New York Journal of Mathematics

New York J. Math. 12 (2006) 47–62.

The isolated ideal of a correspondence associated with a topological quiver

Berndt Brenken

Abstract. For a general C*-correspondence E a canonical saturated invariant ideal, on which the correspondence is not supported, is identified. The quotient correspondence is formed and the Cuntz–Pimsner C*-algebra of it is identified both as a relative Cuntz–Pimsner algebra for E, and as a quotient of the Cuntz– Pimsner algebra for E. For the C*-correspondence arising from a topological quiver this process amounts to restricting the base space of vertices to the closed subspace supporting the space of edges.

Contents

Introduction 47 1. The isolated ideal 48 2. Topological quivers 55 References 62

Introduction

Associated with a correspondence E over a C*-algebra A is the Cuntz–Pimsner C*-algebra O_{E}. This is a universal C*-algebra for representations of the correspondence E subject to relations determined by an ideal of A. The algebra contains an isomorphic copy of the algebra A even though the actual correspondence E may only involve a part of A. In the following we form the Cuntz–Pimsner C*-algebra associated with the correspondence restricted to the part of A on which E lives, and show it is a relative Cuntz–Pimsner algebra for the correspondence E. Since this relative Cuntz–Pimsner C*-algebra is a quotient of the Cuntz–Pimsner algebra O_{E} by the part of A independent of E, it is an algebra meriting consideration. Certainly the algebra O_{E} is not simple if E is not based on all of A. If the correspondence E lives on all of A this relative Cuntz–Pimsner algebra is just the usual C*-algebra O_{E}. In the discrete case, namely when the correspondence E is associated with

47^{ISSN 1076-9803/06}

Received October 28, 2005.

Mathematics Subject Classification. 46L08, 46L05.

Key words and phrases. C*-correspondence, Cuntz–Pimsner algebra, Hilbert module, isolated ideal, topological quiver, source, sink, simple.

The author acknowledges support, in connection with this research, from the Natural Sciences and Engineering Research Council of Canada.

a directed graph, this relative Cuntz–Pimsner C*-algebra just ignores the abelian C*-algebra summand generated by the isolated points of the graph.

The organization of the article is as follows. In Section 1, after some preliminary definitions and concepts mainly following the literature, we introduce the ideal N in A of isolated points for a correspondence E over A, intrinsically described as the annihilating ideal of E, and show that restricting this correspondence yields a new correspondence whose Cuntz–Pimsner algebra is a quotient of the Cuntz–Pimsner algebra O_{E}. This quotient algebra is seen to be a relative Cuntz–Pimsner algebra of the correspondence E. The Cuntz–Pimsner algebra of the correspondence can never be simple if there are isolated points and, in a sense, there is no dynamical content for the part of the correspondence over this isolated ideal.

In the second section we apply this to the correspondence associated with a topological quiver ([MT]), or what could also be called a directed topological graph. The new correspondence is then associated with an altered topological quiver, namely one restricted over the topological space of nonisolated vertices. In [MT] two conditions, (L) and (K), are introduced on topological quivers as analogues of these conditions on graphs. The restricted topological quiver is then shown to satisfy condition (L), or condition (K), if and only if the original topological quiver satisfies the same condition. Since these two conditions reflect representation theoretic aspects of the correspondence this illustrates that restricting a correspondence to ignore the isolated points does not affect these crucial properties. As is the case in [MT], the topological constraints of the topological quiver context results in proofs that can be intricate.

Notation. If D is a subset of a topological space Y then the closure of D is denoted by Cl_{Y} D, or if there is no ambiguity by D, while ∂_{Y} D = D ∩ (\overline{Y ⧹D}) is the boundary of D. The interior of D is Int_{Y} D. The algebra of continuous functions on Y is C(Y), while if Y is locally compact Hausdorff C_{c}(Y) is the algebra of continuous functions with compact support. Its closure in the uniform sup norm is the algebra of continuous functions that vanish at infinity, C_{0}(Y). The supports of a function f or of a measure λ are denoted supp(f) and supp(λ) respectively. If f : Y → Z is a continuous map of topological spaces then dom(f) and ran(f) denote the domain and range respectively of f, and the dual map f ^{♯} : C(Z) → C(Y) is given by f ^{♯}(h) = h ∘ f. By an ideal of a C*-algebra A we shall mean a closed two sided ideal, and if B is a subset of a C*-algebra A then I(B) denotes the ideal of A generated by B. For an ideal J of A, J^{⊥} denotes the ideal {a ∈ A | ab = 0, (b ∈ J)}.

1. The isolated ideal

For results and conventions on C*-modules we follow Lance [L]; so if A is a C*-algebra a Hilbert A-module E is a Banach space which is a right A-module with an A-valued inner product ⟨, ⟩_{A}, denoted ⟨, ⟩ if the context is clear. The norm on E is given by ∥x∥^{2} = ∥⟨x, x⟩∥, (x ∈ E); L(E) denotes the C*-algebra of adjointable operators on E while K(E), in analogy with the case when A is the complex numbers, is the closed two-sided ideal of compact operators \overline{span}\{θ_{x,y}^{E}|x, y ∈ E\} where θ_{x,y}^{E}(z) = x ⟨y, z⟩, (z ∈ E). If E is a Hilbert A-module the linear span of {⟨x, y⟩ |x, y ∈ E}, denoted ⟨E,E⟩, has closure a two-sided ideal of A. Note that E ⟨E,E⟩ is dense in E ([L]). The Hilbert module E is called full if ⟨E,E⟩ is dense in A. If A is a C*-algebra then A_{A} refers to the Hilbert module A over itself, where ⟨a, b⟩ = a^{∗}b for a, b ∈ A.

Definition 1.1. If A is a C*-algebra then a C*-correspondence E over A is a right Hilbert A-module E together with a left action of A on E defined by a ∗- homomorphism ϕ_{A} : A → L(E), a · x = ϕ(a)x, for a ∈ A, x ∈ E, where just ϕ is used if the context is clear. The correspondence is called faithful if ϕ is injective, and nondegenerate (or essential [MT]) if span {ϕ(a)x | a ∈ A, x ∈ E} is dense in E.

In the literature a correspondence E over A is also commonly referred to as a Hilbert bimodule over A, although this may also refer to a particular type of correspondence. The identity correspondence A over A is A viewed as a Hilbert module over itself with the left action given by left multiplication.

If E is a C*-correspondence over A, and B is a C*-algebra, we say (T, π) is a representation of E in B, written (T, π) : E → B, if T : E → B is a linear map, π : A → B a ∗-homomorphism with

(1) T^{∗}(x)T (y) = π(⟨x, y⟩),

(2) T (ϕ(a)x) = π(a)T (x),

(3) T (x · a) = T (x)π(a), for all x, y ∈ E, a ∈ A.

The C*-subalgebra of B generated by T (E) ∪ π(A) is denoted C^{∗}(T, π). Note that the first condition ensures that T is an isometry not only if π is injective as is usually noted in the literature, but also if only the restriction of π to the ideal \overline{⟨E,E⟩} of A is injective, since ∥T (x)∥^{2} = ∥π ⟨x, x⟩∥ = ∥⟨x, x⟩∥ = ∥x∥^{2}, (x ∈ E). If ρ : B → C is a ∗-homomorphism of C*-algebras then (ρ∘T, ρ∘π) is a representation of E in C, denoted ρ∘(T, π). If E, F are correspondences over A, B respectively then a morphism from E to F is a pair (T, Π) with Π a ∗-homomorphism from A to B, T: E → F a linear map with ⟨T x, T y⟩_{B} = Π(⟨x, y⟩_{A}) and ϕ_{F}(Π(a))T (x) = T (ϕ_{E}(a)x) for x, y ∈ E, a ∈ A. Thus a representation (T, π) of E is a morphism from E to the identity correspondence of B over B. A morphism (T, Π) from E to F yields a ∗- homomorphism Ψ_{T} : K(E) → K(F) by Ψ_{T} (θ_{x,y}) = θ_{T (x),T (y)} for x, y ∈ E ([KPW]), so using the identification of K(B) with B, a representation (T, π) of E in a C*- algebra B yields a ∗-homomorphism Ψ_{T} : K(E) → B given by θ_{x,y} → T (x)T^{∗}(y). The argument of Lemma 2.2 [KPW] showing that Ψ_{T} is injective if π is injective also serves to show that Ψ_{T} is injective if only the restriction of π to the ideal \overline{⟨E,E⟩} of A is injective.

For E a C*-correspondence over A use J(E) to denote the ideal ϕ^{−1}(K(E)) of A and J_{E} to denote the ideal J(E) ∩ (ker ϕ)^{⊥}.

Definition 1.2. For E a C*-correspondence over A, K an ideal in J(E), a representation (T, π) of E in a C*-algebra B is coisometric on K if Ψ_{T} (ϕ(a)) = π(a) for all a ∈ K.

Given a C*-correspondence E over A and K an ideal in J(E) there is a representation (T_{E}, π_{E}) of E which is coisometric on K and universal among all such representations ([FMR]), in the sense that if (T, π) is a representation of E in a C*-algebra B which is coisometric on K then there is a ∗-homomorphism ρ : C^{∗}(T_{E}, π_{E}) → B with (T, π) = ρ ∘ (T_{E}, π_{E}). The C*-algebra C^{∗}(T_{E}, π_{E}) is called the relative Cuntz– Pimsner algebra of E determined by K and denoted O(K, E). When K = 0 the C*-algebra O(K, E) is denoted T (E) and called the universal Toeplitz C*-algebra for E.

For C*-correspondences Pimsner ([P]) originally introduced the (augmented) C*-algebra O_{E}, where ϕ was injective, as O(J(E), E). The algebra O(J(E), E) was then used as the Cuntz–Pimsner algebra of a correspondence with general ϕ (Remark 2.14 of [MT]). The coisometric condition on the smaller ideal J_{E} first arose in Theorem 1.1 of (the preprint of) [B1], where the graph C*-algebra C^{∗}(E) for a general directed graph E — with any (countable) number of edges, sources, sinks, and isolated vertices — was obtained as a relative Cuntz–Pimsner C*-algebra using J_{E}. In [K1] the ideal J_{E} was viewed as the maximal ideal on which ϕ is an injection into K(E), and the C*-algebra O(J_{E}, E) was investigated as the appropriate analogue of the Cuntz–Pimsner algebra for general C*-correspondences E. In this paper we investigate a relative Cuntz–Pimsner algebra of a general C*-correspondence, where the ideal used to define a universal C*-algebra for coisometric representations is in general larger than the ideal J_{E}; thus, since π(a) = Ψ_{T} (ϕ(a)) = Ψ_{T} (0) = 0 for those a in this larger ideal of coisometry satisfying ϕ(a) = 0, the representation of E into this universal C*-algebra will not be injective in general. What this amounts to, for a general C*-correspondence E over A, is to view the parts of A that E is not supported on as superfluous to the Cuntz–Pimsner algebra of the correspondence.

Definition 1.3 ([MT]). For E a C*-correspondence over A, an ideal I in A is E-invariant if ϕ(I)E ⊆ EI . Such an invariant ideal is called E-saturated if

{a ∈ J_{E} | ϕ(a)E ⊆ EI} ⊆ I.

Proposition 1.4. Let N ⊆ A and E be a C*-correspondence over A.

(1) N satisfies NE = 0 if and only if N ⊆ ker ϕ. If N is an ideal then it is E-invariant.

(2) EN = 0 if and only if N ⊆ ⟨E, E⟩^{⊥}. In this case if N is an invariant ideal

then it is E-saturated. The ideal N = ker(ϕ) ∩ ⟨E,E⟩^{⊥} is E-invariant and E-saturated.

Proof. Part (1) is clear. If EN = 0 and n ∈ N then fn = 0, (f ∈ E), so 0 = ⟨e, fn⟩ = ⟨e, f⟩ n for all e, f ∈ E, and thus n ∈ ⟨E, E⟩^{⊥}. Conversely, let N ⊆ ⟨E, E⟩^{⊥}. Then E ⟨E,E⟩ N = 0, and since E ⟨E,E⟩ is dense in E it follows that EN = 0. If EN = 0 then {a ∈ J_{E} | ϕ(a)E ⊆ EN} ⊆ J_{E} ∩ ker(ϕ) ⊆ ker(ϕ)^{⊥} ∩ ker(ϕ) = 0 which is contained in N, so N is E-saturated. □

Corollary 1.5. Let E be a C*-correspondence over A. The maximal ideal N of A satisfying NE = EN = 0 is ker(ϕ) ∩ ⟨E,E⟩^{⊥}, and N is E-invariant and E-saturated.

For nontrivial E, so for E ≠ 0, the nonzero ideal \overline{⟨E, E⟩} is contained in N^{⊥}, so N is never an essential ideal in A. It is worth noticing that in many examples N = 0; indeed the point of view taken is that if N ≠ 0 then we should consider a new correspondence where N = 0, cf. Proposition 1.9. If α is an automorphism of a C*- algebra A the correspondence associated with this ([P]) is given by the left action A on the Hilbert module A_{A} where ϕ(a)b = α(a)b for a, b ∈ A. Clearly ker ϕ = 0 so N = 0. If α is an endomorphism of A the correspondence is E = \overline{α(A)A}, where the closure is taken in A, with the same inner product as before and with ϕ(a)b = α(a)b for a ∈ A, b ∈ E ([MS1]). Here ker ϕ = ker α, a closed α-invariant ideal of A. Since it is usually the case that one only considers injective endomorphisms, N = 0 here also. The correspondence giving rise to the Cuntz algebra also has N = 0, as will soon be clear.

Let q_{N} : A → A/N (or just q) denote the quotient map. With the E-invariant ideal N of A, and the fact, noted above, that EN = 0, we may form the new C*- correspondence E/EN = E over the C*-algebra A/N where ϕ_{A/N} : A/N → L(E) given by ϕ_{A/N} ∘ q = ϕ is clearly well-defined since N ⊂ ker ϕ. The right action of A/N on E is given by x · q(a) = xa, and ⟨x, y⟩_{A/N} = q(⟨x, y⟩_{A}) for x, y ∈ E and a ∈ A.

Definition 1.6. For E a C*-correspondence over A let N (or N(E) if required) denote the ideal ker(ϕ_{A}) ∩ ⟨E,E⟩^{⊥} of A and E_{N} denote the C*-correspondence over A/N.

The pair (I, q) is a morphism of the correspondence E over A to the correspondence E_{N} over A/N, where I is the identity map on E.

We have q_{N}^{−1}ϕ_{A/N}^{−1}(K(E)) = ϕ^{−1}(K(E)) and q_{N}^{−1}(ker ϕ_{A/N}) = ker ϕ so, since q_{N} is a surjection, ker ϕ_{A/N} = q_{N}(ker ϕ) and q_{N}(J(E)) = J(E_{N}). In general if I is an ideal of a C*-algebra A and q : A → B is a surjective ∗-homomorphism then q(I) is an ideal of B and q(I^{⊥}) ⊆ q(I)^{⊥}. With this observation it is clear that q_{N}(J_{E}) ⊆ [ker ϕ_{A/N}]^{⊥} ∩ ϕ_{A/N}^{−1}(K(E)) = J_{E_{N}}.

Proposition 1.7. (1) If I is an ideal in A then I is E-invariant if and only if

q_{N}(I) is E_{N}-invariant.

(2) If H is an ideal in A/N and H is E_{N}-saturated then q_{N}^{−1}(H) is E-saturated.

Proof. By definition q(I) is E_{N}-invariant if and only if ϕ_{N}(q(I))E ⊆ Eq(I). Since ϕ(I)E = ϕ_{N}(q(I))E and Eq(I) = EI the first part follows.

Given a ∈ J_{E} with ϕ(a)E ⊆ Eq^{−1}(H) we need to show that a ∈ q^{−1}(H). However q(a) ∈ q(J_{E}) ⊆ J_{E_{N}} by the preceding comment and ϕ_{N}(q(a))E = ϕ(a)E ⊆ Eq^{−1}(H) = EH, and since H is E_{N} -saturated we have q(a) ∈ H . Thus a ∈ q^{−1}(H). □

Definition 1.8. For E a C*-correspondence over A and E_{N} the correspondence over A/N with N = N(E) let J(N) denote the ideal q_{N}^{−1}(J_{E_{N}}) of A. Let (T_{E}, π_{E}) be the universal representation of E coisometric on J(N).

The comments preceding the previous proposition show J_{E} ⊆ J(N) ⊆ J(E). The later inclusion is crucial as it allows us to define the C*-algebra O(J(N), E), the relative Cuntz–Pimsner algebra generated by the images of T_{E} and π_{E}. If N = 0 then E_{N} = E and J(N) = J_{E} ∩(ker ϕ)^{⊥} = J_{E}, so O(J(N), E) is the usual C*-algebra O_{E} = O(J_{E}, E) characterized in [K2].

Proposition 1.9. If E is a C*-correspondence over A and E_{N} the correspondence over A/N then the ideal N(E_{N}) of A/N is zero and so

O(J(N(E_{N})), E_{N}) = O(J_{E_{N}} , E_{N}).

Proof. The ideal N(E_{N}) = ker(ϕ_{A/N}) ∩ ⟨E,E⟩_{A/N}^{⊥} where N = ker(ϕ_{A}) ∩ ⟨E,E⟩^{⊥}. Since ker ϕ_{A/N} = q_{N}(ker ϕ) it is enough to show that if a ∈ ker(ϕ) with q_{N}(a) ∈ (⟨E,E⟩_{A/N})^{⊥} then a ∈ ⟨E,E⟩^{⊥}. Now (⟨E,E⟩_{A/N})^{⊥} = (q_{N} ⟨E,E⟩)^{⊥}, so for such el-

ements a we have q_{N}(a ⟨E,E⟩) = 0. Thus a ⟨E,E⟩ ⊆ ker q_{N} = N ⊆ ⟨E,E⟩^{⊥}, so a(⟨E,E⟩ ⟨E,E⟩) = 0 from which it follows that a(⟨E,E⟩) = 0, i.e., a ∈ ⟨E,E⟩^{⊥}. □

For K an ideal of A contained in J(E) and (T, π) a universal covariant representation of E in O(K, E) coisometric on K there is a ∗-homomorphism δ (= δ_{(T,π)}) : J(E) → O(K, E) defined by δ(a) = π(a) − Ψ_{T} (ϕ(a)), (a ∈ J(E)) ([MT]). Since J_{E} ⊆ J(N), the representation (T_{E}, π_{E}) of E is also coisometric on J_{E} and the universal property yields a surjective ∗-homomorphism τ : O(J_{E}, E) → O(J(N), E) satisfying τ ∘ (T, π) = (T_{E}, π_{E}) where (T, π) is a universal representation of E in O(J_{E}, E) coisometric on J_{E}. The kernel of τ is the ideal in O(J_{E}, E) generated by δ(J(N)) where δ = δ_{(T,π)} (Lemma 8.21 of [MT]).

The observations that N ⊆ q^{−1}(J_{E_{N}}) = J(N) , that the equality π_{E} = Ψ_{T_{E}} ∘ ϕ holds on J(N), and that N ⊆ ker(ϕ), together imply that N ⊆ ker π_{E} and so π(N) ⊆ ker τ. Since π is injective on A it follows that ker τ ≠ 0 and O(J_{E}, E) can never be simple if N ≠ 0.

Theorem 1.10. If E is a C*-correspondence over A and E_{N} the correspondence over A/N then the relative Cuntz–Pimsner C*-algebra O(J(N), E), which is a quotient of the Cuntz–Pimsner C*-algebra O(J_{E}, E) of E, is isomorphic to the Cuntz– Pimsner C*-algebra O(J_{E_{N}} , E_{N}) of E_{N}.

Proof. If (T, π) is a representation of the C*-correspondence E_{N} over A/N in a C*- algebra B then (T, π∘q) is a representation of the C*-correspondence E over A in B. Furthermore if (T, π) is coisometric on an ideal J with J_{E_{N}} ⊆ J ⊆ ϕ_{E_{N}}^{−1}(K(E_{N})) then (T, π ∘ q) is coisometric on q^{−1}(J). Applying this to the universal representation (T, π) of E_{N} coisometric on J_{E_{N}} and using the universal representation (T_{E}, π_{E}) of E coisometric on J(N) yields a ∗-homomorphism ρ : O(J(N), E) → O(J_{E_{N}} , E_{N}) with ρ ∘ T_{E} = T and ρ ∘ π_{E} = π ∘ q.

Since N ⊆ ker π_{E} there is a well-defined map π_{0} : A/N → O(J(N), E) with π_{E} = π_{0} ∘ q. It is straightforward to check that (T_{E}, π_{0}) is a representation of the C*-correspondence E_{N} coisometric on J_{E_{N}}, so there is a ∗-homomorphism σ : O(J_{E_{N}} , E_{N}) → O(J(N), E) with σ ∘ T = T_{E} and σ ∘ π = π_{0}. By checking that ρ ∘ σ is the identity on the images of T and π, and similarly that σ ∘ ρ is the identity on the images of T_{E} and π_{E}, we see that ρ = σ^{−1}. □

Note that [K2] has some general conditions under which a relative Cuntz– Pimsner C*-algebra is itself the Cuntz–Pimsner C*-algebra for another correspondence, however it is not clear how to apply this here. At the very least one would need to apply several results from [K2] and also prove, for example, that the ‘Opair (N, J(N))’ is the same as the ‘O-pair ω(π, t)’ where (π, t) is the universal representation of O(J(N), E).

Theorem 1.11. Let E be a C*-correspondence over A and K an ideal of A contained in J(E). If (T_{K}, π_{K}) is a universal covariant representation of E coisometric on K then π_{K}(N) is an ideal in the relative Cuntz–Pimsner C*-algebra O(K, E) and the quotient C*-algebra O(K, E)/π_{K}(N) is isomorphic to O(q(K), E_{N}). In particular the Toeplitz C*-algebra T (E_{N}) is isomorphic to the quotient of T (E) by the ideal N.

Proof. Since N is an ideal in A, π_{K}(N) is an ideal in π_{K}(A). Also ϕ(N)E = EN = 0 implies that π_{K}(N)T_{K}(E) = T_{K}(E)π_{K}(N) = 0 and so π_{K}(N) is an ideal in C^{∗}(T_{K}, π_{K}).

Let q_{E} : O(K, E) → O(K, E)/π_{K}(N) denote the canonical quotient map and define T_{0} = q_{E} ∘ T_{K} and π_{0} : A/N → O(K, E)/π_{K}(N) the well-defined map sending q(a) to q_{E}π_{K}(a), (a ∈ A). One can check that (T_{0}, π_{0}) is a covariant representation of the correspondence E_{N} in the C*-algebra O(K, E)/π(N) and that q_{E} ∘δ_{(T_{K},π_{K})} = δ_{(T_{0},π_{0})} ∘ q. Since (T_{K}, π_{K}) is coisometric on K we have δ_{(T_{K},π_{K})}(K) = 0, so δ_{(T_{0},π_{0})}q(K) = 0 and therefore (T_{0}, π_{0}) is coisometric on q(K).

For (T_{u}, π_{u}) a universal representation of E_{N} coisometric on q(K) the universal property yields a surjective ∗-homomorphism ρ : O(q(K), E_{N}) → O(K, E)/π(N) with ρ ∘ (T_{u}, π_{u}) = (T_{0}, π_{0}). Since (T_{u}, π_{u} ∘ q) is a covariant representation of E coisometric on K, the universal property yields a ∗-homomorphism

σ : O(K, E) → O(q(K), E_{N})

with (T_{u}, π_{u} ∘ q) = σ ∘ (T_{K}, π_{K}). Thus σ contains π(N) in its kernel and therefore defines a ∗-homomorphism σ̃ : O(K, E)/π(N) → O(q(K), E_{N}) satisfying

(T_{u}, π_{u} ∘ q) = σ̃ ∘ q_{E} ∘ (T_{K}, π_{K}).

We see that σ̃ ∘ ρ is the identity map on O(q(K), E_{N}) by checking that

σ̃ ∘ ρ(T_{u}(x)) = σ̃(T_{0}(x)) = σ̃(q_{E} ∘ T_{K}(x)) = σ(T_{K}(x)) = T_{u}(x)

and that

σ̃ ∘ ρ(π_{u}(q(a))) = σ̃(π_{0}(q(a))) = σ̃(q_{E}π_{K}(a)) = σ(π_{K}(a)) = π_{u}(q(a)),

(x ∈ E, a ∈ A). This implies that ρ is injective, and so an isomorphism. Thus (T_{0}, π_{0}) is a universal representation of E_{N} coisometric on q(K).

When K = 0, π_{K} is injective and the last statement follows. □

Note that if π_{K} is injective then π_{0} is injective.

To see that the map ρ in the above proof is an isomorphism one could conceivably have applied the relative gauge invariant uniqueness theorem of [K2] since it is clear that there is a gauge action on O(K, E)/π_{K}(N). However, it is not straightforward to verify the second condition of this result in our context, so a self contained approach was used.

Recall the surjective ∗-homomorphism τ : O(J_{E}, E) → O(J(N), E) described after Proposition 1.9.

Corollary 1.12. If E is a C*-correspondence over A and (T, π) a universal representation of E coisometric on J_{E} then π(N) is an ideal in O(J_{E}, E) contained in ker τ. The quotient C*-algebra O(J_{E}, E)/π(N) is isomorphic to O(q(J_{E}), E_{N}). Furthermore, J_{E} = 0 if and only if q(J_{E}) = 0.

Proof. N is an ideal of J(N) and δ = δ_{(T,π)} is a ∗-homomorphism so δ(N) is an ideal in δ(J(N)). Now N ⊆ ker ϕ so δ = π on N, and I(π(N)) = I(δ(N)) is an ideal in I(δ(J(N))) = ker τ. However π(N) = I(π(N)).

Apply the previous theorem with K = J_{E}, so (T, π) = (T_{K}, π_{K}). Here (T_{0}, π_{0}) is now a universal representation of E_{N} coisometric on q(J_{E}). Note that π_{0} is injective since π is injective.

Since N ⊆ ker ϕ we have J_{E} ⊆ (ker ϕ)^{⊥} ⊆ N^{⊥}, so q is injective on J_{E} . It follows that q(J_{E}) = 0 implies J_{E} = 0. □

It is worth pointing out that the map T_{0} in the corollary is an isometry. To see this first note that the map q restricted to the ideal \overline{⟨E, E⟩} is injective so an isometry, since \overline{⟨E, E⟩} ∩ N ⊆ \overline{⟨E, E⟩} ∩ \overline{⟨E, E⟩}^{⊥} = 0. For (T, π) a covariant representation of a correspondence we had noted above that T is an isometry as long as π|_{⟨E,E⟩} is injective. Since q(⟨E, E⟩) = ⟨E, E⟩_{A/N} it follows that T_{0} = q_{E} ∘ T must also be an isometry.

The C*-algebra O(J(N), E), which is isomorphic to the quotient of O(J_{E}, E) by ker τ = I(δ(J(N))), is therefore isomorphic to the quotient of O(J_{E}, E)/π(N) by the ideal generated by δ(J(N))/π(N) where δ = δ_{(T,π)}, so by the ideal q_{E}I(δ(J(N))) = I(δ_{(T_{0},π_{0})}q(J(N))) = I(δ_{(T_{0},π_{0})}(J_{E_{N}})). Since (T, π) is coisometric on J_{E}, we have δ(J_{E}) = 0, so q(J_{E}) ⊆ ker δ_{(T_{0},π_{0})} and δ_{(T_{0},π_{0})}(J_{E_{N}}) = δ_{(T_{0},π_{0})}(J_{E_{N}} \q(J_{E})). Note that there is a well-defined map δ̃_{(T_{0},π_{0})} from the quotient space ϕ_{N}^{−1}(K)/q(J_{E}) to O(J_{E}, E)/π(N) with δ̃_{(T_{0},π_{0})}([a]) = δ_{(T_{0},π_{0})}(a), a ∈ ϕ_{N}^{−1}(K).

Corollary 1.13. O(J_{E_{N}} , E_{N}) is isomorphic to the quotient of O(J_{E}, E)/π(N) by the ideal generated by δ_{(T_{0},π_{0})}(J_{E_{N}} \q(J_{E})) = δ̃_{(T_{0},π_{0})}(J_{E_{N}} /q(J_{E})). In particular O(J_{E_{N}} , E_{N}) is isomorphic to O(J_{E}, E)/π(N) if q(J_{E}) = J_{E_{N}}.

Proof. The C*-algebra O(J_{E_{N}} , E_{N}) is isomorphic to O(J_{E}, E)/π(N) if and only if δ_{(T_{0},π_{0})}(J_{E_{N}} \q(J_{E})) = 0. □

One condition that ensures q(J_{E}) = J_{E_{N}} is if A = N ⊕ M as a direct sum of C*-algebras. In this case the ideal M = N^{⊥}, and since N ⊆ ker ϕ we have J_{E} ⊆ (ker ϕ)^{⊥} ⊆ N^{⊥} = M. For any ideal J of A we have J = (N ∩ J) ⊕ (M ∩ J); for if a ∈ J with a = n + m, n ∈ N, m ∈ M then for e_{λ} an approximate unit of N, e_{λ}a = e_{λ}n + e_{λ}m = e_{λ}n → n . However, e_{λ}a ∈ J so n ∈ J, and m ∈ J also. Identifying M with A/N, the ∗-homomorphism ϕ_{N} becomes the restriction of ϕ to M, and so ϕ_{N}^{−1}(K) = M ∩ ϕ^{−1}(K), (ker ϕ_{N})^{⊥} = M ∩ (ker ϕ)^{⊥}, and J_{E_{N}} = ϕ_{N}^{−1}(K) ∩ (ker ϕ_{N})^{⊥} = M ∩ J_{E} = J_{E}. Thus O(J_{E_{N}} , E_{N}) is isomorphic to O(J_{E}, E)/π(N) if A = N ⊕ M.

Corollary 1.14. If A = N ⊕ M as a direct sum of C*-algebras then

O(J_{E}, E) ≅ N ⊕ O(J_{E_{N}} , E_{N}).

Proof. Let (T, π) denote a universal covariant representation of E coisometric on J_{E}. Note again that π is injective on A. Theorem 1.11 applied with K = J_{E} noted that the covariant representation (T_{0}, π_{0}) of E_{N} is coisometric on q(J_{E}) which is equal to J_{E_{N}} under our hypothesis. One can check that (T_{0}, π|_{N} ⊕ π_{0}) is then a covariant representation of E in the C*-algebra π(N) ⊕ O(J_{E_{N}} , E_{N}) which is coisometric on J_{E}. Since this representation admits a gauge action, and since π|_{N} ⊕π_{0} is injective on N ⊕M = A, the surjection of O(J_{E}, E) to π(N)⊕O(J_{E_{N}} , E_{N}) given by the universal property is, by the gauge invariant uniqueness theorem ([K3]), actually an injection. □

2. Topological quivers

In [MT] the authors show that a certain topological condition, called Condition (K), on a topological quiver implies that the Cuntz–Pimsner C*-algebra of the topological quiver has only gauge invariant ideals ([MT, Theorem 9.10]). Furthermore, the Cuntz–Pimsner C*-algebra enjoys the Cuntz–Krieger uniqueness property if the topological quiver satisfies Condition (L) ([MT, Theorem 6.16]). We show that the restricted topological quiver satisfies Condition (K), or Condition (L), if and only if the original topological quiver does.

Following [MT], G = (X, E, r, s, λ) is a topological quiver when X, E are a pair of second countable locally compact Hausdorff spaces, r : E → X and s : E → X a pair of continuous maps (the range and source maps) with r open, and λ a family {λ_{x} | x ∈ X} of Radon measures on E with

(1) supp(λ_{x}) = r^{−1}(x), (x ∈ X),

(2) x → ∫_{E} f(α)dλ_{x}(α) ∈ C_{c}(X) for f ∈ C_{c}(X).

A topological quiver G defines a C*-correspondence E (or E(G)) over the C*- algebra A = C_{0}(X) as follows: for f, g ∈ C_{c}(E) an A-valued inner product given by

⟨f, g⟩ (x) = ∫_{r^{−1}(x)} \overline{f(α)}g(α)dλ_{x}(α), (x ∈ X),

defines a norm on C_{c}(E) with completion E, a Hilbert module over C_{0}(X). The left action coming from a ∗-homomorphism ϕ : A → L(E) and the right action of A on an element h of C_{c}(E) are given by

h · g = h(r^{♯}(g))
ϕ(g)h = (s^{♯}(g))h,

for g ∈ A.

We briefly include some comments regarding ideals in an abelian C*-algebra. If D is a closed subset of a compact space Y then D is compact in the subspace topology and the dual i^{♯} of the inclusion i : D → Y is a surjective ∗-homomorphism i^{♯} : C(Y) → C(D) with kernel the ideal I_{D} = {f ∈ C(Y)| f|_{D} = 0} determined by D. We obtain the exact sequence

0 → C_{0}(Y ⧹D) → C(Y) →^{i^{♯}} C(D) → 0.

For the situation that D is closed in a locally compact space Y, thus locally compact in the subspace topology, then Y ⧹D is open in Y and also in its one point compactification Y_{{∞}}. Thus Y_{{∞}}⧹D = D ∪ {∞} is closed, so compact in Y_{{∞}} and may be identified with the one point compactification of the locally compact space D, since Y ⧹D = Y_{{∞}}⧹D_{{∞}}. Now 0 → C_{0}(D) → C(D_{{∞}}) → ℂ → 0 is exact for any locally compact space D, so applying this to the spaces Y ⧹D, Y, and D, and arranging these exact sequences in an array we obtain via the 5-lemma the exact sequence

0 → C_{0}(Y ⧹D) → C_{0}(Y) →^{i^{♯}} C_{0}(D) → 0.

In [MT], X_{sink} denotes the open set of X with ker ϕ ≅ C_{0}(X_{sink}) and it is shown that X_{sink} = X⧹\overline{s(E)}, or equivalently that \overline{s(E)} = \{x ∈ X | f(x) = 0, (f ∈ ker ϕ)\}, so ker ϕ = \{f ∈ C_{0}(X) | f|\overline{s(E)} ≡ 0\}. Recall that \overline{⟨E,E⟩} is an ideal of A.

Proposition 2.1. The ideal \overline{⟨E,E⟩} of A is \{f ∈ A | f ≡ 0 on X⧹r(E)\}.

Proof. Since r is an open map, X⧹r(E) is closed. If x ∈ X with r^{−1}(x) = ϕ then ⟨f, g⟩ (x) = 0 for f, g ∈ C_{c}(E), a dense subspace of E. Thus

X⧹r(E) ⊆ \{x ∈ X | f(x) = 0, (f ∈ \overline{⟨E,E⟩})\} .

To show the reverse inclusion it is enough to show that given x ∈ r(E) there is h ∈ C_{c}(E) with ⟨h, h⟩ (x) ≠ 0 . Since supp(λ_{x}) = r^{−1}(x) ≠ ϕ there is a positive g ∈ C(r^{−1}(x)) with λ_{x}(|g|^{2}) ≩ 0. If K is the compact support of g in r^{−1}(x), then K must also be compact in E, so by Urysohn’s Lemma there is an f ∈ C_{c}(E) with f|_{K} ≡ 1 . By Tietze’s Extension Theorem there is a continuous function l : supp(f) → ℝ with l|_{K} = g. Setting h to be the element of C_{c}(E) which is l · f on supp(f) and 0 on E⧹supp(f) we have

\begin{aligned}⟨h, h⟩ (x) & = ∫_{r^{−1}(x)} |l|^{2} (α)dλ_{x}(α) \\ & ≥ ∫_{r^{−1}(x)∩K} |l|^{2} (α)dλ_{x}(α) \\ & = ∫_{r^{−1}(x)∩K} |g|^{2} (α)dλ_{x}(α) \\ & = ∫_{r^{−1}(x)} |g|^{2} (α)dλ_{x}(α) ≩ 0. \end{aligned}  □

In general for U ⊆ X an open set and I = {f ∈ C_{0}(X) | f|_{X⧹U} ≡ 0} the ideal C_{0}(U) of C_{0}(X) determined by the closed set X⧹U we have that the ideal I^{⊥} is determined by the closed set \overline{U}. Thus the ideal [\overline{⟨E,E⟩}]^{⊥}of A is determined by the closed set \overline{r(E)}. The next proposition follows.

Proposition 2.2. For (X, E, r, s, λ) a topological quiver and E the associated correspondence over A = C_{0}(X), the ideal N = ker(ϕ)∩ [\overline{⟨E,E⟩}]^{⊥} of A is determined by the closed set \overline{s(E) ∪ r(E)}; namely

N = \{f ∈ C_{0}(X) | f|\overline{s(E)∪r(E)} ≡ 0\} .

Recall the terminology from [MT], where X_{fin} and X_{reg} = X_{fin}⧹\overline{X_{sink}} are open sets so that the ideals C_{0}(X_{fin}) and C_{0}(X_{reg}) of A are equal to J(E) = ϕ^{−1}(K(E)) and J_{E} = ϕ^{−1}(K(E)) ∩ (ker ϕ)^{⊥} respectively.

Definition 2.3. Let X_{source} be the open set X⧹\overline{r(E)}, so C_{0}(X_{source}) is isomorphic to the ideal

\{f ∈ C_{0}(X) | f|\overline{r(E)} ≡ 0\} = [\overline{⟨E,E⟩}]^{⊥}.

Define X_{isol} = X_{source}∩X_{sink} = X⧹\overline{s(E) ∪ r(E)}. and set D (or D_{G}) = \overline{s(E) ∪ r(E)}.

Since ker ϕ = C_{0}(X_{sink}) and X_{sink} = X⧹\overline{s(E)}, [ker ϕ]^{⊥} = C_{0}(Int(\overline{s(E)})). From the definitions we have C_{0}(X_{reg}) = J_{E} ⊆ [ker ϕ]^{⊥}, so it follows that X_{reg} ⊆ Int(\overline{s(E))}. In particular X_{reg} is contained in the closed set D = \overline{s(E) ∪ r(E)}.

If E is the C*-correspondence over A = C_{0}(X) associated with the topological quiver G = (X, E, r, s, λ) and N = ker ϕ ∩ [\overline{⟨E,E⟩}]^{⊥} is the ideal C_{0}(X_{isol}) of A, form the correspondence E_{N} over A/N ≅ C_{0}(X)/C_{0}(X_{isol}) = C_{0}(D) as in Section 1. The correspondence E_{N} over A/N is the same as the correspondence associated with the restricted topological quiver G^{N} = (D, E, r, s, λ) where we continue to use r and s to denote the appropriate maps, now viewed with ranges in D. The natural quotient map q : A → A/N is the map i^{♯} : C_{0}(X) → C_{0}(D) where i : D → X is the inclusion of the closed subset D in X . We have ϕ_{N} ∘ q = ϕ where ϕ : C_{0}(X) → L(E) and ϕ_{N} : C_{0}(D) → L(E) define the left actions of A and C_{0}(D) respectively on E. This implies that q(ker ϕ) = ker ϕ_{N}. Similarly q(f) ∈ ϕ_{N}^{−1}(K(E)) if and only if f ∈ ϕ^{−1}(K(E)), so q(ϕ^{−1}(K(E))) = ϕ_{N}^{−1}(K(E)). For any ideal I = {f ∈ C_{0}(X) | f|_{X⧹U} ≡ 0} of C_{0}(X) where U ⊆ X is open, so I = C_{0}(U), the ideal q(I) = {g ∈ C_{0}(D) | f|_{D} ≡ g, f|_{D∩(X⧹U)} ≡ 0} = C_{0}(D ∩ U) since D⧹D∩(X⧹U) = D∩U. It follows that D_{fin} = X_{fin}∩D and D_{sink} = X_{sink}∩D.

Recall that the ideal J_{E} = ϕ^{−1}(K(E)) ∩ [ker ϕ]^{⊥} of A = C_{0}(X) is contained in the ideal J(N) = q^{−1}(J_{E_{N}}) . In the present situation J_{E} = C_{0}(X_{reg}) and J_{E_{N}} = C_{0}(D_{reg}). Thus the inclusion q(J_{E}) ⊆ J_{E_{N}} is equivalent to i^{♯}(C_{0}(X_{reg})) ⊆ C_{0}(D_{reg}). Since i^{♯}(C_{0}(X_{reg})) = C_{0}(D ∩ X_{reg}), the above inclusion of ideals yields D ∩ X_{reg} ⊆ D_{reg}. However we have already seen that X_{reg} ⊆ D, so we have that X_{reg} ⊆ D_{reg}, where both are open subsets of D.

Apply the results of the first section to A = C_{0}(X) and N = C_{0}(X\D), and view the restriction of the map q = i^{♯} : A → A/N to domain J_{E} = C_{0}(X_{reg}) and codomain J_{E_{N}} = C_{0}(D_{reg}) as the natural inclusion of C_{0}(X_{reg}) in C_{0}(D_{reg}). Then

J_{E_{N}} /q(J_{E}) = C_{0}(D_{reg})/C_{0}(X_{reg}) ≅ C_{0}(D_{reg}\X_{reg})

and the C*-algebra O(C_{0}(D_{reg}), E_{N}) is isomorphic to the quotient of

O(C_{0}(X_{reg}), E)/C_{0}(X\D)

by the ideal generated by δ̃_{(T_{0},π_{0})}(C_{0}(D_{reg}\X_{reg})), where E is the correspondence over A associated with a topological quiver, or in fact any correspondence over A.

We briefly recall some terminology from [MT].

Definition 2.4. If G = (X, E, r, s, λ) is a topological quiver, a path (of length n) in G is a finite sequence α := α_{1}...α_{n} with α_{k} ∈ E and r(α_{k}) = s(α_{k+1}) for 1 ≤ k ≨ n. Denote the paths of length n by E^{n} , viewed as a subspace of ΠE and extend r, s to E^{n} by r^{n} : E^{n} → X, s^{n} : E^{n} → X where r^{n}(α) = r(α_{n}) and s^{n}(α) = s(α_{1}). A path α ∈ E^{n} is a loop if s(α_{1}) = r(α_{n}) with base point s^{n}(α). A loop α ∈ E^{n} has an exit if there is a β ∈ E and a k ∈ {1, ..., n} with s(β) = s(α_{k}) and β ≠ α_{k}. We say G satisfies condition (L) if the set of base points of loops in G with no exits has empty interior. Let B = {x ∈ X | x is a base point of a loop in G with no exits} and B_{N} = {x ∈ D | x is a base point of a loop in G^{N} with no exits}.

Theorem 2.5. The topological quiver G = (X, E, r, s, λ) satisfies condition (L) if and only if the restricted topological quiver G^{N} = (D, E, r, s, λ) satisfies condition Proof. We show that Int_{X}B = Int_{D}B using that D is closed in X and that r is an open map. In general Cl_{D}B = D ∩ Cl_{X}B ⊆ Cl_{X}B for any B ⊆ D, so ∂_{D}B = Cl_{D}B ∩ Cl_{D}(D⧹B) ⊆ Cl_{X}B ∩ Cl_{X}(D⧹B) ⊆ Cl_{X}B ∩ Cl_{X}(X⧹B) = ∂_{X}B. Now D is closed in X and since B ⊆ D, Cl_{X}B is contained in D and Cl_{D}B = Cl_{X}B. Thus Int_{X}B = Cl_{X}B⧹∂_{X}B = Cl_{D}B⧹∂_{X}B ⊆ Cl_{D}B⧹∂_{D}B = Int_{D}B.

To see the other inclusion first note that base points of loops lie in r(E), so B ⊆ r(E). Also r(E) is open in X, so open in D. Thus B ⊆ Int_{X}D. If z ∈ Int_{D}B then there is a neighbourhood N_{z} of z in X with (N_{z} ∩ D) ⊆ B ⊆ Int_{X}D. Thus N_{z} ∩ Int_{X}D ⊆ B ∩ Int_{X}D ⊆ Int_{X}D. Since z ∈ B ⊆ Int_{X}D, the set N_{z} ∩ Int_{X}D is an open subset of B containing z and so z ∈ Int_{X}B. □

In order to discuss condition (K) on topological quivers we first recall some material from [MT]. If G = (X, E, r, s, λ) is a topological quiver and U ⊆ X is open, U is called hereditary (for G) if r^{−1}(s(u)) ⊆ U. A hereditary (open) set U is called saturated (for G) if {v ∈ X_{reg} | r(s^{−1}(v)) ⊆ U} ⊆ U. If E is the C*- correspondence over A = C_{0}(X) associated with the topological quiver G, then U is hereditary if and only if the ideal I = C_{0}(U) is E-invariant. In this case, the new C*-correspondence E = E/EI over the C*-algebra A/I is the correspondence associated with the topological quiver G_{U} = (X⧹U, E⧹r^{−1}(U), r_{U}, s_{U}, λ^{U}), where r_{U}, s_{U}, and λ^{U} are just the respective restrictions of r, s, to E⧹r^{−1}(U) and λ to X⧹U. Note also ([MT]) that U is saturated for G is equivalent to C_{0}(U) being an E-saturated ideal of A. The topological quiver G is said to satisfy condition (K) if the topological quiver G_{U} satisfies condition (L) for all open, hereditary, saturated sets U in X.

Clearly a union or finite intersection of open hereditary subsets of X is open and hereditary. Since N = C_{0}(X⧹D) is an invariant saturated ideal in E(G), X⧹D is an open hereditary subset of X. Therefore, if U is an open hereditary set for G = (X, E, r, s, λ) then so is U ∪ (X⧹D). In this case, if U is also saturated for G then, since r(s^{−1}(v)) ⊆ U if and only if r(s^{−1}(v)) ⊆ U ∪ (X⧹D), and X_{reg} ∩ (X⧹D) = ϕ, we see that U ∪ (X⧹D) is also saturated.

Lemma 2.6. Let V ⊆ X be open and V_{D} = D ∩ V. Then:

(1) V_{D} is hereditary in G^{N} if and only if V is hereditary in G.

(2) If V_{D} is saturated in G^{N} then V is saturated in G.

Proof. The first part follows from Proposition 1.4. Suppose V is saturated in G^{N}. To show V is saturated in G we need to show that if v ∈ X with r(s^{−1}(v)) ⊆ V then v ∈ V. However X_{reg} ⊆ D_{reg}, so v is in D_{reg}, and r(s^{−1}(v)) = r(s^{−1}(v))∩D ⊆ V ∩ D = V_{D}. Since V_{D} is saturated in G^{N} we have v ∈ V_{D} ⊆ V. □

Lemma 2.7. If z ∈ D_{reg}⧹X_{reg}, then:

(1) z ∈ ∂_{X} \overline{s(E)}⧹\overline{[\overline{r(E)} ∩ X_{sink}]}.

(2) There is a neighbourhood N_{z} of z such that N_{z} ∩ Int_{X} \overline{s(E)} ⊆ X_{reg} and N_{z} ∩ \overline{r(E)} = ϕ.

Proof. Since D is closed we see that Cl_{D}(A) = Cl_{X}(A) for any subset A of D. If z ∈ D_{reg} = D_{fin}⧹\overline{D_{sink}} then z ∈ D_{fin} = X_{fin} ∩ D ⊆ X_{fin}. Since z ∉ X_{reg} = X_{fin}⧹\overline{X_{sink}} then z must be in \overline{X_{sink}}. However z ∉ X_{sink} since z ∉ D_{sink} = X_{sink} ∩ D. Thus z ∈ ∂_{X}(X_{sink}) = ∂_{X}(\overline{s(E)}). Since D_{sink} = X_{sink} ∩ D = X_{sink} ∩ \overline{r(E)} and z ∉ \overline{D_{sink}}, statement (1) follows.

We noted that z ∈ X_{fin} which is open, so there is a neighbourhood N of z with N ⊆ X_{fin}. Since z ∈ ∂_{X}(s(E)) we have N ∩ Int_{X} \overline{s(E)} is a nonempty open subset in X_{fin}. Since Int_{X} \overline{s(E)} ∩ \overline{X_{sink}} = ϕ this open subset is in X_{reg} = X_{fin}⧹\overline{X_{sink}}. To show that N can be chosen disjoint from \overline{r(E)} it is enough to see that z ∉ \overline{r(E)}, or since z ∈ ∂_{X} \overline{s(E)}, that z ∉ \overline{r(E)} ∩ ∂_{X} \overline{s(E)} = \overline{r(E)} ∩ ∂_{X}(X_{sink}). However, the latter is a subset of \overline{[X_{sink} ∩ \overline{r(E)}]} since any neighbourhood of a point in \overline{r(E)}∩∂_{X}(X_{sink}) meets both X_{sink} and r(E). By (1) we are done. □

Lemma 2.8. Let V ⊆ X be open, V_{D} = V ∩ D, and

B_{V} = {z ∈ D_{reg}⧹X_{reg} | r(s^{−1}(z)) ⊆ V }.

For z ∈ B_{V} there is a neighbourhood M_{z} of z with r(s^{−1}(M_{z})) ⊆ V, M_{z} ∩ \overline{r(E)} = ϕ.

Proof. Let M_{k}, k ∈ ℕ be a decreasing countable neighbourhood base of z. We may choose M_{k} ∩ \overline{r(E)} = ϕ by Lemma 2.7. By Proposition 3.15 of [MT], since z ∈ D_{fin}, there is a neighbourhood W of Z with both \overline{W ∩ D} and s^{−1}(\overline{W ∩ D}) compact in D, and r|_{s^{−1}(W ∩D)} a local homeomorphism. Suppose rs^{−1}(M_{k}) = rs^{−1}(M_{k} ∩ D) ⫅̸ V for all k. Then for each k there is an α_{k} ∈ s^{−1}(M_{k}) with r(α_{k}) ∉ V. There is k_{0} such that W ∩ D ⊇ M_{k} ∩ D for k ≥ k_{0}, so α_{k} ∈ s^{−1}(\overline{W ∩ D}) for k ≥ k_{0}. By compactness there is α ∈ s^{−1}(\overline{W ∩ D}) an accumulation point of α_{k}, so α_{k_{j}} → α. By continuity s(α_{k_{j}}) → s(α). Since s(α_{k_{j}}) ∈ M_{k_{j}} for all j, s(α_{k_{j}}) → z also, and so z = s(α) i.e., α ∈ s^{−1}(z). Since z ∈ B_{V}, r(α) ∈ V. Now by the choice of α_{k}, r(α_{k_{j}}) ∈ X⧹V, so r(α) = lim r(α_{k_{j}}) ∈ X⧹V, a contradiction. □

Definition 2.9. Let G = (X, E, r, s, λ) be a topological quiver and

D = \overline{r(E) ∪ s(E)}.

For V an open hereditary subset of X define Ṽ = V ∪ {∪ {M_{z} | z ∈ B_{V} }} where M_{z} is chosen as in the previous lemma.

Lemma 2.10. If V is hereditary then Ṽ is also; if V is saturated then Ṽ is also.

Proof. First suppose V is hereditary. If α ∈ s^{−1}(Ṽ) then α ∈ s^{−1}(V) or α ∈ s^{−1}(M_{z}) for some z ∈ B_{V}. In the case that α ∈ s^{−1}(V) we have r(α) ∈ V ⊆ Ṽ since V is hereditary. For α ∈ s^{−1}(M_{z}) with z ∈ B_{V} then r(α) ∈ rs^{−1}(M_{z}) ⊆ V ⊆ Ṽ. Thus rs^{−1}(Ṽ) ⊆ Ṽ. To show the remaining claim it is enough, by Lemma 2.6, to show that Ṽ ∩ D is saturated in G^{N} = (D, E, r, s, λ). Suppose v ∈ D_{reg} and r(s^{−1}(v)) ⊆ Ṽ ∩D. We need to show that v ∈ Ṽ ∩D. Since the sets M_{z} added to V are all disjoint from \overline{r(E)} the condition r(s^{−1}(v)) ⊆ Ṽ ∩ D implies r(s^{−1}(v)) ⊆ V. We consider two cases; v ∉ X_{reg} and v ∈ X_{reg}. In the first case v ∈ D_{reg}⧹X_{reg}. Since r(s^{−1}(v)) ⊆ V we have v ∈ B_{V}, and so v ∈ Ṽ ∩ D. On the other hand if v ∈ X_{reg} , then since r(s^{−1}(v)) ⊆ V and V is saturated, we have v ∈ V; and since v ∈ D we have v ∈ Ṽ ∩ D. □

Theorem 2.11. The topological quiver G = (X, E, r, s, λ) satisfies condition (K) if and only if the restricted topological quiver G^{N} = (D, E, r, s, λ) satisfies condition (K).

Proof. First assume G satisfies condition (K). Let V ⊆ D be an arbitrary open, hereditary, saturated set. If W ⊆ X open with W ∩ D = V, then W ∪ (X⧹D) = Ṽ ⊆ X is open with Ṽ ∩ D = V. We have

GṼ = (X⧹Ṽ, E⧹r^{−1}(Ṽ), rṼ , sṼ , λṼ) = (D⧹V, E⧹r^{−1}(V), r_{V} , s_{V} , λ^{V}) = (G^{N})_{V} .

Since Ṽ is saturated and hereditary by Lemma 2.6, GṼ, and therefore also (G^{N})_{V} satisfies condition (L). Since V was arbitrary, G^{N} satisfies condition (K).

Now suppose G^{N} satisfies condition (K). For V ⊆ X open, hereditary, and saturated for G, we need to show G_{V} has condition (L); namely

Int_{X⧹V} {v a base point for a loop in G with no exit} = ϕ.

By Lemma 2.10 Ṽ is open, hereditary, and saturated, so by Lemma 2.6 Ṽ ∩ D is open and hereditary. The proof of Lemma 2.10 showed Ṽ ∩ D is saturated for G^{N}. Since G^{N} satisfies condition (K), G_{Ṽ ∩D}^{N} satisfies condition (L). We note that G_{Ṽ ∩D}^{N} = G_{Ṽ ∪(X⧹D)}, where, by the comments preceding Lemma 2.6, Ṽ ∪ (X⧹D) is open, hereditary, and saturated; so G_{Ṽ ∪(X⧹D)} also satisfies condition (L). Thus

Int_{X⧹(Ṽ ∪(X⧹D))}{v a base point for a loop in G_{Ṽ ∪(X⧹D)} with no exit} = ϕ.

However, since X⧹D and the sets M_{z} are disjoint from \overline{r(E)}, r^{−1}(Ṽ ∪ (X⧹D)) = r^{−1}(V) and E⧹r^{−1}(Ṽ ∪ (X⧹D)) = E⧹r^{−1}(V), so any edge under consideration in G_{Ṽ ∪(X⧹D)} is also under consideration in G, and

{v a base point for an exitless loop in G_{Ṽ ∪(X⧹D)}}

= {v a base point for an exitless loop in G_{V} }.

In general the interior of a set decreases if we take the interior with respect to a larger set, so

Int_{X⧹V} {v a base point for an exitless loop in G_{V} } = ϕ.  □

The following example illustrates some of the concepts above. The quiver G we consider is basically a discrete topological graph where the base space has been enlarged to have a nondiscrete component. For E the C*-correspondence associated with G, the two C*-algebras O(J_{E}, E) and O(J_{E_{N}} , E_{N}) are compared.

Let X be a nondiscrete second countable locally compact Hausdorff space with p_{0}, p_{1} ∈ X, X = {p_{0}} ∪ B, and p_{0} ∉ B, p_{1} ∈ B. The edge space E is the singleton {e} with r and s continuous maps from E to X defined by r(e) = p_{0}, s(e) = p_{1}. Note that the map r is open. Set λ_{x} = 0 for x ∈ X⧹ {p_{0}}, and λ_{p_{0}} a probability measure concentrated at {e}. Then G = (X, E, r, s, λ) is a topological quiver, in fact a topological relation defined by a function ([B2]) since E may be viewed as the subset {(p_{0}, p_{1})} of X×X. The correspondence E = ℂ ·δ_{e} over C_{0}(X) is described as follows: the map ϕ : C(X) → L(E) giving the left action is defined by multiplication by the scalar f(p_{1}), while the right action of C_{0}(X) on E is given by multiplication by the scalar f(p_{0}), f ∈ C_{0}(X). We have ⟨δ_{e}, δ_{e}⟩ = δ_{p_{0}}, so the element δ_{e} ⊗ δ_{e} of K(E) is the identity map of E. Thus ϕ(f) = f(p_{1})δ_{e} ⊗ δ_{e} for f ∈ C_{0}(X), ϕ^{−1}(K(E)) = C_{0}(X), and X_{fin} = X. Since ker ϕ = {f ∈ C_{0}(X) | f(p_{1}) = 0} we have X_{sink} = X⧹ \{p_{1}\}. Thus \overline{X_{sink}} = X and X_{reg} = ϕ; the ideal J_{E} = 0.

The closed subset D = \overline{r(E) ∪ s(E)} of X is \{p_{0}, p_{1}\}, so the correspondence E_{N} over C_{0}(D) = C(D) is the same as the correspondence associated with the discrete graph given by a single directed edge from the vertex p_{1} to the vertex p_{0}. We also note that D_{sink} = {p_{0}} , D_{fin} = D, D_{reg} = {p_{1}}. Thus the C*-algebra O(J_{E_{N}} , E_{N}) is the universal C*-algebra generated by a partial isometry S with orthogonal initial and final ranges which sum up to the identity.

Since J_{E} = 0 the coisometric condition on the universal representation vanishes and the Cuntz–Pimsner C*-algebra O(J_{E}, E) is just T (E) the Toeplitz C*- algebra for E. This is the universal C*-algebra generated by a partial isometry T (= T (e)) and the abelian C*-algebra C_{0}(X) with the initial space T^{∗}T equal to the projection δ_{p_{0}}of C_{0}(X) and f · T = f(p_{1})T for f ∈ C_{0}(X). We have T^{∗}T T = (δ_{p_{0}}) · T = δ_{p_{0}}(p_{1})T = 0, so the final space of T is orthogonal to the initial space T^{∗}T. The quotient C*-algebra T (E)/C_{0}(D) is the Toeplitz C*-algebra T (E_{N}) for the correspondence E_{N}, so for the discrete directed graph, while its quotient by the ideal generated by δ̃_{(T_{0},π_{0})}(C_{0}(D_{reg})) is the C*-algebra O(J_{E_{N}} , E_{N}), namely the C*-algebra of this discrete directed graph. Since C_{o}(D_{reg}) = C({p_{1}}) = ℂδ_{p_{1}} and ϕ_{N}(δ_{p_{1}}) = δ_{e} ⊗ δ_{e} we see that δ̃_{(T_{0},π_{0})}(δ_{p_{1}}) = δ_{p_{1}} − T T^{∗}.

A very special case of a topological quiver occurs within the context of a function f : dom(f) → X where dom(f) is a subspace of a locally compact space X. If π_{1} and π_{2}: X × X → X denote the canonical continuous projections onto the first and second coordinates respectively, set r = π_{1}|_{graph(f)} and s = π_{2}|_{graph(f)}, continuous maps of the subspace graph(f) of X × X to X.

Proposition 2.12. Let X be a locally compact Hausdorff space. The map r : graph(f) → X is open if and only if f : dom(f) → X is continuous and dom(f) is open in X. In this case dom(f) is homeomorphic with graph(f), and graph(f) is locally compact in X × X.

Proof. First note that for A, B open subsets of X then

graph(f) ∩ (A × B) = {(x, f(x)) | x ∈ A ∩ f^{−1}(B)} , so
r(graph(f) ∩ (A × B)) = A ∩ f^{−1}(B) = A ∩ f^{−1}(B) ∩ dom(f).

If f is continuous then f^{−1}(B) is open in dom(f) and r is an open map to the subspace dom(f) of X ; since r is continuous, one to one, and onto dom(f), r is a homeomorphism of graph(f) with the subspace dom(f) of X. If, in addition, dom(f) is open in X, then r is an open map to X . Since dom(f) is open in X it is also locally compact. Thus graph(f) must also be locally compact in X × X, so is therefore the intersection of a closed and open set in X × X. In fact one can show directly that graph(f) is the intersection of the closure of graph(f) with the open set dom(f) × X.

Conversely, if r : graph(f) → X is open, then r(graph(f)) = dom(f), so dom(f) is open in X and therefore locally compact. If B is open in X, f^{−1}(B) = r(graph(f) ∩ (X × B)) which is open in X and a subset of dom(f), so also open in dom(f). Thus f is continuous. □

Thus (cf. [B2]), for f : dom(f) → X a continuous function with dom(f) open in X we may form the topological quiver (X, graph(f), r, s, μ), or, using the homeomorphism r : graph(f) → dom(f), G = (X, dom(f), i, f, λ) where i is the inclusion 62 Berndt Brenken

of dom(f) in X and λ is the normalized measure on X with support {f(x)}. Here the restricted topological quiver G^{N} = (D, E, r, s, λ) where D = \overline{dom(f) ∪ ran(f)}.

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Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4 bbrenken@math.ucalgary.ca

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