The ideal structure of the $C\sp *$-algebras of infinite graphs

Bates, Teresa; Hong, Jeong Hee; Raeburn, Iain; Szymański, Wojciech

doi:10.1215/ijm/1258138472

Cited by 168 publications

(378 citation statements)

References 22 publications

(43 reference statements)

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“…In Example 5.2, the three displayed graphs are all essential one-sink extensions of the graph with one vertex and four edges, whose C * -algebra is O 4 . We also have that Ext O 4 ∼ = Z 3 , and the first two graphs in Example 5.2 determine the classes [1] and [2] in Z 3 , respectively. Consequently, we cannot apply [26,Theorem 4.1], and we see that the methods of this paper have applications to situations not covered by [26,Theorem 4.1].…”

Section: Remark 53mentioning

confidence: 87%

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On the classification of nonsimple graph C *-algebras

Eilers

Tomforde

2009

Math. Ann.

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We prove that a graph C * -algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K -theory. We prove that a similar classification also holds for a graph C * -algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C * -algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C * -algebra is stable.

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Section: Remark 53mentioning

confidence: 87%

“…Because C * ( H E ∅ ) is a simple, separable, purely infinite, and nonunital C * -algebra, Zhang's Theorem [31] implies that I ∼ = C * ( H E ∅ ) is stable. Thus we are in the situation described in (1).…”

Section: Lemma 63 Let E Be a Graph Such That Cmentioning

confidence: 99%

On the classification of nonsimple graph C *-algebras

Eilers

Tomforde

2009

Math. Ann.

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“…, for all σ ∈ S n , where S n is a symmetric group for n ∈ N, and where K σ(1) σ(2) means the full-subgraph of K σ(1) determined by K σ (2) , and K σ(1),σ (2) σ (3) means the full- (3) , and, inductively, K…”

Section: Corollary 22 (Also See [9]mentioning

confidence: 99%

Operator Algebraic Quotient Structures Induced by Directed Graphs

Cho

2008

Complex Anal. Oper. Theory

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Our result is about inclusions for (finite or infinite) countable directed graphs. In the proof, we use Free Probability Theory, groupoids, and algebras of operators (von Neumann algebras). We show that inclusions of directed graphs induce quotients for associated groupoid actions. With the use of operator thechniques, we then establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is that each connected graph induces a quotient generated by a free group. This is a generalization of the notion of induced representations in the context of unitary representations of groups, i.e., the induction from the representations of a subgroup of an ambient group. The analogue is to systems of imprimitivity based on the homogeneous space. The parallel of this is the more general context of graphs (extending from groups to groupoids): We first prove that inclusions for connected graphs correspond to free group quotients, and we then build up the general case via connected components of given graphs.Recent work in Free Probability Theory involves algebras of operators (in particular, von Neumann algebras) based on countable directed graphs. We introduce quotients, in this context. We begin by showing that inclusions of directed graphs induce a quotient of associated groupoid actions. With the use of operator techniques, we establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is a generalization of the notion of induced representations in the context of the left regular unitary representations of groups, i.e., induction from representations of a subgroup Υ of an ambient group Γ. In deciding which representations of a group Γ are induced from 662 l. ChoComp.an.op.th.the representations of the subgroup Υ, one must look at systems of imprimitivity based on the homogeneous space Γ/Υ. We show that there is a parallel of this in the more general context of graphs (extending from groups to groupoids): in Corollary 3.4, we do this first for connected graphs, and in Theorem 3.5, we build up the general case via connected components.In Chapter 2, we introduce and include two results of [9] we will need, for the benefit of the readers, and in order to make the paper selfcontained. They are the base stones of Chapter 3.One of our main objects are directed graphs. Directed graphs are combinatorial graphs consisting of vertices, which are represented as points (or nodes), and directed edges, which are represented as the arrowed arcs connecting the vertices, where the arrows represent the direction (or the orientation) of edges. And the other main object is operator algebras. In particular, we will work on Hilbert spaces. Let H be a Hilbert space and let B(H) be the collection of all bounded (or continuous) linear operators. Then, by considering (subspace) topologies of B(H), we can define topological * -subalgebras of B(H). If the given topology is a weak operator topology and the * -subalgebra M of B(H) is close...

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“…By [2, Theorem 11] and Lemma 2.8, L(E/K) is a unital purely infinite simple ring, a contradiction. The following definition is a particular case of that of [10]: Let E be a graph. A nonempty subset M ⊆ E 0 is a maximal tail if it satisfies the following properties: Since every vertex is an idempotent, the reverse inclusion is clear.…”

Section: The Class Of Leavitt Path Algebras Is Closed Under Quotientsmentioning

confidence: 99%

“…Some of the motivating ideas for our characterization of the exchange property are contained in the works of Jeong and Park [13] and Bates, Hong, Raeburn and Szymański [10], while ideas regarding the stable rank grew from the paper by Deicke, Hong and Szymański [11]. The proofs presented here significantly differ from those of the analytic setting of C * -algebras and the arguments are necessarily different in the purely algebraic context since many of the tools used there are not available in our setting.…”

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confidence: 99%