is identified with the closed unit disc \overline{𝔻} by

M_{A_{q}} ∋ ρ ⟷ ρ(S_{2}^{A_{q}}) ∈ \overline{𝔻}.

We write ρ_{z} for the character that sends S_{2}^{A_{q}} to z ∈ \overline{𝔻}. This identifies the vacuum vector ρ_{0} with the point 0. Recall also that if φ : A_{A_{q}} → A_{A_{r}} is an isometric isomorphism, then it induces a homeomorphism φ_{∗} : M_{A_{r}} → M_{A_{q}} given by φ_{∗}ρ = ρ ∘ φ. We write F_{φ} for the homeomorphism \overline{𝔻} → \overline{𝔻} induced by φ, that is, F_{φ} is the unique self map of \overline{𝔻} that satisfies

φ_{∗}ρ_{z} = ρ_{F_{φ}(z)} , z ∈ \overline{𝔻}.

Let us introduce the notation

O(0; q, r) = {F_{φ}(0)|φ : A_{A_{q}} → A_{A_{r}} is an isometric isomorphism},

and

O(0; q) = O(0; q, q).

Lemma 11.9. Let q, r ≥ 0. If q ≠ r then 0 does not lie in O(0; q, r).

Proof. Assume that 0 ∈ O(0; q, r). Then there is some isometric isomorphism φ : A_{A_{q}} → A_{A_{q}} that preserves the character ρ_{0}. It follows from Theorem 9.7 and Proposition 11.1 that, for some unitary 2 × 2 matrix U and some λ ∈ ℂ, A_{q} = λU^{t}A_{r}U. But, as noted in Lemma 11.8, this is impossible if r ≠ q. □

Lemma 11.10. The sets O(0; q, r) are invariant under rotations around 0.

Proof. For λ with |λ| = 1, write φ_{λ} for the isometric isomorphism mapping S_{i}^{A_{q}} to λS_{i}^{A_{q}} (i = 1, 2). For b = F_{φ}(0) ∈ O(0; q, r), consider φ ∘ φ_{λ}. We have ρ_{0}((φ ∘ φ_{λ})(S_{2}^{A_{q}})) = ρ_{0}(φ(λS_{2}^{A_{q}})) = λρ_{0}(φ(S_{2}^{A_{q}})) = λb. Thus λb ∈ O(0; q, r). □

Lemma 11.11. Let q, r ≥ 0. If q ≠ r then A_{A_{q}} is not isometrically isomorphic to A_{A_{r}}.

Proof. Assume that φ : A_{A_{q}} → A_{A_{r}} is an isometric isomorphism. We have ρ_{0} ∘ φ = ρ_{b}, with b = F_{φ}(0), and F_{φ} is a homeomorphism of \overline{𝔻} onto itself. By definition, b ∈ O(0; q, r). By Lemma 11.9, b ≠ 0. Denote C := {z : |z| = |b|}. By Lemma 11.10, C ⊆ O(0; q, r). Consider C′ := F_{φ}^{−1}(C). We have that C′ ⊆ O(0; r). C′ is a simply connected closed path in 𝔻 that goes through the origin. By Lemma 11.10, the interior of C′, int(C′), is in O(0; r). But then F_{φ}(int(C′)) is the interior of C, and it is in O(0; q, r). But then 0 ∈ O(0; q, r), contradicting Lemma 11.9. □

That concludes the proof of Theorem 11.2.

12. The representation theory of Matsumoto’s subshift C^{∗}-algebras

In [28] Kengo Matsumoto introduced a class of C^{∗}-algebras that arise from symbolic dynamical systems called “subshifts” (we note that in the later paper [17] Carlsen and Matsumoto suggest another way of associating a C^{∗}-algebra with a subshift. Here we are discussing only the algebras originally introduced in [28]). These subshift algebras, as we shall call them, are strict generalizations of Cuntz-Krieger algebras and have been extensively studied by Matsumoto, T. M. Carlsen and others. For example, the following have been studied: criteria for simplicity and pure-infiniteness; conditions on the underlying dynamical systems for subshift algebras to be isomorphic; the automorphisms of the subshift algebras; K-theory of the subshift algebras; and much more. In this section we will use the framework constructed in the previous sections to give a complete description of all representations of a subshift algebra when the subshift is of finite type.

12.1. Subshifts and the corresponding subproduct systems and C^{∗}- algebras. Our references for subshifts are [28] and [16, Chapter 3]. Let I = {1, 2, …, d} be a fixed finite set. I^{ℤ} is the space of all two-sided infinite sequences, endowed with the product topology. The left shift (or simply the shift) on I^{ℤ} is the homeomorphism σ : I^{ℤ} → I^{ℤ} given by (σ(x))_{k} = x_{k+1}. Let Λ be a shift invariant closed subset of I^{ℤ}. By this we mean σ(Λ) = Λ. The topological dynamical system (Λ, σ|_{Λ}) is called a subshift. Sometimes Λ is also referred to as the subshift. If W is a set of words in 1, 2, …, d, one can define a subshift by forbidding the words in W as follows:

Λ_{W} = {x ∈ I^{ℤ} : no word in W occurs as a block in x}.

Conversely, every subshift arises this way: i.e., for every subshift Λ there exists a collection of words W, called the set of forbidden words, such that Λ = Λ_{W}. In this context, if W can be chosen finite then Λ = Λ_{W} is called a subshift of finite type, or SFT for short. By replacing I if needed, we may always assume that W has no words of length one. If W can be chosen such that the longest word in W has length k + 1 then Λ is called a k-step SFT. A 1-step SFT is also called a topological Markov chain. A basic result is that every SFT is isomorphic to a topological Markov chain ([16, Proposition 3.2.1]). For a fixed subshift (Λ, σ|_{Λ}), we set

Λ^{k} = {α : α is a word with length k occurring in some x ∈ Λ},

and Λ_{l} = ∪_{k=0}^{l}Λ^{k}, Λ^{∗} = ∪_{k=0}^{∞}Λ^{k}. With the subshift (Λ, σ|_{Λ}) we associate a subproduct system X_{Λ} as follows. Let {e_{i}}_{i∈I} be an orthonormal basis of a Hilbert space E. We define

X_{Λ}(0) = ℂ,

and for n ≥ 1 we define

X_{Λ}(n) = span{e_{α} : α ∈ Λ^{n}}.

We define a product U_{m,n} : X_{Λ}(m) ⊗ X_{Λ}(n) → X_{Λ}(m + n) by

U_{m,n}(e_{α} ⊗ e_{β}) = \begin{cases}e_{αβ}, & if αβ ∈ Λ^{m+n} \\ 0, & else. \end{cases}

Since Λ^{m+n} ⊆ Λ^{m} · Λ^{n}, X_{Λ} is a standard subproduct system.

Definition 12.1. The C^{∗}-algebra associated with a subshift (Λ, σ|_{Λ}) is defined as the quotient algebra

O_{Λ} := O_{X_{Λ}} = T_{X_{Λ}}/K(𝔉_{X_{Λ}}).

Remark 12.2. Just to prevent confusion: In [28], O_{Λ} was defined as the quotient by the compacts of the C^{∗}-algebra generated by the “creation operators” (that is, the X-shift) on 𝔉_{X}, without using the language of subproduct systems.

12.2. Subproduct systems that come from subshifts.

Proposition 12.3. Let X be a standard subproduct system such that there is an orthonormal basis {e_{i}}_{i∈I} of X(1), with I finite, such that

(1) Every X(n), n ≥ 1, is spanned by vectors of the form e_{α} with |α| = n.

(2) For all m, n ∈ ℕ, |α| = n and e_{α} ∈ X(n), implies that there is some β, γ ∈ I^{m} such that e_{β} ⊗ e_{α} and e_{α} ⊗ e_{γ} are in X(m + n).

Then there is a shift invariant closed subset Λ of I^{ℤ} such that X = X_{Λ}. X is the maximal standard subproduct system with prescribed fibers X(1), X(2), …, X(k + 1) if and only if Λ is k-step SFT.

Proof. For all k ∈ ℕ, define

Λ^{(k)} = {α ∈ I^{k} : e_{α} ∈ X(k)}.

For all m ∈ ℤ, k ∈ ℕ, define the closed sets

A_{m,k} = {x ∈ I^{ℤ} : (x_{m}, x_{m+1}, …, x_{m+k−1}) ∈ Λ^{(k)}}.

Condition (2) implies that X(k) always contains a nonzero vector of the form e_{α}, |α| = k. That implies that the family {A_{m,k}}_{m,k} has the finite intersection property. Indeed,

A_{m_{1},k_{1}} ∩ A_{m_{2},k_{2}} ⊇ A_{M,K} ≠ ∅,

where M = min{m_{1}, m_{2}}, K = max{m_{2} + k_{2}, m_{1} + k_{1}} − M. By compactness of I^{ℤ} we conclude that the closed set

Λ := ⋂_{m,k} A_{m,k}

is non-empty. Λ is invariant under the left and the right shifts, so σ(Λ) = Λ, so (Λ, σ|_{Λ}) is a subshift. By condition (2), Λ^{k} = Λ^{(k)}. Condition (1) together with the definition of X_{Λ} now imply that X = X_{Λ}. The final assertion follows from the following facts, together with X = X_{Λ}. Fact number one:

E^{⊗n} ⊖ X_{Λ}(n) = span{e_{α} : α is a forbidden word of length n}.

Fact number two: X is the maximal standard subproduct system with prescribed fibers X(1), …, X(k + 1) if and only if for every n > k + 1,

X(n) = ⋂_{i+j=n} X(i) ⊗ X(j),

or in other words, if and only if

E^{⊗n} ⊖ X(n) = ⋁_{i+j=n} (E^{⊗n} ⊖ (X(i) ⊗ X(j)))

= ⋁_{i+j=n} (E^{⊗i} ⊗ (E^{⊗j} ⊖ X(j)) + (E^{⊗i} ⊖ X(i)) ⊗ E^{⊗j}) .

Fact number three: Λ is a k-step SFT if and only if for every n > k + 1,

{forbidden words of length n} =

⋃_{i+j=n}(I^{i} · {forbidden words of length j} ∪ {forbidden words of length i} · I^{j}) .

These facts assemble together to complete the proof. □

Not every subproduct system is isomorphic to one that comes from a subshift. Indeed, in the symmetric subproduct system SSP (see Example 1.3) for any basis {e_{i}}_{i∈I} of X(1), the product e_{i} ⊗ e_{j} for i ≠ j is never in X(2), and thus the images f_{i} and f_{j} of e_{i} and e_{j} in any isomorphic subproduct system X can never be such that f_{i} ⊗ f_{j} is mapped isometrically to U_{1,1}^{X}(f_{i} ⊗ f_{j}). Thus if SSP is isomorphic to X_{Λ} for some subshift Λ, then Λ must be the subshift containing only constant sequences. But such X_{Λ} is clearly not isomorphic to SSP. As another example, the subproduct system X(0) = ℂ, X(1) = ℂ^{2}, and X(n) = 0 for n > 1, cannot be of the form X_{Λ} for any Λ ⊆ I^{ℤ}.

12.3. The representation theory of the C^{∗}-algebra associated with a subshift of finite type. Let Λ be a fixed subshift in I^{ℤ} (with I = {1, 2, …, d}), and let X = X_{Λ} be the associated subproduct system. We will denote the X-shift by S (instead of S^{X}) to make some formulas more readable. Let Z_{i} be the image of S_{i} in the quotient O_{Λ}. We define for i ∈ I, k ∈ ℕ the sets

E_{i}^{k} = {α ∈ Λ^{k} : iα ∈ Λ^{∗}}.

Lemma 12.4. If Λ is a k-step SFT, then for all i ∈ I,

{γ ∈ Λ^{∗} : |γ| ≥ k, iγ ∈ Λ^{∗}} = {αβ ∈ Λ^{∗} : α ∈ E_{i}^{k}, β ∈ Λ^{∗}}.

Proof. Assume that γ ∈ Λ^{∗} is such that |γ| ≥ k and iγ ∈ Λ^{∗}. Defining α = γ_{1} ⋯ γ_{k} and β = γ_{k+1} ⋯ γ_{k+l}, we have that γ = αβ where α ∈ E_{i}^{k} and β ∈ Λ^{∗}. Conversely, if γ = αβ ∈ Λ^{∗} where α ∈ E_{i}^{k} and β ∈ Λ^{∗}, then iγ must be in Λ^{∗}. Indeed, if not, then iγ must contain a forbidden word. But γ ∈ Λ^{∗}, thus the forbidden word must be in iα (since Λ is a k-step SFT). But that is impossible because α ∈ E_{i}^{k}. □

Lemma 12.5. If Λ is a k-step SFT then for all i, j ∈ I, i ≠ j,

S_{i}^{∗}S_{j} = 0,

and

(12.1)  S_{i}^{∗}S_{i} = ∑_{α∈E_{i}^{k}} \underline{S}^{α}\underline{S}^{α∗} mod K_{X}.

Consequently, E_{X} = T_{X}.

Proof. Since the S_{i} are partial isometries with orthogonal ranges, we have S_{i}^{∗}S_{j} = 0 for all i ≠ j. Since K_{X} ⊆ E_{X} ⊆ T_{X} (Proposition 8.1), E_{X} = T_{X} will be established once we prove (12.1). S_{i}^{∗}S_{i} is the projection onto the initial space of S_{i}. Call this space G. We have

G = span{e_{α} : α ∈ Λ^{∗} such that iα ∈ Λ^{∗}}.

The space

G′ = span{e_{α} : α ∈ Λ^{∗} such that iα ∈ Λ^{∗} and |α| ≥ k}

has finite codimension in G. But by Lemma 12.4,

G′ = {e_{αβ} : αβ ∈ Λ^{∗}, α ∈ E_{i}^{k}},

that is, G′ is spanned by e_{γ} where γ runs through all legal words beginning with some α ∈ E_{i}^{k}. Thus, G′ is the range of the projection ∑_{α∈E_{i}^{k}} \underline{S}^{α}\underline{S}^{α^{∗}}. Since G′ has finite codimension in G, we have (12.1). □

Proposition 12.6. For every subshift Λ, the d-tuple \underline{Z} = (Z_{1}, …, Z_{d}) satisfies the following relations:

(12.2)  p(\underline{Z}) = 0 , for all p ∈ I^{X},
(12.3)  Z_{i}^{∗}Z_{j} = 0 , for all i, j ∈ I , i ≠ j,

and

(12.4)  ∑_{i=1}^{d} Z_{i}Z_{i}^{∗} = 1.

In particular, Z_{i} is a partial isometry for all i ∈ I. If Λ is a k-step SFT, the \underline{Z} also satisfies

(12.5)  Z_{i}^{∗}Z_{i} = ∑_{α∈E_{i}^{k}} \underline{Z}^{α}\underline{Z}^{α∗} , for all i ∈ I.

Proof. The quotient map T_{X} → O_{Λ} is a ∗-homomorphism, so (12.2) follows from Theorem 7.5. (12.3) and (12.5) follow from the previous lemma, and (12.4) follows ¿¿from equation (8.1). □

Theorem 12.7. Let Λ be a k-step SFT. Every unital representation π : O_{Λ} → B(H) is determined by a row-contraction \underline{T} = (T_{1}, …, T_{d}) satisfying relations (12.2)-(12.5) such that π(Z_{i}) = T_{i} for all i ∈ I. Conversely, every row contraction in B(H)^{d} satisfying the relations (12.2)-(12.5) gives rise to a unital representation π : O_{Λ} → B(H) such π(Z_{i}) = T_{i} for all i ∈ I.

Proof. It is the second assertion that is non-trivial, and we will try to convince that it is true. By Theorem 8.2, there is unital completely positive map

Ψ : E_{X} → B(H)

sending \underline{S}^{α}\underline{S}^{β∗} to \underline{T}^{α}\underline{T}^{β∗}. Since enough of the rank one operators on 𝔉_{X} arise as \underline{S}^{α}(I − ∑_{i=1}^{d} S_{i}S_{i}^{∗})\underline{S}^{β∗} (see equation (8.1)), and because \underline{T} satisfies (12.4), we must have that Ψ(K) = 0 for every K ∈ K(𝔉_{X}). By Lemma 12.5, E_{X} = T_{X}, and it follows that Ψ induces a positive and unital (hence contractive) mapping

π : O_{Λ} → B(H)

that sends \underline{Z}^{α}\underline{Z}^{β∗} to \underline{T}^{α}\underline{T}^{β∗}. Roughly speaking: π must be multiplicative because \underline{Z} and \underline{T} satisfy the same relations. In more detail: every product (\underline{Z}^{α}\underline{Z}^{β∗})(\underline{Z}^{α′} \underline{Z}^{β′∗}) may be written, using the relations (12.2)-(12.5) as some sum ∑_{γ,δ} \underline{Z}^{γ} \underline{Z}^{δ∗}. The mapping π then takes this sum to ∑_{γ,δ} \underline{T}^{γ} \underline{T}^{δ∗}, and this can be rewritten (using the same relations) as

(\underline{T}^{α}\underline{T}^{β∗})(\underline{T}^{α′} \underline{T}^{β′∗}) = π(\underline{Z}^{α}\underline{Z}^{β∗})π(\underline{Z}^{α′} \underline{Z}^{β′∗}).

This shows that

π ((\underline{Z}^{α}\underline{Z}^{β∗})(\underline{Z}^{α′} \underline{Z}^{β′∗})) = π(\underline{Z}^{α}\underline{Z}^{β∗})π(\underline{Z}^{α′} \underline{Z}^{β′∗}),

and since the elements of the form \underline{Z}^{α}\underline{Z}^{β∗} span O_{Λ}, and since π is a positive linear map, it follows that π is in fact a ∗-representation. □

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Orr Moshe Shalit Department of Pure Mathematics University of Waterloo 200 University Avenue West Waterloo, ON, Canada N2L 3G1. oshalit@math.uwaterloo.ca Baruch Solel Department of Mathematics, Technion, 32000, Haifa, Israel. mabaruch@tx.technion.ac.il