Adding Tails to C^{∗}-Correspondences

Paul S. Muhly and Mark Tomforde

Received: September 22, 2003 Revised: March 9, 2004

Communicated by Joachim Cuntz

Abstract. We describe a method of adding tails to C^{∗}- correspondences which generalizes the process used in the study of graph C^{∗}-algebras. We show how this technique can be used to extend results for augmented Cuntz-Pimsner algebras to C^{∗}-algebras associated to general C^{∗}-correspondences, and as an application we prove a gauge-invariant uniqueness theorem for these algebras. We also define a notion of relative graph C^{∗}-algebras and show that properties of these C^{∗}-algebras can provide insight and motivation for results about relative Cuntz-Pimsner algebras.

2000 Mathematics Subject Classification: 46L08, 46L55

Keywords and Phrases: C^{∗}-correspondence, Cuntz-Pimsner algebra, relative Cuntz-Pimsner algebra, graph C^{∗}-algebra, adding tails, gauge-invariant uniqueness

1 Introduction

In [18] Pimsner introduced a way to construct a C^{∗}-algebra O_{X} from a pair (A, X), where A is a C^{∗}-algebra and X is a C^{∗}-correspondence (sometimes called a Hilbert bimodule) over A. Throughout his analysis Pimsner assumed that his correspondence was full and that the left action of A on X was injective. These Cuntz-Pimsner algebras have been found to compose a class of C^{∗}- algebras that is extraordinarily rich and includes numerous C^{∗}-algebras found in the literature: crossed products by automorphisms, crossed products by endomorphisms, partial crossed products, Cuntz-Krieger algebras, C^{∗}-algebras of graphs with no sinks, Exel-Laca algebras, and many more. Consequently, the study of Cuntz-Pimsner algebras has received a fair amount of attention by the operator algebra community in recent years, and because information about O_{X} is very densely codified in (A, X), determining how to extract it has been the focus of much current effort.

One interesting consequence of this effort has been the introduction of the so-called relative Cuntz-Pimsner algebras, denoted O(K, X), that have Cuntz-Pimsner algebras as quotients. Very roughly speaking, a relative Cuntz-Pimsner algebra arises by relaxing some of the relations that must hold among the generators of a Cuntz-Pimsner algebra. These relations are codified in an ideal K of A. (The precise definition will be given shortly.) Relative Cuntz-Pimsner algebras arise quite naturally, particularly when trying to understand the ideal structure of a Cuntz-Pimsner algebra (See, e.g., [15, 6]). It turns out, in fact, that not only are Cuntz-Pimsner algebras quotients of relative Cuntz-Pimsner algebras, but quotients of Cuntz-Pimsner algebras are often relative Cuntz-Pimsner algebras [6, Theorem 3.1]. Although in his initial work Pimsner assumed that his C^{∗}-correspondences were full and had injective left action, in recent years there have been efforts to remove these restrictions. Pimsner himself described how to deal with the case when X was not full, defining the so-called augmented Cuntz-Pimsner algebras [18, Remark 1.2(3)]. However, the case when the left action is not injective has been more elusive. In [6] it was shown that for any C^{∗}-correspondence X and for any ideal K of A consisting of elements that act as compact operators on the left of X, one may define O(K, X) to be a C^{∗}-algebra which satisfies a certain universal property [6, Proposition 1.3]. In the case that X is full with injective left action, this definition agrees with previously defined notions of relative Cuntz-Pimsner algebras, and the Cuntz-Pimsner algebra O_{X} is equal to O(J(X), X), where J(X) denotes the ideal consisting of all elements of A which act on the left of X as compact operators. In [6] it was proposed that for a general C^{∗}-correspondence X, the C^{∗}-algebra O(J(X), X) is the proper analogue of the Cuntz-Pimsner algebra. However, upon further analysis it seems that this is not exactly correct. To see why, consider the case of graph C^{∗}-algebras. If E = (E^{0}, E^{1}, r, s) is a graph, then there is a natural C^{∗}-correspondence X(E) over C_{0}(E^{0}) associated to E (see [7, Example 1.2]). If E has no sinks, then the C^{∗}-algebra O(J(X(E)), X(E)) is isomorphic to the graph C^{∗}-algebra C^{∗}(E). However, when E has sinks this will not necessarily be the case. It is worth mentioning that graphs with sinks play an important role in the study of graph C^{∗}-algebras. Even if one begins with a graph E containing no sinks, an analysis of C^{∗}(E) will often necessitate considering C^{∗}-algebras of graphs with sinks. For example, quotients of C^{∗}(E) will often be isomorphic to C^{∗}-algebras of graphs with sinks even when E has no sinks. Consequently, one needs a theory that incorporates these objects. This deficiency in the generalization of Cuntz-Pimsner algebras was addressed by Katsura in [10] and [11]. If X is a C^{∗}-correspondence over a C^{∗}-algebra A with left action ϕ : A → L(X), then Katsura proposed that the appropriate analogue of the Cuntz-Pimsner algebra is O_{X} := O(J_{X}, X), where

J_{X} := {a ∈ J(X) : ab = 0 for all b ∈ ker ϕ}.

Adding Tails to C^{∗}-Correspondences 81

(Note that when ϕ is injective J_{X} = J(X).) It turns out that when ϕ is injective, O_{X} is equal to the augmented Cuntz-Pimsner algebra of X, and when X is also full O_{X} coincides with the Cuntz-Pimsner algebra of X. Furthermore, if E is a graph (possibly containing sinks), then O_{X(E)} is isomorphic to C^{∗}(E). In addition, as with graph algebras, the class of O_{X}’s is closed under quotients by gauge-invariant ideals. These facts, together with the analysis described in [11] and [12], provide strong arguments for using O_{X} := O(J_{X}, X) as the analogue of the Cuntz-Pimsner algebra. We shall adopt this viewpoint here, and for a general C^{∗}-correspondence X we define O_{X} := O(J_{X}, X) to be the C^{∗}-algebra associated to X. In this paper we shall describe a method which will allow one to “bootstrap” many results for augmented Cuntz-Pimsner algebras to C^{∗}-algebras associated to general correspondences. This method is inspired by a technique from the theory of graph C^{∗}-algebras, where one can often reduce to the sinkless case by the process of “adding tails to sinks”. Specifically, if E is a graph and v is a vertex of E, then by adding a tail to v we mean attaching a graph of the form

\xymatrix{v \ar[r] & • \ar[r] & • \ar[r] & • \ar[r] & ⋯}

to E. It is well known that if F is the graph formed by adding a tail to every sink of E, then F is a graph with no sinks and C^{∗}(E) is canonically isomorphic to a full corner of C^{∗}(F). Thus in the proofs of many theorems about graph C^{∗}-algebras, one can reduce to the case of no sinks. In this paper we describe a generalization of this process for C^{∗}- correspondences. More specifically, if X is a C^{∗}-correspondence over a C^{∗}-algebra A, then we describe how to construct a C^{∗}-algebra B and a C^{∗}- correspondence Y over B with the property that the left action of Y is injective and O_{X} is canonically isomorphic to a full corner of O_{Y}. Thus many questions about C^{∗}-algebras associated to correspondences can be reduced to questions about augmented Cuntz-Pimsner algebras, and many results characterizing properties of augmented Cuntz-Pimsner algebras may be easily generalized to C^{∗}-algebras associated to general correspondences. As an application of this technique, we use it in the proof of Theorem 5.1 to extend the Gauge-Invariant Uniqueness Theorem for augmented Cuntz-Pimsner algebras to C^{∗}-algebras of general correspondences. This paper is organized as follows. We begin in Section 2 with some preliminaries. In Section 3 we analyze graph C^{∗}-algebras in the context of Cuntz-Pimsner and relative Cuntz-Pimsner algebras, and describe a notion of a relative graph C^{∗}-algebra. Since graph algebras provide much of the impetus for our analysis of C^{∗}-correspondences, we examine these objects carefully in order to provide a framework which will motivate and illuminate the results of subsequent sections. In Section 4 we describe our main result — a process of “adding tails” to general C^{∗}-correspondences. We also prove that this process preserves the Morita equivalence class of the associated C^{∗}-algebra. In Section 5 we provide an application of our technique of “adding tails” by using it to extend the Gauge-Invariant Uniqueness Theorem for augmented Cuntz-Pimsner algebras to C^{∗}-algebras associated to general correspondences. We also interpret this theorem in the context of relative Cuntz-Pimsner algebras, and in Section 6 we use it to classify the gauge-invariant ideals in C^{∗}-algebras associated to certain correspondences. Finally, we conclude in Section 7 by discussing other possible applications of our technique. The authors would like to thank Takeshi Katsura for pointing out an error in a previous draft of this paper, and for many useful conversations regarding these topics.

2 Preliminaries

For the most part we will use the notation and conventions of [6], augmenting them when necessary with the innovations of [10] and [11].

Definition 2.1. If A is a C^{∗}-algebra, then a right Hilbert A-module is a Banach space X together with a right action of A on X and an A-valued inner product ⟨·, ·⟩_{A} satisfying

(i) ⟨ξ, ηa⟩_{A} = ⟨ξ, η⟩_{A}a

(ii) ⟨ξ, η⟩_{A} = ⟨η, ξ⟩_{A}^{∗}

(iii) ⟨ξ, ξ⟩_{A} ≥ 0 and ∥ξ∥ = ⟨ξ, ξ⟩_{A}^{1/2}

for all ξ, η ∈ X and a ∈ A. For a Hilbert A-module X we let L(X) denote the C^{∗}-algebra of adjointable operators on X, and we let K(X) denote the closed two-sided ideal of compact operators given by

K(X) := \overline{span}\{Θ_{ξ,η}^{X} : ξ, η ∈ X\}

where Θ_{ξ,η}^{X} is defined by Θ_{ξ,η}^{X}(ζ) := ξ⟨η, ζ⟩_{A}. When no confusion arises we shall often omit the superscript and write Θ_{ξ,η} in place of Θ_{ξ,η}^{X}.

Definition 2.2. If A is a C^{∗}-algebra, then a C^{∗}-correspondence is a right Hilbert A-module X together with a ∗-homomorphism ϕ : A → L(X). We consider ϕ as giving a left action of A on X by setting a · x := ϕ(a)x.

Definition 2.3. If X is a C^{∗}-correspondence over A, then a representation of X into a C^{∗}-algebra B is a pair (π, t) consisting of a ∗-homomorphism π : A → B and a linear map t : X → B satisfying

(i) t(ξ)^{∗}t(η) = π(⟨ξ, η⟩_{A})

(ii) t(ϕ(a)ξ) = π(a)t(ξ)

(iii) t(ξa) = t(ξ)π(a)

for all ξ, η ∈ X and a ∈ A. Note that Condition (iii) follows from Condition (i) due to the equation

∥t(ξ)π(a) − t(ξa)∥^{2} = ∥(t(ξ)π(a) − t(ξa))^{∗}(t(ξ)π(a) − t(ξa))∥ = 0.

If (π, t) is a representation of X into a C^{∗}-algebra B, we let C^{∗}(π, t) denote the C^{∗}-subalgebra of B generated by π(A) ∪ t(X). A representation (π, t) is said to be injective if π is injective. Note that in this case t will also be isometric since

∥t(ξ)∥^{2} = ∥t(ξ)^{∗}t(ξ)∥ = ∥π(⟨ξ, ξ⟩_{A})∥ = ∥⟨ξ, ξ⟩_{A}∥ = ∥ξ∥^{2}.

When (π, t) is a representation of X into B(H) for a Hilbert space H, we say that (π, t) is a representation of X on H. In the literature a representation (π, t) is sometimes referred to as a Toeplitz representation (See, e.g., [7] and [6].), and as an isometric representation [15]. However, here, all representations considered will be at least Toeplitz or isometric and so we drop the additional adjective. We note that in [7] the authors show that given a correspondence X over a C^{∗}-algebra A, there is a C^{∗}-algebra, denoted T_{X} and a representation (π_{X}, t_{X}) of X in T_{X} that is universal in the following sense: T_{X} is generated as a C^{∗}-algebra by the ranges of π_{X} and t_{X}, and given any representation (π, t) in a C^{∗}-algebra B, then there is a C^{∗}- homomorphism of T_{X} into B, denoted ρ_{(π,t)}, that is unique up to an inner automorphism of B, such that π = ρ_{(π,t)} ∘ π_{X} and t = ρ_{(π,t)} ∘ t_{X}. The C^{∗}- algebra T_{X} and the representation (π_{X}, t_{X}) are unique up to an obvious notion of isomorphism. We call T_{X} the Toeplitz algebra of the correspondence X, but we call (π_{X}, t_{X}) a universal representation of X in T_{X}, with emphasis on the indefinite article, because at times we want to consider more than one. Definition 2.4. For a representation (π, t) of a C^{∗}-correspondence X on B there exists a ∗-homomorphism π^{(1)} : K(X) → B with the property that

π^{(1)}(Θ_{ξ,η}) = t(ξ)t(η)^{∗}.

See [18, p. 202], [9, Lemma 2.2], and [7, Remark 1.7] for details on the existence of this ∗-homomorphism. Also note that if (π, t) is an injective representation, then π^{(1)} will be injective as well. Definition 2.5. For an ideal I in a C^{∗}-algebra A we define

I^{⊥} := {a ∈ A : ab = 0 for all b ∈ I}.

If X is a C^{∗}-correspondence over A, we define an ideal J(X) of A by J(X) := ϕ^{−1}(K(X)). We also define an ideal J_{X} of A by

J_{X} := J(X) ∩ (ker ϕ)^{⊥}.

Note that J_{X} = J(X) when ϕ is injective, and that J_{X} is the maximal ideal on which the restriction of ϕ is an injection into K(X). Definition 2.6. If X is a C^{∗}-correspondence over A and K is an ideal in J(X), then we say that a representation (π, t) is coisometric on K, or is K-coisometric if

π^{(1)}(ϕ(a)) = π(a) for all a ∈ K.

In [6, Proposition 1.3] the authors show that given a correspondence X over a C^{∗}-algebra A, and an ideal K of A contained in J(X), there is a C^{∗}-algebra, denoted O(K, X), and a representation (π_{X}, t_{X}) of X in O(K, X) that is coisometric on K and is universal with this property, in the following sense: O(K, X) is generated as a C^{∗}-algebra by the ranges of π_{X} and t_{X}, and given any representation (π, t) of X in a C^{∗}-algebra B that is K-coisometric, then there is a C^{∗}-homomorphism of O(K, X) into B, denoted ρ_{(π,t)}, that is unique up to an inner automorphism of B, such that π = ρ_{(π,t)} ∘ π_{X} and t = ρ_{(π,t)} ∘ t_{X}. Definition 2.7. The algebra O(K, X), associated with an ideal K in J(X), is called the relative Cuntz-Pimsner algebra determined by X and the ideal K. Further, a representation (π_{X}, t_{X}) that is coisometric on K and has the universal property just described is called a universal K-coisometric representation of X. Remark 2.8. When the ideal K is the zero ideal in J(X), then the algebra O(K, X) becomes T_{X} and a universal 0-coisometric representation of X is simply a representation of X. Furthermore, if X is a C^{∗}-correspondence in which ϕ is injective, then O_{X} := O(J_{X}, X) is precisely the augmented Cuntz-Pimsner algebra of X defined in [18]. If X is full, i.e., if \overline{span}\{⟨ξ, η⟩_{A} : ξ, η ∈ X\} = A, then the augmented Cuntz-Pimsner algebra of X and the Cuntz-Pimsner algebra of X coincide. Thus O_{X} coincides with the Cuntz-Pimsner algebra of [18] when ϕ is injective and X is full. Whether or not ϕ is injective, a universal J(X)-coisometric representation is sometimes called a universal Cuntz-Pimsner covariant representation [6, Definition 1.1]. Remark 2.9. If O(K, X) is a relative Cuntz-Pimsner algebra associated to a C^{∗}-correspondence X, and if (π, t) is a universal K-coisometric representation of X, then for any z ∈ 𝕋 (π, zt) is also a universal K-coisometric representation. Hence by the universal property, there exists a homomorphism γ_{z} : O(K, X) → O(K, X) such that γ_{z}(π(a)) = π(a) for all a ∈ A and γ_{z}(t(ξ)) = zt(ξ) for all ξ ∈ X. Since γ_{z^{−1}} is an inverse for this homomorphism, we see that γ_{z} is an automorphism. Thus we have an action γ : 𝕋 → Aut O(K, X) with the property that γ_{z}(π(a)) = π(a) and γ_{z}(t(ξ)) = zt(ξ). Furthermore, a routine ϵ/3 argument shows that γ is strongly continuous. We call γ the gauge action on O(K, X).

3 Viewing graph C^{∗}-algebras as Cuntz-Pimsner algebras

Let E := (E^{0}, E^{1}, r, s) be a directed graph with countable vertex set E^{0}, countable edge set E^{1}, and range and source maps r, s : E^{1} → E^{0}. A Cuntz-Krieger E-family is a collection of partial isometries {s_{e} : e ∈ E^{1}} with commuting range projections together with a collection of mutually orthogonal projections {p_{v} : v ∈ E^{0}} that satisfy

1. s_{e}^{∗}s_{e} = p_{r(e)} for all e ∈ E^{1}

2. s_{e}s_{e}^{∗} ≤ p_{s(e)} for all e ∈ E^{1}

3. p_{v} = ∑_{{e:s(e)=v}} s_{e}s_{e}^{∗} for all v ∈ E^{0} with 0 < |s^{−1}(v)| < ∞

The graph algebra C^{∗}(E) is the C^{∗}-algebra generated by a universal Cuntz-Krieger E-family (see [14, 13, 2, 5, 1]). Example 3.1 (The Graph C^{∗}-correspondence). If E = (E^{0}, E^{1}, r, s) is a graph, we define A := C_{0}(E^{0}) and

X(E) := {x : E^{1} → ℂ : the function v ↦ ∑_{{f∈E^{1}:r(f)=v}} |x(f)|^{2} is in C_{0}(E^{0}) }.

Then X(E) is a C^{∗}-correspondence over A with the operations

(x · a)(f) := x(f)a(r(f)) for f ∈ E^{1}
⟨x, y⟩_{A}(v) := ∑_{\{f∈E^{1}:r(f)=v\}} \overline{x(f)}y(f) for f ∈ E^{1}
(a · x)(f) := a(s(f))x(f) for f ∈ E^{1}

and we call X(E) the graph C^{∗}-correspondence associated to E. Note that we could write X(E) = ⨁_{v∈E^{0}}^{0} ℓ^{2}(r^{−1}(v)) where this denotes the C_{0} direct sum (sometimes called the restricted sum) of the ℓ^{2}(r^{−1}(v))’s. Also note that X(E) and A are spanned by the point masses {δ_{f} : f ∈ E^{1}} and {δ_{v} : v ∈ E^{0}}, respectively.

Theorem 3.2 ([5, Proposition 12]). If E is a graph with no sinks, and X(E) is the associated graph C^{∗}-correspondence, then O(J(X(E)), X(E)) ≅ C^{∗}(E). Furthermore, if (π_{X}, t_{X}) is a universal J(X(E))-coisometric representation, then {t_{X}(δ_{e}), π_{X}(δ_{v})} is a universal Cuntz-Krieger E-family in O(J(X(E)), X(E)).

It was shown in [7, Proposition 4.4] that

J(X(E)) = \overline{span}\{δ_{v} : |s^{−1}(v)| < ∞\}

and if v emits finitely many edges, then

ϕ(δ_{v}) = ∑_{{f∈E^{1}:s(f)=v}} Θ_{δ_{f} ,δ_{f}} and π_{X}(ϕ(δ_{v})) = ∑_{{f∈E^{1}:s(f)=v}} t_{X}(δ_{f})t_{X}(δ_{f})^{∗}.

Furthermore, one can see that δ_{v} ∈ ker ϕ if and only if v is a sink in E. Also δ_{v} ∈ \overline{span}\{⟨x, y⟩_{A}\} if and only if v is a source, and since δ_{s(f)} · δ_{f} = δ_{f} we see that \overline{span} A · X = X and X(E) is essential. These observations show that we have the following correspondences between the properties of the graph E and the properties of the graph C^{∗}-correspondence X(E).

Property of X(E)  Property of E
ϕ(δ_{v}) ∈ K(X(E))  v emits a finite number of edges
ϕ(A) ⊆ K(X(E))  E is row-finite
ϕ is injective  E has no sinks
X(E) is full  E has no sources
X(E) is essential  always

Remark 3.3. If E is a graph with no sinks, then O(J(X(E)), X(E)) is canonically isomorphic to C^{∗}(E). When E has sinks, this will not be the case. If (π, t) is the universal J(X(E))-coisometric representation of X(E), then it will be the case that {t(δ_{e}), π(δ_{v})} is a Cuntz-Krieger E-family. However, when v is a sink in E, ϕ(δ_{v}) = 0 and thus π(δ_{v}) = π^{(1)}(ϕ(δ_{v})) = 0. Consequently, {t(δ_{e}), π(δ_{v})} will not be a universal Cuntz-Krieger E-family when E has sinks. However, if E is a graph with sinks, then we see that ϕ(δ_{v}) = 0 if and only if v is a sink, and δ_{v} ∈ (ker ϕ)^{⊥} if and only if v is not a sink. Thus

J_{X(E)} = \overline{span}\{δ_{v} : 0 < |s^{−1}(v)| < ∞\}

and a proof similar to that in [7, Proposition 4.4] shows that O_{X(E)} := O(J_{X(E)}, X(E)) is isomorphic to C^{∗}(E). Furthermore, if (π_{X}, t_{X}) is a universal J(X(E))-coisometric representation of X(E), then {t_{X}(δ_{e}), π_{X}(δ_{v})} is a universal Cuntz-Krieger E-family in O_{X(E)}.

3.1 Relative Graph Algebras

We shall now examine relative Cuntz-Pimsner algebras in the context of graph algebras. If E is a graph and X(E) is the associated graph C^{∗}-correspondence, then J_{X(E)} := \overline{span}\{δ_{v} : 0 < |s^{−1}(v)| < ∞\}. If K is an ideal in J_{X(E)}, then K = \overline{span}\{δ_{v} : v ∈ V \} for some subset V of vertices which emit a finite and nonzero number of edges. If (O(K, X(E)), t_{X}, π_{X}) is the relative Cuntz-Pimsner algebra determined by K, then the relation π_{X}(δ_{v}) = ∑_{s(e)=v} t_{X}(δ_{e})t_{X}(δ_{e})^{∗} will hold only for vertices v ∈ V. This motivates the following definition.

Definition 3.4. Let E = (E^{0}, E^{1}, r, s) be a graph and define R(E) := {v ∈ E^{0} : 0 < |s^{−1}(v)| < ∞}. For any V ⊆ R(E) we define a Cuntz-Krieger (E, V)- family to be a collection of mutually orthogonal projections {p_{v} : v ∈ E^{0}} together with a collection of partial isometries {s_{e} : e ∈ E^{1}} that satisfy

1. s_{e}^{∗}s_{e} = p_{r(e)} for e ∈ E^{1}

2. s_{e}s_{e}^{∗} < p_{s(e)} for e ∈ E^{1}

3. p_{v} = ∑_{s(e)=v} s_{e}s_{e}^{∗} for all v ∈ V

We refer to a Cuntz-Krieger (E, R(E))-family as simply a Cuntz-Krieger Efamily, and we refer to a Cuntz-Krieger (E, ∅)-family as a Toeplitz-Cuntz-Krieger family.

Definition 3.5. If E is a graph and V ⊆ R(E), then we define the relative graph algebra C^{∗}(E, V) to be the C^{∗}-algebra generated by a universal Cuntz-Krieger (E, V)-family.

The existence of C^{∗}(E, V) can be proven by adapting the argument for the existence of graph algebras in [13], or by realizing C^{∗}(E, V) as a relative Cuntz-Pimsner algebra.

Note that C^{∗}(E, R(E)) is the graph algebra C^{∗}(E), and C^{∗}(E, ∅) is the Toeplitz algebra defined in [7, Theorem 4.1] (but different from the Toeplitz algebra defined in [4]). It is also the case that if {s_{e}, p_{v}} is a universal Cuntz-Krieger (E, V)-family, then whenever v ∈ R(E)\V we have p_{v} > ∑_{s(e)=v} s_{e}s_{e}^{∗}.

Definition 3.6. Let E = (E^{0}, E^{1}, r, s) be a graph and V ⊆ R(E). We define the graph E_{V} to be the graph with vertex set E_{V}^{0} := E^{0} ∪ {v′ : v ∈ R(E)\V }, edge set E^{1} ∪ {e′ : e ∈ E^{1} and r(e) ∈ R(E)\V }, and r and s extended to E_{V}^{1} by defining s(e′) := s(e) and r(e′) := r(e)′.

Roughly speaking, when forming E_{V} one takes E and adds a sink for each element v ∈ R(E)\V as well as edges to this sink from each vertex that feeds into v.

Theorem 3.7. If E is a graph and V ⊆ R(E), then the relative graph algebra C^{∗}(E, V) is canonically isomorphic to the graph algebra C^{∗}(E_{V}).

Proof. Let {s_{e}, p_{v} : e ∈ E^{1}, v ∈ E^{0}} be a generating Cuntz-Krieger (E, V)- family in C^{∗}(E, V). For w ∈ E_{V}^{0} and f ∈ E_{V}^{1} define

q_{w} := \begin{cases}p_{v} & if w ∉ R(E)\V \\ ∑_{\{e∈E^{1}:s(e)=w\}} s_{e}s_{e}^{∗} & if w ∈ R(E)\V \\ p_{v} − ∑_{\{e∈E^{1}:s(e)=v\}} s_{e}s_{e}^{∗} & if w = v′ for some v ∈ R(E)\V. \end{cases}

t_{f} := \begin{cases}s_{f}q_{r(f)} & if f ∈ E^{1} \\ s_{e}q_{r(e)′} & if f = e′ for some e ∈ E^{1}. \end{cases}

It is straightforward to check that {t_{f}, q_{w} : f ∈ E_{V}^{1} , w ∈ E_{V}^{0} } is a Cuntz-Krieger E_{V} -family in C^{∗}(E, V). Thus by the universal property there exists a homomorphism α : C^{∗}(E_{V}) → C^{∗}(E, V) taking the generators of C^{∗}(E_{V}) to {t_{f}, q_{w}}. By the gauge-invariant uniqueness theorem [1, Theorem 2.1] α is injective. Furthermore, whenever v ∈ R(E)\V we see that p_{v} = q_{v} + q_{v′} and whenever r(e) ∈ R(E)\V we see that s_{e} = t_{e} + t_{e′}. Thus {q_{w}, t_{f}} generates C^{∗}(E, V) and α is surjective. Consequently α is an isomorphism.

This theorem shows that the class of relative graph algebras is the same as the class of graph algebras. Thus we gain no new C^{∗}-algebras by considering relative graph algebras in place of graph algebras. However, we maintain that relative graph algebras are still useful and arise naturally in the study of graph algebras. In particular, we give three examples of common situations in which relative graph algebras prove convenient.

Example 3.8 (Subalgebras of Graph Algebras). Let E = (E^{0}, E^{1}, r, s) be a graph and let {s_{e}, p_{v} : e ∈ E^{1}, v ∈ E^{0}} be a generating Cuntz-Krieger E-family in C^{∗}(E). If F = (F^{0}, F^{1}, r_{F} , s_{F}) is a subgraph of E, and A denotes the C^{∗}- subalgebra of C^{∗}(E) generated by {s_{e}, p_{v} : e ∈ F^{1}, v ∈ F^{0}}, then it is wellknown that A is a graph algebra (but not necessarily the C^{∗}-algebra associated to F). In fact, we see that for any v ∈ F^{0}, the sum ∑_{{e∈F^{1}:s_{F} (e)=v}} s_{e}s_{e}^{∗} may not add up to p_{v} because some of the edges in s^{−1}(v) may not be in F. However, if we let V := {v ∈ R(F) : s_{F}^{−1}(v) = s^{−1}(v)}. Then {s_{e}, p_{v} : e ∈ F^{1}, v ∈ F^{0}} is a Cuntz-Krieger (F, V)-family and A ≅ C^{∗}(F, V). These subalgebras arise often in the study of graph algebras. In [8, Lemma 2.4] they were realized as graph algebras by the method shown in the proof of Theorem 3.7, and in [19, Lemma 1.2] these subalgebras were realized as graph algebras by using the notion of a dual graph. In both of these instances it would have been convenient to have used relative graph algebras. Realizing the subalgebra as C^{∗}(F, V) would have provided an economy of notation as well as a more direct analysis of the subalgebras under consideration.

Example 3.9 (Spielberg’s Toeplitz Graph Algebras). In [21] Spielberg introduced a notion of a Toeplitz graph groupoid and a Toeplitz graph algebra. The Toeplitz graph algebras defined in [21, Definition 2.17] are relative graph algebras as defined in Definition 3.5 (see [21, Theorem 2.9]). Spielberg also made use of his Toeplitz graph algebras in [22] to construct graph algebras with a specified K-theory.

Example 3.10 (Quotients of Graph Algebras). If E = (E^{0}, E^{1}, r, s) is a rowfinite graph and H is a saturated hereditary subset of vertices of E, then it follows from [2, Theorem 4.1(b)] that C^{∗}(E)/I_{H} ≅ C^{∗}(F) where F is the subgraph defined by

F^{0} := E^{0}\H F^{1} := {e ∈ E^{1} : r(e) ∉ H}.

If E is not row-finite, then this is not necessarily the case. The obstruction is due to the vertices in the set

B_{H} = {v ∈ E^{0} | v is an infinite emitter and 0 < |s^{−1}(v) ∩ r^{−1}(E^{0} \ H)| < ∞}.

In fact, if {s_{e}, p_{v}} is a generating Cuntz-Krieger E-family in C^{∗}(E), then the cosets {s_{e} + I_{H}, p_{v} + I_{H} : v ∉ H, r(e) ∉ H} will have the property that p_{v} + I_{H} ≥ ∑_{e∈E\H:s(e)=v}}(s_{e} + I_{H})(s_{e} + I_{H})^{∗} with equality occurring if and only if v ∈ R(F)\B_{H}. Thus it turns out that {s_{e} + I_{H}, p_{v} + I_{H} : v ∉ H, r(e) ∉ H} will be a Cuntz-Krieger (F, R(F)\B_{H})-family and C^{∗}(E)/I_{H} ≅ C^{∗}(F, R(F)\B_{H}). The quotient C^{∗}(E)/I_{H} was realized as a graph algebra in [1, Proposition 3.4] by a technique similar to that used in the proof of Theorem 3.7. However, relative graph algebras provide a more natural context for describing these quotients.

In addition to their applications in the situations mentioned above, relative graph algebras can be useful for another reason. Since any relative graph algebra is canonically isomorphic to a graph algebra, we see that for every theorem about graph algebras there will be a corresponding theorem for relative graph algebras. Thus the relative graph algebras provide a class of relative Cuntz-Pimsner algebras that are well understood. With this in mind, we shall now state a version of the Gauge-Invariant Uniqueness Theorem for relative graph algebras.

Theorem 3.11 (Gauge-Invariant Uniqueness for Relative Graph Algebras). Let E = (E^{0}, E^{1}, r, s) be a graph and V ⊆ R(E). Also let {s_{e}, p_{v} : e ∈ E^{1}, v ∈ E^{0}} and let γ : 𝕋 → Aut C^{∗}(E, V) denote the gauge action on C^{∗}(E, V). If ρ : C^{∗}(E, V) → A is a ∗-homomorphism between C^{∗}- algebras that satisfies

1. ρ(p_{v}) ≠ 0 for all v ∈ E^{0}

2. ρ(p_{v} − ∑_{s(e)=v} s_{e}s_{e}^{∗}) ≠ 0 for all v ∈ V

3. there exists a strongly continuous action β : 𝕋 → Aut A such that β_{z} ∘ρ = ρ ∘ γ_{z} for all z ∈ 𝕋.

then ρ is injective.

Proof. By Theorem 3.7 there exists an isomorphism α : C^{∗}(E_{V}) → C^{∗}(E, V) and a generating Cuntz-Krieger E_{V} -family {t_{e}, q_{w}} for which

α(q_{w}) := \begin{cases}p_{v} & if w ∉ R(E)\V \\ ∑_{\{e∈E^{1}:s(e)=w\}} s_{e}s_{e}^{∗} & if w ∈ R(E)\V \\ p_{v} − ∑_{\{e∈E^{1}:s(e)=v\}} s_{e}s_{e}^{∗} & if w = v′ for some v ∈ R(E)\V. \end{cases}

α(t_{f}) := \begin{cases}s_{f}q_{r(f)} & if f ∈ E^{1} \\ s_{e}q_{r(e)′} & if f = e′ for some e ∈ E^{1}. \end{cases}

To show that ρ is injective, it suffices to show that ρ ∘ α is injective. We shall do this by applying the gauge-invariant uniqueness theorem for graph algebras [1, Theorem 2.1] to ρ ∘ α. Now clearly if w ∉ R(E)\V, then ρ ∘ α(q_{w}) ≠ 0 by (1). If w = v′, then ρ ∘ α(q_{w}) ≠ 0 by (2). Furthermore, if w ∈ R(E)\V then ρ ∘ α(q_{w}) = 0 implies that ρ(∑_{s(e)=w} s_{e}s_{e}^{∗}) = 0 and thus for any f ∈ s^{−1}(v) we have

ρ(s_{f}) = ρ( ∑_{s(e)=w} s_{e}s_{e}^{∗})ρ(s_{f}) = 0.

But then ρ(p_{r(f)}) = ρ(s_{f}^{∗}s_{f}) = 0 which contradicts (1). Hence we must have ρ ∘ α(q_{w}) ≠ 0. Finally, if γ′ denotes the gauge action on C^{∗}(E_{V}), then by checking on generators we see that β_{z} ∘ (ρ ∘ α) = (ρ ∘ α) ∘ γ′_{z}. Therefore, ρ ∘ α is injective by the gauge invariant uniqueness theorem for graph algebras, and consequently ρ is injective.

We have shown in Theorem 3.7 that every relative graph algebra is isomorphic to a graph algebra. More generally, Katsura has shown in [12] that every relative Cuntz-Pimsner algebra is isomorphic to the C^{∗}-algebra associated to a correspondence; that is, if O(K, X) is a relative Cuntz-Pimsner algebra, then there exists a C^{∗}-correspondence X′ such that O_{X′} := O(J_{X′}, X′) is isomorphic to O(K, X). In Theorem 5.1 we shall prove a gauge-invariant uniqueness theorem for C^{∗}-algebras associated to correspondences. Afterwards, in Remark 5.3, we shall use Katsura’s analysis in [12] to give an interpretation of Theorem 3.11 in the context of relative Cuntz-Pimsner algebras.

4 Adding Tails to C^{∗}-correspondences

If E is a graph and v is a vertex of E, then by adding a tail to v we mean attaching a graph of the form

\xymatrix{v \ar[r]^{e_{1}} & v_{1} \ar[r]^{e_{2}} & v_{2} \ar[r]^{e_{3}} & v_{3} \ar[r]^{e_{4}} & ⋯}

to E. It was shown in [2, §1] that if F is the graph formed by adding a tail to every sink of E, then F is a graph with no sinks and C^{∗}(E) is canonically isomorphic to a full corner of C^{∗}(F). The technique of adding tails to sinks is a simple but powerful tool in the analysis of graph algebras. In the proofs of many results it allows one to reduce to the case in which the graph has no sinks and thereby avoid certain complications and technicalities. Our goal in this section is to develop a process of “adding tails to sinks” for C^{∗}-correspondences, so that given any C^{∗}-correspondence X we may form a C^{∗}-correspondence Y with the property that the left action of Y is injective and O_{X} is canonically isomorphic to a full corner in O_{Y}. Definition 4.1. Let X be a C^{∗}-correspondence over A with left action ϕ : A → L(X), and let I be an ideal in A. We define the tail determined by I to be the C^{∗}-algebra

T := I^{(ℕ)}

where I^{(ℕ)} denotes the c_{0}-direct sum of countably many copies of the ideal I. We shall denote the elements of T by

f⃗:= (f_{1}, f_{2}, f_{3}, …)

where each f_{i} is an element of I. We shall consider T as a right Hilbert C^{∗}-module over itself (see [20, Example 2.10]). We define Y := X ⊕ T and B := A ⊕ T. Then Y is a right Hilbert B-module in the usual way; that is, the right action is given by

(ξ, f⃗) · (a, g⃗) := (ξ · a, f⃗g⃗) for ξ ∈ X, a ∈ A, and f⃗, g⃗ ∈ T

and the inner product is given by

⟨(ξ, f⃗), (ν, g⃗)⟩_{B} := (⟨ξ, ν⟩_{A}, f⃗^{∗}g⃗) for ξ, ν ∈ X and f⃗, g⃗ ∈ T.

Furthermore, we shall make Y into a C^{∗}-correspondence over B by defining a left action ϕ_{B} : B → L(Y) as

ϕ_{B}(a, f⃗)(ξ, g⃗) := (ϕ(a)(ξ), (ag_{1}, f_{1}g_{2}, f_{2}g_{3}, …)) for a ∈ A, ξ ∈ X, and f⃗, g⃗ ∈ T.

We call Y the C^{∗}-correspondence formed by adding the tail T to X.

Lemma 4.2. Let X be a C^{∗}-correspondence over A, and let T := (ker ϕ)^{(ℕ)} be the tail determined by ker ϕ. If Y := X ⊕T is the C^{∗}-correspondence over B := A ⊕ T formed by adding the tail T to X, then the left action ϕ_{B} : B → L(Y) is injective. Consequently, J_{Y} = J(Y) and O_{Y} = O(J(Y), Y) is equal to the C^{∗}-algebra defined by Pimsner in [18].

Proof. If (a, f⃗) ∈ ker ϕ_{B}, then for all ξ ∈ X we have

(ϕ(a)ξ,0) = ϕ_{B}(a, f⃗)(ξ,0) = (0,0)

so that ϕ(a)ξ = 0 and a ∈ ker ϕ. Thus (0, (a, f_{1}, f_{2}, …)) ∈ X ⊕ T and

(0, (aa^{∗}, f_{1}f_{1}^{∗}, f_{2}f_{2}^{∗}, …)) = ϕ_{B}(a, f⃗)(0, (a^{∗}, f_{1}, f_{2}, …)) = (0,0)

so that ∥a∥^{2} = ∥aa^{∗}∥ = 0 and ∥f_{i}∥^{2} = ∥f_{i}f_{i}^{∗}∥ = 0 for all i ∈ ℕ. Consequently, a = 0 and f⃗ = 0 so that ϕ_{B} is injective.

Theorem 4.3. Let X be a C^{∗}-correspondence over A, and let T := (ker ϕ)^{(ℕ)} be the tail determined by ker ϕ. Also let Y := X ⊕ T be the C^{∗}-correspondence over B := A ⊕ T formed by adding the tail T to X.

(a) If (π, t) is a J_{X}-coisometric representation of X on a Hilbert space H_{X}, then there is a Hilbert space H_{Y} = H_{X} ⊕ H_{T} and a J(Y)-coisometric representation (π̃, t̃) of Y on H_{Y} with the property that π̃|_{X} = π and t̃|_{A} = t.

(b) If (π̃, t̃) is a J(Y)-coisometric representation of Y into a C^{∗}-algebra C, then (π̃|_{A}, t̃|_{X}) is a J_{X}-coisometric representation of X into C. Furthermore, if π̃|_{A} is injective, then π̃ is injective.

(c) Let (π_{Y} , t_{Y}) be a universal J(Y)-coisometric representation of Y. Then (π, t) := (π_{Y} |_{A}, t_{Y} |_{X}) is a J_{X}-coisometric representation of X in C^{∗}(π_{Y} , t_{Y}). Furthermore, ρ_{(π,t)} : O_{X} → C^{∗}(π_{X}, t_{X}) ⊆ O_{Y} is an isomorphism onto the C^{∗}-subalgebra of O_{Y} generated by

{π_{Y} (a,0), t_{Y} (ξ,0) : a ∈ A and ξ ∈ X}

and this C^{∗}-subalgebra is a full corner of O_{Y}. Consequently, O_{X} is naturally isomorphic to a full corner of O_{Y}.

Corollary 4.4. If X is a C^{∗}-correspondence and (π_{X}, t_{X}) is a universal J(X)-coisometric representation of X, then (π_{X}, t_{X}) is injective.

Proof. By the theorem (π_{X}, t_{X}) extends to a universal J(Y)-coisometric representation (π_{Y} , t_{Y}) of Y. Since ϕ_{B} is injective by Lemma 4.2 it follows from [6, Corollary 6.2] that (π_{Y} , t_{Y}) is injective. Consequently, π_{X} = π_{Y} |_{A} is injective.

To prove this theorem we shall need a number of lemmas.

Lemma 4.5. Let X be a C^{∗}-correspondence and let T := (ker ϕ)^{(ℕ)} be the tail determined by ker ϕ. Also let Y := X ⊕ T be the C^{∗}-correspondence over B := A ⊕ T formed by adding the tail T to X. Then for any (a, f⃗) ∈ Y we have that (a, f⃗) ∈ J(Y) if and only if a = a_{1} + a_{2} with a_{1} ∈ J_{X} and a_{2} ∈ ker ϕ.

Proof. Suppose a = a_{1} + a_{2} with a_{1} ∈ J_{X} and a_{2} ∈ ker ϕ. Then we may write ϕ(a_{1}) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{ξ_{n,k},η_{n,k}}^{X} for some ξ_{n,k}, η_{n,k} ∈ X. But then

ϕ_{B}(a_{1},0) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},0),(η_{n,k},0)}^{Y} ∈ K(Y).

In addition, since a_{2} ∈ ker ϕ we see that if we let {e⃗_{λ}}_{λ∈Λ} be an approximate unit for T with e⃗_{λ} = (e_{λ}^{1}, e_{λ}^{2}, …) for each λ, then

ϕ_{B}(a_{2}, f⃗) = lim_{λ} Θ_{(0,(a_{2},f_{1},f_{2},...)),(0,(e_{λ}^{1},e_{λ}^{2},...))}^{Y} ∈ K(Y).

Thus ϕ_{B}(a, f⃗) = ϕ_{B}(a_{1},0) + ϕ_{B}(a_{2}, f⃗)) ∈ K(Y).

Conversely, suppose that ϕ_{B}(a, f⃗) ∈ K(Y). Then we may write

ϕ_{B}(a, f⃗) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y}.

If we write f⃗_{n,k} = (f_{n,k}^{1}, f_{n,k}^{2}, …) and g⃗_{n,k} = (g_{n,k}^{1}, g_{n,k}^{2}, …) then for any (ξ, g⃗) ∈ X ⊕ T we have that

(ϕ(a)ξ, (ag_{1}, f_{1}g_{2}, …)) = ϕ_{B}(a, f⃗)(ξ, g⃗)

= lim_{n} ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y}(ξ, g⃗)

= lim_{n} ∑_{k=1}^{N_{n}}((ξ_{n,k}⟨η_{n,k}, ξ⟩_{A}, (f_{n,k}^{1}g_{n,k}^{1}^{∗}g_{1}, f_{n,k}^{2}g_{n,k}^{2}^{∗}g_{2}, …))

= lim_{n} ∑_{k=1}^{N_{n}}((Θ_{ξ_{n,k},η_{n,k}}^{X}ξ, (f_{n,k}^{1}g_{n,k}^{1}^{∗}g_{1}, f_{n,k}^{2}g_{n,k}^{2}^{∗}g_{2}, …)).(1)

Now since the operator norm on L(Y) dominates the operator norm on L(X), we see that lim_{n} ∑_{k=1}^{N_{n}} Θ_{ξ_{n,k},η_{n,k}}^{X} converges and ϕ(a) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{ξ_{n,k},η_{n,k}}^{X}. Thus a ∈ K(X).

Furthermore, if {e_{λ}}_{λ∈Λ} is an approximate unit for ker ϕ, then for any n, m ∈ ℕ we have

∥(∑_{k=1}^{N_{n}} f_{n,k}^{1}g_{n,k}^{1}^{∗} − ∑_{k=1}^{N_{m}} f_{m,k}^{1}g_{m,k}^{1}^{∗})e_{λ}∥

= ∥( ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y} − ∑_{k=1}^{N_{m}} Θ_{(ξ_{m,k},f⃗_{m,k}),(η_{m,k},g⃗_{m,k})}^{Y})(0, (e_{λ}, 0, 0, …))∥

≤ ∥ ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y} − ∑_{k=1}^{N_{m}} Θ_{(ξ_{m,k},f⃗_{m,k}),(η_{m,k},g⃗_{m,k})}^{Y}∥ ∥(0, (e_{λ}, 0, 0, …))∥

= ∥ ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y} − ∑_{k=1}^{N_{m}} Θ_{(ξ_{m,k},f⃗_{m,k}),(η_{m,k},g⃗_{m,k})}^{Y}∥

for all λ ∈ Λ. Taking the limit with respect to λ shows that

∥ ∑_{k=1}^{N_{n}} f_{n,k}^{1}g_{n,k}^{1}^{∗} − ∑_{k=1}^{N_{m}} f_{m,k}^{1}g_{m,k}^{1}^{∗}∥

≤ ∥ ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y} − ∑_{k=1}^{N_{m}} Θ_{(ξ_{m,k},f⃗_{m,k}),(η_{m,k},g⃗_{m,k})}^{Y})∥.

Since the ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},f⃗_{n,k}),(η_{n,k},g⃗_{n,k})}^{Y}’s converge in the operator norm on L(Y),

this inequality implies that ∑_{k=1}^{N_{n}} f_{n,k}^{1}g_{n,k}^{1}^{∗} converges to an element in ker ϕ. If we let a_{2} = lim_{n} ∑_{k=1}^{N_{n}} f_{n,k}^{1}g_{n,k}^{1}^{∗} ∈ ker ϕ, then Eq.(1) shows that ag = a_{2}g for all g ∈ ker ϕ. But then a_{1} := a − a_{2} ∈ (ker ϕ)^{⊥}, and consequently a_{1} ∈ J_{X}. Since a = a_{1} + a_{2} the proof is complete.

Lemma 4.6. Let (π̃, t̃) be a representation of Y which is coisometric on ker ϕ ⊕ T, and suppose that π̃|_{A} is injective. For any f ∈ ker ϕ we define ϵ_{i}(f) := (0, …, 0, f, 0, …) ∈ T where f appears in the i^{th} position. Then for every i ∈ ℕ and for every f ∈ ker ϕ, the equation π̃(0, ϵ_{i}(f)) = 0 implies that f = 0.

Proof. First note that it suffices to prove the lemma for f ≥ 0, because if π̃(0, ϵ_{i}(f)) = 0 then π̃(0, ϵ_{i}(ff^{∗})) = π̃(0, ϵ_{i}(f))π̃(0, ϵ_{i}(f))^{∗} = 0, and ff^{∗} = 0 if and only if f = 0. If π̃(0, ϵ_{i}(f)) = 0 and f ≥ 0, then

∥t̃(0, ϵ_{i}(\sqrt{f}))∥^{2} = ∥t̃(0, ϵ_{i}(\sqrt{f}))^{∗}t̃(0, ϵ_{i}(\sqrt{f}))∥

= ∥π̃(⟨(0, ϵ_{i}(\sqrt{f})), (0, ϵ_{i}(\sqrt{f}))⟩_{B}∥ = ∥π̃(0, ϵ_{i}(f))∥ = 0 so that t̃(0, ϵ_{i}(\sqrt{f})) = 0 and consequently

0 = t̃(0, ϵ_{i}(\sqrt{f}))t̃(0, ϵ_{i}(\sqrt{f}))^{∗} = π̃^{(1)}(Θ_{(0,ϵ_{i}(\sqrt{f})),(0,ϵ_{i}(\sqrt{f}))}^{Y})

= \begin{cases}π̃^{(1)}(ϕ_{B}(f,0)) & if i = 1 \\ π̃^{(1)}(ϕ_{B}(0, ϵ_{i−1}(f))) & if i ≥ 2 \end{cases} = \begin{cases}π̃(f,0) & if i = 1 \\ π̃(0, ϵ_{i−1}(f)) & if i ≥ 2. \end{cases}

If i = 1, the fact that π̃|_{A} is injective implies that f = 0. Furthermore, an inductive argument combined with the above equality shows that for all i ∈ ℕ we have f = 0.

Lemma 4.7. Let (π̃, t̃) be a representation of Y which is coisometric on J_{Y} = J(Y). If f⃗ = (f_{1}, f_{2}, …) ∈ T and g⃗ = (g_{1}, g_{2}, …) ∈ T, then

t̃(0, f⃗)t̃(0, g⃗)^{∗} = π̃(f_{1}g_{1}^{∗}, (f_{2}g_{2}^{∗}, f_{3}g_{3}^{∗}, …)).

Proof. For any (ξ, h⃗) ∈ Y = X ⊕ T we have

Θ_{(0,f⃗),(0,g⃗)}(ξ, h⃗) = (0, f⃗)⟨(0, g⃗), (ξ, h⃗)⟩_{B} = (0, f⃗g^{∗}h⃗)
= ϕ_{B}(f_{1}g_{1}^{∗}, ((f_{2}g_{2}^{∗}, …))(ξ, h⃗)

so that Θ_{(0,f⃗),(0,g⃗)} = ϕ_{B}(f_{1}g_{1}^{∗}, (f_{2}g_{2}^{∗}, …)). Thus

t̃(0, f⃗)t̃(0, g⃗)^{∗} = π̃^{(1)}(Θ_{(0,f⃗),(0,g⃗)}) = π̃^{(1)}(ϕ_{B}(f_{1}g_{1}^{∗}, (f_{2}g_{2}^{∗}, …))
= π̃(f_{1}g_{1}^{∗}, (f_{2}g_{2}^{∗}, f_{3}g_{3}^{∗}, …)).

Lemma 4.8. Let (π̃, t̃) be a representation of Y. If ξ ∈ X, a ∈ A, and f⃗ ∈ T, then the following relations hold:

(1) t̃(0, f⃗)π̃(a,0) = 0
(2) t̃(0, f⃗)t̃(ξ,0) = 0
(3) t̃(0, f⃗)t̃(ξ,0)^{∗} = 0

Proof. To see (1) we note that t̃(0, f⃗)π̃(a,0) = t̃((0, f⃗)(a,0)) = t̃(0, 0) = 0. To see (2) and (3) let {e⃗_{λ}}_{λ∈Λ} be an approximate unit for T. Then

t̃(0, f⃗)t̃(ξ,0) = lim_{λ} t̃(0, f⃗e⃗_{λ})t̃(ξ,0) = lim_{λ} t̃(0, f⃗)π̃(0, e⃗_{λ})t̃(ξ,0) = 0

which shows that (2) holds, and

t̃(0, f⃗)t̃(ξ,0)^{∗} = lim_{λ} t̃(0, f⃗e⃗_{λ})t̃(ξ,0)^{∗} = lim_{λ} t̃(0, f⃗)π̃(0, e⃗_{λ})t̃(ξ,0)^{∗}

= lim_{λ} t̃(0, f⃗)(t̃(ξ,0)π̃(0, e⃗_{λ}))^{∗} = 0

which shows that (3) holds.

Lemma 4.9. Let (π̃, t̃) be a representation of Y, and define (π, t) := (π̃|_{A}, t̃|_{X}). If c ∈ C^{∗}(π, t) and f⃗ ∈ T, then

t̃(0, f⃗)c = 0.

Proof. Since C^{∗}(π, t) is generated by elements of the form π̃(a,0) and t̃(ξ,0), the result follows from the relations in Lemma 4.8.

Lemma 4.10. Let (π̃, t̃) be a representation of Y which is coisometric on J_{Y} = J(Y), and define (π, t) := (π̃|_{A}, t̃|_{X}). If n ∈ {0, 1, 2, …}, then any element of the form

t̃(ξ_{1}, f⃗_{1}) … t̃(ξ_{n}, f⃗_{n})π̃(a, h⃗)t̃(η_{n}, g⃗_{n})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}

will be equal to c + π̃(0, k⃗) for some c ∈ C^{∗}(π, t) and some k⃗ ∈ T.

Proof. We shall prove this by induction on n. Base Case: n = 0. Then the term above is equal to π̃(a, h⃗) = π̃(a,0)+π̃(0, h⃗) and the claim holds trivially. Inductive Step: Assume the claim holds for n. Given an element

t̃(ξ_{1}, f⃗_{1}) … t̃(ξ_{n+1}, f⃗_{n+1})π̃(a, h⃗)t̃(η_{n+1}, g⃗_{n+1})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}

it follows from the inductive hypothesis that

t̃(ξ_{2}, f⃗_{2}) … t̃(ξ_{n+1}, f⃗_{n+1})π̃(a, h⃗)t̃(η_{n+1}, g⃗_{n+1})^{∗} … t̃(η_{2}, g⃗_{2})^{∗}

has the form c + π̃(0, k⃗) for c ∈ C^{∗}(π, t) and k⃗ ∈ T. Thus using Lemma 4.9 gives

t̃(ξ_{1}, f⃗_{1}) … t̃(ξ_{n+1}, f⃗_{n+1})π̃(a, h⃗)t̃(η_{n+1}, g⃗_{n+1})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}
= t̃(ξ_{1}, f⃗_{1})(c + π̃(0, k⃗))t̃(η_{1}, g⃗_{1})^{∗}
= (t̃(ξ_{1},0) + t̃(0, f⃗_{1}))(c + π̃(0, k⃗))(t̃(η_{1},0) + t̃(0, g⃗_{1})^{∗})
= t̃(ξ_{1},0)ct̃(η_{1},0)^{∗} + t̃(0, f⃗_{1})π̃(0, k⃗)t̃(0, g⃗_{1})^{∗}
= t̃(ξ_{1},0)ct̃(η_{1},0)^{∗} + t̃(0, f⃗_{1}k⃗)t̃(0, g⃗_{1})^{∗}.

It follows from Lemma 4.7 that t̃(0, f⃗_{1}k⃗)t̃(0, g⃗_{1})^{∗} is of the form c′ + π̃(0, k⃗′) with c′ ∈ im π ⊆ C^{∗}(π, t). Since t̃(ξ_{1},0)ct̃(η_{1},0)^{∗} is also in C^{∗}(π, t) the proof is complete.

We wish to show that if (π̃, t̃) is a representation of Y and if we restrict to obtain (π, t) := (π̃|_{X}, t̃|_{A}), then C^{∗}(π, t) is a corner of C^{∗}(π̃, t̃). If A is unital and X is left essential, then this corner will be determined by the projection π(1,0). However, in the following lemma we wish to consider the general case and must make use of approximate units to define the projection that determines the corner.

Lemma 4.11. Let X be a C^{∗}-correspondence over A and let T := (ker ϕ)^{ℕ} be the tail determined by ker ϕ. If Y := X ⊕ T is the C^{∗}-correspondence over B := A ⊕ T formed by adding the tail T to X, and if (π̃, t̃) is a representation of Y, then there exists a projection p ∈ M(C^{∗}(π̃, t̃)) with the property that for all a ∈ A, ξ ∈ X, and f⃗ ∈ T the following relations hold:

(1) pt̃(ξ, f⃗) = t̃(ξ, (f_{1}, 0, 0, …))
(2) t̃(ξ, f⃗)p = t̃(ξ,0)
(3) pπ̃(a, f⃗) = π̃(a, f⃗)p = π̃(a,0)

Proof. Let {e⃗_{λ}}_{λ∈Λ} be an approximate unit for T, and for each λ ∈ Λ let e⃗_{λ} = (e_{λ}^{1}, e_{λ}^{2}, …). Consider {π̃(0, e⃗_{λ})}_{λ∈Λ}. For any element

t̃(ξ_{1}, f⃗_{1}) … t̃(ξ_{n}, f⃗_{n})π̃(a, h⃗)t̃(η_{m}, g⃗_{m})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}  (2)

we have

lim_{λ} π̃(0, e⃗_{λ})t̃(ξ_{1}, f⃗_{1}) … t̃(ξ_{n}, f⃗_{n})π̃(a, h⃗)t̃(η_{m}, g⃗_{m})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}

= lim_{λ} t̃(0, (0, e_{λ}^{1}f_{12}, e_{λ}^{2}f_{13}, …) … t̃(ξ_{n}, f⃗_{n})π̃(a, h⃗)t̃(η_{m}, g⃗_{m})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}
= t̃(0, (0, f_{12}, f_{13}, …)) … t̃(ξ_{n}, f⃗_{n})π̃(a, h⃗)t̃(η_{m}, g⃗_{m})^{∗} … t̃(η_{1}, g⃗_{1})^{∗}

so this limit exists. Now since any c ∈ C^{∗}(π̃, t̃) can be approximated by a finite sum of elements of the form shown in (2), it follows that lim_{λ} π̃(0, e⃗_{λ})c exists for all c ∈ C^{∗}(π̃, t̃). Let us view C^{∗}(π̃, t̃) as a C^{∗}-correspondence over itself (see [20, Example 2.10]). If we define q : C^{∗}(π̃, t̃) → C^{∗}(π̃, t̃) by q(c) = lim_{λ} π̃(0, e⃗_{λ})c then we see that for any c, d ∈ C^{∗}(π̃, t̃) we have

d^{∗}q(c) = lim_{λ} d^{∗}π̃(0, e⃗_{λ})c = lim_{λ}(π̃(0, e⃗_{λ})d)^{∗}c = q(d)^{∗}c

and hence q is an adjointable operator on C^{∗}(π̃, t̃). Therefore q defines (left multiplication by) an element in the multiplier algebra M(C^{∗}(π̃, t̃)) [20, Theorem 2.47]. It is easy to check that q^{2} = q^{∗} = q so that q is a projection. Now if we let p := 1 − q in M(C^{∗}(π̃, t̃)), then it is easy to check that relations (1), (2), and (3) follow from the definition of q.

Proof of Theorem 4.3. (a) Let I := ker ϕ, set H_{0} := π(I)H_{X}, and define H_{T} := ⨁_{i=1}^{∞} H_{i} where H_{i} = H_{0} for all i = 1, 2, . . .. We define t̃: Y → B(H_{X} ⊕ H_{T}) and π̃ : B → B(H_{X} ⊕ H_{T}) as follows: Viewing Y as Y = X ⊕ T and B as B = A ⊕ T, for any (h, (h_{1}, h_{2}, …)) ∈ H_{Q} ⊕ H_{T} we define

t̃(ξ, (f_{1}, f_{2}, …))(h, (h_{1}, h_{2}, …)) = (t(ξ)h + π(f_{1})h_{1}, (π(f_{2})h_{2}, π(f_{3})h_{3}, …))

and

π̃(a, (f_{1}, f_{2}, …))(h, (h_{1}, h_{2}, …)) = (π(a)h, (π(f_{1})h_{1}, π(f_{2})h_{2}, …)).

Then it is straightforward to show that (π̃, t̃) is a representation of Y on H_{Q} ⊕ H_{T}. To see that (π̃, t) is coisometric on J(Y), choose an element (a, (f_{1}, f_{2}, …)) ∈ J(Y). By Lemma 4.5 we know that a = a_{1} + a_{2} for a_{1} ∈ J_{X} and a_{2} ∈ ker ϕ. Furthermore, since a_{1} ∈ J(X) we may write ϕ(a_{1}) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{ξ_{n,k},η_{n,k}}^{X} for some ξ_{n,k}, η_{n,k} ∈ X. It follows that

ϕ_{B}(a_{1},0) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},0),(η_{n,k},0)}^{Y} ∈ K(Y).

In addition, since a_{2} ∈ ker ϕ we see that if we let {e⃗_{λ}}_{λ∈Λ} be an approximate unit for T with e⃗_{λ} = (e_{λ}^{1}, e_{λ}^{2}, …) for each λ, then

ϕ_{B}(a_{2}, f⃗) = lim_{λ} Θ_{(0,(a,f_{1},f_{2},...)),(0,(e_{λ}^{1},e_{λ}^{2},...))}^{Y} ∈ K(Y).

Now for any n ∈ ℕ we see that {e_{λ}^{n}}_{λ∈Λ} is an approximate unit for ker ϕ. Furthermore, we see that for all (ξ, f⃗), (η, g⃗) ∈ Y = X ⊕ T we have

t̃(ξ, f⃗)t̃(η, g⃗)^{∗} = (t(ξ)t(η)^{∗} + π(f_{1}g_{1}^{∗}), (π(f_{2}g_{2}^{∗}), π(f_{3}g_{3}^{∗}), …))

and thus

π̃^{(1)}(ϕ_{B}(a, f⃗))
= π̃^{(1)}(ϕ_{B}(a_{1},0)) + π̃^{(1)}(ϕ_{B}(a_{2}, f⃗))
= lim_{n} ∑_{k=1}^{N_{n}} t̃(ξ_{n,k},0)t̃(η_{n,k},0)^{∗} + lim_{λ} t̃(0, (a_{2}, f_{1}, f_{2}, …))t̃(0, (e_{λ}^{1}, e_{λ}^{2}, …))^{∗}

= lim_{n} ∑_{k=1}^{N_{n}}(t(ξ_{n,k})t(η_{n,k})^{∗},0) + lim_{λ}(π(a_{2}e_{λ}^{1}), (π(f_{1}e_{λ}^{2}), π(f_{2}e_{λ}^{3}), …))
= (π^{(1)}(ϕ(a_{1})),0) + (π(a_{2}), (π(f_{1}), π(f_{2}), …))
= (π(a_{1}),0) + (π(a_{2}), (π(f_{1}), π(f_{2}), …))
= π̃(a, f⃗)

so (π̃, t̃) is coisometric on J(Y).

(b) If (π̃, t̃) is a representation of Y in a C^{∗}-algebra C which is coisometric on J(Y), then it is straightforward to see that the restriction (π̃|_{A}, t̃|_{X}) is a representation. To see that (π̃|_{A}, t̃|_{X}) is coisometric on J_{X}, choose an element a ∈ J_{X}. Since J_{X} ⊆ J(X) we may write ϕ(a) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{ξ_{n,k},η_{n,k}}^{X} for some ξ_{n,k}, η_{n,k} ∈ X. In addition, since a ∈ (ker ϕ)^{⊥} ⊆ J_{X} we have that

ϕ_{B}(a,0) = lim_{n} ∑_{k=1}^{N_{n}} Θ_{(ξ_{n,k},0),(η_{n,k},0)}^{Y} ∈ K(Y)

and we have

π̃|_{A}^{(1)}(ϕ_{A}(a)) = lim_{n} ∑_{k=1}^{N_{n}} t̃|_{X}(ξ_{n,k})t̃|_{X}(η_{n,k})^{∗} = lim_{n} ∑_{k=1}^{N_{n}} t̃(ξ_{n,k},0)t̃(η_{n,k},0)^{∗}
= π̃^{(1)}(ϕ_{B}(a,0)) = π̃(a,0) = π̃|_{A}(a)

so (π̃|_{A}, t̃|_{X}) is coisometric on J_{X}.

Furthermore, suppose that the restriction π̃|_{A} is injective. If (a, f⃗) ∈ B := A⊕T and π̃(a, f⃗) = 0, let {g_{λ}}_{λ∈Λ} be an approximate unit for ker ϕ, and for any f ∈ ker ϕ and i ∈ ℕ let ϵ_{i}(f) := (0, …, 0, f, 0, …) where f is in the i^{th} position. Since π̃(a, f⃗) = 0 we see that if we write f⃗ = (f_{1}, f_{2}, …), then for all i ∈ ℕ we have

π̃(0, ϵ_{i}(g_{λ}f_{i})) = π̃(0, ϵ_{i}(g_{λ}))π̃(a, f⃗) = 0,

and taking limits with respect to λ shows that π̃(0, ϵ_{i}(f_{i})) = 0 for all i ∈ ℕ. From Lemma 4.6 it follows that f_{i} = 0 for all i ∈ ℕ. Thus f⃗ = 0, and since π̃|_{A} is injective we also have that a = 0. Hence π̃ is injective.

(c) The fact that (π, t) := (π_{Y} |_{A}, t_{Y} |_{X}) is a representation which is coisometric on J_{X} follows from Part (b). Furthermore, the fact that ρ_{(π,t)} is injective follows from Part (a) which shows that any ∗-representation of O_{X} factors through a ∗-representation of O_{Y}. All that remains is to show that im ρ_{(π,t)} = C^{∗}(π, t) is a full corner of O_{Y}. Let p ∈ M(O_{Y}) be the projection described in Lemma 4.11. We shall first show that C^{∗}(π, t) = pO_{Y} p. To begin, we see from the relations in Lemma 4.11 that for all a ∈ A we have pπ(a)p = pπ_{Y} (a,0)p = π_{Y} (a,0) = π(a) and for all ξ ∈ X we have pt(ξ)p = p(t_{Y} (ξ,0))p = t_{Y} (ξ,0) = t(ξ). Thus C^{∗}(π, t) ⊆ pO_{Y} p. To see the reverse inclusion, note that any element in O_{Y} is the limit of sums of elements of the form

t_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n}, f⃗_{n})π_{Y} (a, h⃗)t_{Y} (η_{m}, g⃗_{m})^{∗} … t_{Y} (η_{1}, g⃗_{1})^{∗}

and thus any element of pO_{Y} p is the limit of sums of elements of the form

pt_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n}, f⃗_{n})π_{Y} (a, h⃗)t_{Y} (η_{m}, g⃗_{m})^{∗} … t_{Y} (η_{1}, g⃗_{1})^{∗}p

Therefore, it suffices to show that each of these elements is in C^{∗}(π, t). Now if n ≥ m, then we may use Lemma 4.10 to write

t_{Y} (ξ_{n−m+1}, f⃗_{n−m+1}) … t_{Y} (ξ_{n}, f⃗_{n})π_{Y} (a, h⃗)t_{Y} (η_{m}, g⃗_{m})^{∗} … t_{Y} (η_{1}, g⃗_{1})^{∗}

as c + π_{Y} (0, k⃗) for c ∈ C^{∗}(π, t) and k⃗ ∈ T. Then

pt_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n}, f⃗_{n})π_{Y} (a, h⃗)t_{Y} (η_{m}, g⃗_{m})^{∗} … t_{Y} (η_{1}, g⃗_{1})^{∗}p
= pt_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n−m}, f⃗_{n−m})(c + π_{Y} (0, k⃗))p
= pt_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n−m}, f⃗_{n−m})cp
= pt_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n−m}, f⃗_{n−m})pcp
= pt_{Y} (ξ_{1}, f⃗_{1}) … t_{Y} (ξ_{n−m−1}, f⃗_{n−m−1})pt_{Y} (ξ_{n−m},0)pcp
⋮
= pt_{Y} (ξ_{1},0)p … pt_{Y} (ξ_{n−m},0)pcp
= t_{Y} (ξ_{1},0) … t_{Y} (ξ_{n−m},0)c
= t(ξ_{1}) … t(ξ_{n−m})c

which is in C^{∗}(π, t). The case when n ≤ m is similar. Hence pO_{Y} p ⊆ C^{∗}(π, t). To see that the corner C^{∗}(π, t) = pO_{Y} p is full, suppose that I is an ideal in O_{Y} that contains C^{∗}(π, t). For f ∈ ker ϕ and n ∈ N define ϵ_{n}(f) := (0, …, 0, f, 0, …) ∈ T, where the term f is in the n^{th} position. Let {e_{λ}}_{λ∈Λ} be an approximate unit for ker ϕ. Now t_{Y} (ξ,0), π_{Y} (a,0) ∈ C^{∗}(π, t) ⊆ I for all a ∈ A and ξ ∈ X, and since T is the c_{0}-direct sum of countably many copies of ker ϕ in order to show that I is all of O_{Y} it suffices to prove that for all n ∈ N and λ ∈ Λ we have t_{Y} (0, ϵ_{n}(e_{λ})) ∈ I and π_{Y} (0, ϵ_{n}(e_{λ})) ∈ I. We shall prove this by induction on n. Base Case: For any β, λ ∈ Λ we have from Lemma 4.7 that

t_{Y} (0, ϵ_{1}(e_{λ}))t_{Y} (0, ϵ_{1}(e_{β}))^{∗} = π_{Y} (e_{λ}e_{β},0) ∈ I.

Also for any α ∈ Λ we have

t_{Y} (0, (ϵ_{1}(e_{λ}e_{β}e_{α})) = t_{Y} (0, ϵ_{1}(e_{λ}))π_{Y} (0, ϵ_{1}(e_{β}^{∗}e_{α}))
= t_{Y} (0, ϵ_{1}(e_{λ}))t_{Y} (0, ϵ_{1}(e_{β}))^{∗}t_{Y} (0, ϵ_{1}(e_{α}))

which is in I. Taking limits with respect to α and β gives

t_{Y} (0, ϵ_{1}(e_{λ})) = lim_{β} lim_{α} t_{Y} (0, ϵ_{1}(e_{λ}e_{β}e_{α})) ∈ I.

Furthermore, since t_{Y} (0, ϵ_{1}(e_{λ})) ∈ I for all λ ∈ Λ, we see that

π_{Y} (0, ϵ_{1}(e_{λ})) = lim_{β} π_{Y} (0, ϵ_{1}(e_{λ}e_{β})) = lim_{β} t_{Y} (0, ϵ_{1}(e_{λ}))^{∗}t_{Y} (0, ϵ_{1}(e_{β})) ∈ I.

Inductive step: Suppose that t_{Y} (0, ϵ_{n}(e_{λ})), π_{Y} (0, ϵ_{n}(e_{λ})) ∈ I for any λ ∈ Λ. Then for all λ, β ∈ Λ we have

t_{Y} (0, ϵ_{n+1}(e_{λ}))t_{Y} (0, ϵ_{n+1}(e_{β}))^{∗} = π_{Y}^{(1)}(Θ_{(0,ϵ_{n+1}(e_{λ})),(0,ϵ_{n+1}(e_{β}))}^{Y})
= π_{Y}^{(1)}(ϕ_{B}(0, ϵ_{n}(e_{β}e_{λ})))
= π_{Y} (0, ϵ_{n}(e_{β}e_{λ}))
= π_{Y} (0, ϵ_{n}(e_{β}))π_{Y} (0, ϵ_{n}(e_{λ}))

which is in I. Thus for any α ∈ Λ we have that

t_{Y} (0, ϵ_{n+1}(e_{λ}e_{β}e_{α})) = t_{Y} (0, ϵ_{n+1}(e_{λ}))π_{Y} (0, ϵ_{n+1}(e_{β}e_{α}))
= t_{Y} (o, ϵ_{n+1}(e_{λ}))t_{Y} (0, ϵ_{n+1}(e_{β}))^{∗}t_{Y} (0, ϵ_{n+1}(e_{α}))

is in I. Taking limits with respect to α and β gives

t_{Y} (0, ϵ_{n+1}(e_{λ})) = lim_{β} lim_{α} t_{Y} (0, ϵ_{n+1}(e_{λ}e_{β}e_{α})) ∈ I.

Furthermore, since t_{Y} (0, ϵ_{n+1}(e_{λ})) ∈ I for all λ ∈ Λ, we have

π_{Y} (0, ϵ_{n+1}(e_{λ})) = lim_{β} π_{Y} (0, ϵ_{n+1}(e_{β}e_{λ})) = lim_{β} t_{Y} (0, ϵ_{n+1}(e_{β}))^{∗}t_{Y} (0, ϵ_{n+1}(e_{λ}))

which is in the ideal I.

5 Gauge-Invariant Uniqueness

Recall that we let γ denote the gauge action of 𝕋 on O_{X}. A gauge-invariant uniqueness was proven in [6, Theorem 4.1] for (augmented) Cuntz-Pimsner algebras. Our method of adding tails, together with Theorem 4.3, will allow us to extend this theorem to the case when ϕ is not injective, and ultimately to all relative Cuntz-Pimsner algebras. The following Gauge-Invariant Uniqueness Theorem was proven by Katsura using direct methods in [11, Theorem 6.4]. We shall now give an alternate proof, showing how the method of adding tails can be used to bootstrap [6, Theorem 4.1] to the general case.

Theorem 5.1 (Gauge-Invariant Uniqueness). Let X be a C^{∗}- correspondence over A, and let (π_{X}, t_{X}) be a universal J(X)-coisometric representation of X. If ρ : O_{X} → C is a homomorphism between C^{∗}-algebras which satisfies the following two conditions:

1. the restriction of ρ to π_{X}(A) is injective

2. there is a strongly continuous action β : 𝕋 → Aut(ρ(O_{X})) such that β_{z} ∘ ρ = ρ ∘ γ_{z} for all z ∈ 𝕋

then ρ is injective.

Remark 5.2. When ϕ is injective, the statement above is actually an equivalent reformulation of [6, Theorem 4.1]. The equivalence relies on the fact that for any C^{∗}-correspondence X, the universal J(X)-coisometric representation (i_{A}, i_{X}) has the property that i_{A} is injective if and only if the left action ϕ is injective.

Proof of Theorem 5.1. Let T := (ker ϕ)^{ℕ} be the tail determined by ker ϕ, and let Y := X ⊕ T be the C^{∗}-correspondence over B := A ⊕ T formed by adding the tail T to X. By Theorem 4.3(c) we may identify (O_{X}, π_{X}, t_{X}) with (S, π_{Y} |_{A}, t_{Y} |_{X}) where S is the C^{∗}-subalgebra of O_{Y} generated by

{π_{Y} (a,0), t_{Y} (ξ,0) : a ∈ A and ξ ∈ X}.

Since β : 𝕋 → Aut(im ρ) is an action of 𝕋 on im ρ, there exists a Hilbert space H_{X}, a faithful representation κ : im ρ → B(H_{X}), and a unitary representation U : 𝕋 → U(H_{X}) such that

κ(β_{z}(x)) = U_{z}κ(x)U_{z}^{∗} for all x ∈ im ρ and z ∈ 𝕋.

In addition, since τ := κ ∘ ρ is a ∗-homomorphism from S into B(H_{X}) which is faithful on π_{X}(A), it follows from Theorem 4.3(a) that τ may be extended to a ∗-homomorphism τ̃ : O_{Y} → B(H_{X} ⊕ H_{T}) with τ̃ faithful on π_{Y} (B). We shall now define a unitary representation W : 𝕋 → B(H_{X} ⊕ H_{T}) as follows. We see from the proof of Theorem 4.3(a) that H_{T} := ⨁_{i=1}^{∞} H_{i}. Thus for (h, (h_{1}, h_{2}, …)) ∈ H_{Q} ⊕ H_{T} we define

W_{z}(h, (h_{1}, h_{2}, …)) := (U_{z}h, (z^{−1}h_{1}, z^{−2}h_{2}, …)) for z ∈ 𝕋.

We may then define β̃ : 𝕋 → Aut(B(H_{X} ⊕ H_{T})) by β̃_{z}(T_{0}) := W_{z}T_{0}W_{z}^{∗}, and we see that β̃ is a strongly continuous gauge action. Furthermore, if γ′ denotes the gauge action of 𝕋 on O_{Y}, then β̃_{z} ∘ τ̃ = τ̃ ∘ γ′_{z} (to see this recall how the extension τ̃ is defined in the proof of Theorem 4.3(a) and then simply check on the generators {t_{Y} (ξ, f⃗), π_{Y} (a, g⃗) : ξ ∈ X, a ∈ A, and f⃗, g⃗ ∈ T }). Thus by [6, Theorem 4.1] we have that τ̃ is injective. Hence τ̃|_{S} = τ = κ ∘ ρ is injective, and ρ is injective.

To conclude this section we shall interpret our result in the relative Cuntz-Pimsner setting. Remark 5.3. Katsura has shown in [12] that if O(K, X) is a relative Cuntz-Pimsner algebra, then there exists a C^{∗}-correspondence X′ with the property that O_{X′} is naturally isomorphic to O(K, X). Using this analysis one can obtain the following interpretation of Theorem 5.1 for relative Cuntz-Pimsner algebras.

Interpretation of Theorem 5.1 for Relative Cuntz-Pimsner Algebras: Let X be a C^{∗}-correspondence with left action ϕ : X → L(X), let K be an ideal in J(X) := ϕ^{−1}(K(X)), and let (π_{X}, t_{X}) be a universal K-coisometric representation of X. If ρ : O_{X} → C is a homomorphism between C^{∗}-algebras which satisfies the following three conditions:

(1) the restriction of ρ to π_{X}(A) is injective

(2) if ρ(π_{X}(a)) ∈ ρ(π_{X}^{(1)}(K(X))), then π_{X}(a) ∈ π_{X}(K)

(3) there is a strongly continuous action β : 𝕋 → Aut(ρ(O_{X})) such that β_{z} ∘ ρ = ρ ∘ γ_{z} for all z ∈ 𝕋 then ρ is injective.

Finally, we mention that if we define a map T_{K} : J(X) → O(K, X) by

T_{K}(a) := π_{X}(a) − π_{X}^{(1)}(ϕ(a))

then the equation

T_{K}(a)T_{K}(b) = (π_{X}(a) − π_{X}^{(1)}(ϕ(a)))(π_{X}(b) − π_{X}^{(1)}(ϕ(b)))

= π_{X}(ab) − π_{X}^{(1)}(ϕ(a))π_{X}(b) − π_{X}(a)π_{X}^{(1)}(ϕ(b)) + π_{X}^{(1)}(ϕ(ab))

= π_{X}(ab) − π_{X}^{(1)}(ϕ(ab))
= T_{K}(ab)

shows that this map is a homomorphism. If π_{X} is injective (which by [15, Proposition 2.21] occurs if and only if K ∩ ker ϕ = ∅), then we may replace Condition (2) in the above statement by the condition

(2′) the restriction of ρ to T_{K}(J(X)) is injective.

6 Gauge-Invariant Ideals

In this section we use Theorem 5.1 to characterize the gauge-invariant ideals in C^{∗}-algebras associated to certain correspondences.

Definition 6.1. Let X be a C^{∗}-correspondence over A. We say that an ideal I ◁ A is X-invariant if ϕ(I)X ⊆ XI. We say that an X-invariant ideal I ◁ A is X-saturated if

a ∈ J_{X} and ϕ(a)X ⊆ XI ⟹ a ∈ I.

Remark 6.2. In [9] the authors only considered Hilbert bimodules (i.e. C^{∗}- correspondences) for which ϕ is injective and ϕ(A) ⊆ K(X), and thus the definition of X-saturated that they gave was that a ∈ A and ϕ(a)X ⊆ XI implies a ∈ I. Since J_{X} = A throughout their paper, this notion is equivalent to the one defined in Definition 6.1. In [6, Remark 3.11] it was suggested that the definition of X-saturated for general C^{∗}-correspondences should also be that a ∈ A and ϕ(a)X ⊆ XI implies a ∈ I. However, after considering how the definition of saturated was extended to (or rather modified for) non-row-finite graphs in [1, §3] and [3, §3] we believe that Definition 6.1 is the appropriate generalization.

Recall that if I is an ideal of A, then

X_{I} := {x ∈ X : ⟨x, y⟩_{A} ∈ I for all y ∈ X}

is a right Hilbert A-module, and by the Hewitt-Cohen Factorization Theorem X_{I} = XI := {x · i : x ∈ X and i ∈ I} (see [6, §2]). Furthermore, X/XI is a right Hilbert A/I-module in the obvious way [6, Lemma 2.1]. In order for X/XI to be a C^{∗}-correspondence, we need the ideal I to be X-invariant. Let q^{I} : A → A/I and q^{XI} : X → X/XI be the appropriate quotient maps. If I is X-invariant, then one may define ϕ_{A/I} : A/I → L(X/XI) by

ϕ_{A/I}(q^{I}(a))(q^{XI}(x)) := q^{XI}(ϕ(a)(x))

and with this action X/XI is a C^{∗}-correspondence over A/I [6, Lemma 3.2].

Lemma 6.3. Let X be a C^{∗}-correspondence over a C^{∗}-algebra A, and let I be an X-saturated X-invariant ideal in A. If q^{I} : A → A/I denotes the quotient map, then

q^{I}(J_{X}) ⊆ J_{X/XI}.

Furthermore, if X has the following two properties:

1. ϕ(A) ⊆ K(X)

2. ker ϕ is complemented in A (i.e. there exists an ideal J of A with the property that A = J ⊕ ker ϕ),

then

q^{I}(J_{X}) = J_{X/XI}.

Proof. Let a ∈ J_{X}. Then a ∈ J(X), and it follows from [6, Lemma 2.7] that q^{I}(a) ∈ J(X/XI). Also, if q^{I}(b) ∈ ker ϕ_{A/I}, then q^{I}(ab) ∈ ker ϕ_{A/I} and for all x ∈ X we have

q^{XI}(ϕ(ab)(x)) = ϕ_{A/I}(ab)q^{XI}(x) = 0

and thus

ϕ(ab)XI ⊆ XI.  (3)

Since a ∈ J_{X} and J_{X} is an ideal, we see that ab ∈ J_{X}. Now since I is X-saturated, (3) implies that ab ∈ I and q^{I}(a)q^{I}(b) = q^{I}(ab) = 0. Thus q^{I}(a) ∈ (ker ϕ_{A/I})^{⊥} and q^{I}(a) ∈ J_{X/XI}. Now suppose that Conditions (1) and (2) in the statement of the lemma hold. Since ϕ(A) ⊆ K(X) it follows that J(X) = A. In addition, [6, Lemma 2.7] shows that q^{I}(J(X)) = J(X/XI). From Condition (2) we know that A = J ⊕ker ϕ for some ideal J of A. However, the definition of J_{X} then implies that J = J_{X}. Thus if a ∈ A and q^{I}(a) ∈ J_{X/XI}, then we may write a = b + c for b ∈ J_{X} and c ∈ ker ϕ. But then q^{I}(b) ∈ J_{X/XI} by the first part of the lemma, and q^{I}(c) = q^{I}(a) − q^{I}(b) ∈ J_{X/XI}. Since c ∈ ker ϕ it follows that for all x ∈ X we have

ϕ_{A/I}(q^{I}(c))q^{XI}(x) = q^{XI}(ϕ(c)(x)) = 0

and thus q^{I}(c) ∈ ker ϕ_{A/I}. Thus q^{I}(c) ∈ J_{X/XI} ∩ ker ϕ_{A/I} = {0} so q^{I}(c) = 0 and q^{I}(a) = q^{I}(b) ∈ q^{I}(J_{X}). Thus J_{X/XI} ⊆ q^{I}(J_{X}).

The following theorem was proven in [9, Theorem 4.3] under the hypotheses that ϕ is injective, A is unital, and X is full and finite projective as a right A-module (so in particular, ϕ(A) ⊆ K(X)). However, Theorem 5.1 allows us to give a fairly simple proof of the result for much more general C^{∗}- correspondences.

Theorem 6.4. Let X be a C^{∗}-correspondence with the following two properties:

1. ϕ(A) ⊆ K(X)

2. ker ϕ is complemented in A (i.e. there exists an ideal J of A with the property that A = J ⊕ ker ϕ),

and let (π_{X}, t_{X}) be a universal J(X)-coisometric representation of X. Then there is a lattice isomorphism from the X-saturated X-invariant ideals of A onto the gauge-invariant ideals of O_{X} given by

I ↦ I(I) := the ideal in O_{X} generated by π_{X}(I)

Proof. To begin we see that I(I) is in fact gauge invariant since

I(I) = \overline{span}\{t_{X}(x_{1}) …t_{X}(x_{n})π_{X}(a)t_{X}(y_{1})^{∗} … t_{X}(y_{m})^{∗}

: a ∈ I, x_{1} … x_{n} ∈ X, y_{1} … y_{m} ∈ X, and n, m ≥ 0}.

In addition, the map I ↦ I(I) is certainly inclusion preserving. To see that the map is surjective, let I be a gauge-invariant ideal in O_{X}. If we define I := π_{X}^{−1}(I), then it is straightforward to show that I is Xinvariant and X-saturated. Now clearly I(I) ⊆ I so there exists a quotient map q : O_{X}/I(I) → O_{X}/I. Furthermore, by [6, Theorem 3.1] we have that O_{X}/I(I) is canonically isomorphic to O(q^{I}(J_{X}), X/XI), which by Lemma 6.3 is equal to O_{X/XI} := O(J_{X/XI}, X/XI). If we identify O_{X}/I(I) with O_{X/XI}, then we see that q(π_{X/XI}(q^{I}(a))) = 0 implies that π_{X}(a) ∈ I so that a ∈ I and q^{I}(a) = 0. Thus q is faithful on π_{X/XI}(A/I). Furthermore, since I is gauge invariant, the gauge action on O_{X} descends to an action on the quotient O_{X}/I, and q intertwines this action and the action on O_{X/XI}. Therefore Theorem 5.1 implies that q is injective and consequently I(I) = I. To see that the above map is injective it suffices to prove that π_{X}(a) ∈ I(I) if and only if a ∈ I. Now O_{X}/I(I) is canonically isomorphic to O_{X/XI} as in the previous paragraph. Hence π_{X}(a) ∈ I(I) implies π_{A/I}(q^{I}(a)) = 0, but since π_{X/XI} is injective by Corollary 4.4 it follows that q^{I}(a) = 0 and a ∈ I.

Remark 6.5. We mention that in [17] we have constructed examples which show that the above theorem does not hold if either of the hypotheses (1) or (2) are removed. We also mention that Katsura [12] has given a description of the gauge-invariant ideals in C^{∗}-algebras associated to general C^{∗}-correspondences in terms certain pairs of ideals in A.

7 Concluding Remarks

In Section 4 we gave a method for “adding tails to sinks” in C^{∗}-correspondences; that is, given a C^{∗}-correspondence X we described how to form a C^{∗}- correspondence Y with the property that the left action of Y is injective and O_{X} is canonically isomorphic to a full corner in O_{Y}. The process of adding tails to C^{∗}-correspondences provides a useful tool for extending results for augmented Cuntz-Pimsner algebras (i.e. C^{∗}-algebras associated to C^{∗}-correspondences in which ϕ is injective) to C^{∗}-algebras associated to general C^{∗}-correspondences. We used this idea in Section 5 to extend the Gauge-Invariant Uniqueness Theorem for augmented Cuntz-Pimsner algebras to the general case. More generally, however, we see that many questions about C^{∗}-algebras associated to correspondences may be reduced to the corresponding questions for augmented Cuntz-Pimsner algebras. For example, we see that for any property that is preserved by Morita equivalence (e.g. simplicity, AF-ness, pure infiniteness), one need only characterize when augmented Cuntz-Pimsner algebras will have this property, and then by adding tails one may easily deduce a theorem for C^{∗}- algebras associated to general C^{∗}-correspondences. In addition, if p ∈ M(O_{Y}) is the projection that determines O_{X} as a full corner of O_{Y} (so that O_{X} ≅ pO_{Y} p), then the Rieffel correspondence from the lattice of ideals of O_{Y} to the lattice of ideals of O_{X} takes the form I ↦ pIp. Furthermore, we see from Lemma 4.11 that p is gauge invariant, and consequently the Rieffel correspondence preserves gauge invariance of ideals. Thus questions about the ideal structure of O_{X}, or about gauge-invariant ideals of O_{X}, may be reduced to the corresponding questions for ideals in the augmented Cuntz-Pimsner algebra O_{Y}. Finally, we mention that in [17, §4] the method of adding tails has proven very useful in the analysis of topological quivers. Topological quivers, which were first introduced in [16, Example 5.4], are generalizations of graphs in which the sets of vertices and edges are replaced by topological spaces. By adding tails to topological quivers in [17] the authors are able to reduce their analyses to the case when there are no sinks, or equivalently, to the case when the left action of the associated C^{∗}-correspondence is injective. This simplifies the proofs of many results for topological quivers and allows one to avoid a number of technicalities.

References

[1] T. Bates, J. H. Hong, I. Raeburn, and W. Szymański, The ideal structure of C^{∗}-algebras of infinite graphs, Illinois J. Math 46 (2002), 1159–1176.

[2] T. Bates, D. Pask, I. Raeburn, and W. Szymański, The C^{∗}-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324.

[3] D. Drinen and M. Tomforde, The C^{∗}-algebras of arbitrary graphs, Rocky Mountain J. Math., to appear.

[4] R. Exel and M. Laca, Cuntz-Krieger algebras for infinite matrices, J. reine angew. Math. 512 (1999), 119–172.

[5] N. Fowler, M. Laca, and I. Raeburn, The C^{∗}-algebras of infinite graphs, Proc. Amer. Math. Soc. 8 (2000), 2319–2327.

[6] N. Fowler, P. Muhly, and I. Raeburn, Representations of Cuntz-Pimsner algebras, Indiana Univ. Math. J. 52 (2003), 569–605.

[7] N. Fowler and I. Raeburn, The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), 155–181.

[8] J. H. Hong and W. Szymański, Purely infinite Cuntz-Krieger algebras of directed graphs, preprint (2002).

[9] T. Kajiwara, C. Pinzari, Y. Watatani, Ideal structure and simplicity of the C^{∗}-algebras generated by Hilbert bimodules, J. Funct. Anal. 159 (1998), 295–322.

[10] T. Katsura, A construction of C^{∗}-algebras from C^{∗}-correspondences, Contemp. Math., Proceedings of the AMS conference “Advances in Quantum Dynamics”, to appear.

[11] T. Katsura, On C^{∗}-algebras associated with C^{∗}-correspondences, preprint (2003).

[12] T. Katsura, Ideal structure of C^{∗}-algebras associated with C^{∗}- correspondences, preprint (2003).

[13] A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), 161–174.

[14] A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), 505–541.

[15] P. Muhly and B. Solel, Tensor algebras over C^{∗}-correspondences (representations, dilations, and C^{∗}-envelopes), J. Funct. Anal. 158 (1998), 389–457.

[16] P. Muhly and B. Solel, On the Morita Equivalence of Tensor algebras, Proc. London Math. Soc. 81 (2000), 113–168.

[17] P. Muhly and M. Tomforde, Topological quivers, preprint (2003).

[18] M. V. Pimsner, A class of C^{∗}-algebras generalizing both Cuntz-Krieger algebras and crossed products by ℤ, Fields Institute Comm. 12 (1997), 189–212.

[19] I. Raeburn and W. Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, preprint (2000).

[20] I. Raeburn and D. P. Williams, Morita Equivalence and Continuous-Trace C^{∗}-Algebras, Math. Surveys & Monographs, vol. 60, Amer. Math. Soc., Providence, 1998.

[21] J. Spielberg, A functorial approach to the C^{∗}-algebras of a graph, Internat. J. Math. 13 (2002), 245–277.

[22] J. Spielberg, Semiprojectivity for certain purely infinite C^{∗}-algebras, preprint (2001).

Paul S. Muhly

Department of Mathematics University of Iowa Iowa City IA 52242-1419 USA muhly@math.uiowa.edu Mark Tomforde Department of Mathematics University of Iowa Iowa City IA 52242-1419 USA tomforde@math.uiowa.edu

The first author was supported by NSF Grant DMS-0070405 and the second author was supported by NSF Postdoctoral Fellowship DMS-0201960.