Exact C*-Algebras, Tensor Products, and the Classification of Purely Infinite Algebras

Kirchberg, Eberhard

doi:10.1007/978-3-0348-9078-6_87

Cited by 95 publications

(124 citation statements)

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“…Theorem 6.1 implies that, when the rule has three invertible corners, the C * -algebra falls into the class which is classified by the celebrated theorem of Kirchberg and Phillips, which says that C * (Λ) is determined up to isomorphism by its K-theory [6,18,25]. So we want to compute the K-groups of C * (Λ).…”

Section: K-theorymentioning

confidence: 99%

A family of 2-graphs arising from two-dimensional subshifts

Pask

Raeburn

Weaver

2009

Ergod. Th. Dynam. Sys.

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Abstract. Higher-rank graphs (or k-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger C * -algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the C * -algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their K-theory. We find examples whose C * -algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to the C * -algebras of ordinary directed graphs.

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Section: K-theorymentioning

confidence: 99%

A family of 2-graphs arising from two-dimensional subshifts

Pask

Raeburn

Weaver

2009

Ergod. Th. Dynam. Sys.

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“…In fact most classification results in mathematics involve some class of countable structures as invariants. Elliott's seminal classification of AF algebras by the ordered K 0 group in [9] is of this sort, as well as the K-theoretical classification of purely infinite simple nuclear C*-algebras in the UCT class obtained by Kirchberg and Phillips in [22] and [37]. Nonetheless, in the last decade a number of natural equivalence relations arising in different areas of mathematics have been shown to be not classifiable by countable structures.…”

Section: Introductionmentioning

confidence: 99%

Unitary equivalence of automorphisms of separable C*-algebras

Lupini

2014

Advances in Mathematics

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Abstract. We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures. IntroductionIf A is a separable C*-algebra, the group Aut(A) of automorphisms of A is a Polish group with respect to the topology of pointwise norm convergence. An automorphism of A is called (multiplier) inner if it is induced by the action by conjugation of a unitary element of the multiplier algebra M (A) of A. Inner automorphisms form a Borel normal subgroup Inn(A) of the group of automorphisms of A. The relation of unitary equivalence of automorphisms of A is the coset equivalence relation on Aut(A) determined by Inn(A). (The reader can find more background on C*-algebras in Section 2.) The main result presented here asserts that if A does not have continuous trace, then it is not possible to effectively classify the automorphisms of A up to unitary equivalence using countable structures as invariants; in particular this rules out classification by K-theoretic invariants. In the course of the proof of the main result we will show that the existence of an outer derivation on a C*-algebra A is equivalent to a seemingly stronger statement, that we will refer to as Property AEP (see Definition 4.4), implying in particular the existence of an outer derivable automorphism of A.The notion of effective classification can be made precise by means of Borel reductions in the framework of descriptive set theory (the monographs [20] and [15] are standard references for this subject). If E and E ′ are equivalence relations on standard Borel spaces X and X ′ respectively, then a Borel reduction from E to F is a Borel function f : X → X ′ such that for every x,y ∈ X, xEy if and only if f (x) E ′ f (y). The Borel function f witnesses an effective classification of the objects of X up to E, with E ′ -equivalence classes of objects of X ′ as invariants. This framework captures the vast majority of concrete classification results in mathematics. (In [11] and [12] the computation of most of the invariants in the theory of C*-algebras is shown to be Borel.)2010 Mathematics Subject Classification. 03E15, 46L40, 46L57.

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“…Since we consider only countable graphs and matrices, the algebras C * (E) and O A are all separable. By the Takesaki-Takai duality theorem, every graph algebra C * (E) is stably isomorphic to a crossed product (C * (E) γ T) γ Z of an AF -algebra by Z, and hence is nuclear (see [4,Corollary 3.2] and [5, Proposition 6.8]) and satisfies the Universal Coefficient Theorem (see [29,Theorem 1.17] and [3,Chapter 23] [17,Theorem 9] or [26,Theorem 4.2.4]) that O A is the unique pi-sun algebra with K 0 = 0 and K 1 ∼ = Z, which is usually denoted P ∞ . In other words, P ∞ can be realised as an Exel-Laca algebra.…”

Section: The K-theory Of Exel-laca Algebrasmentioning

confidence: 99%

Cuntz-Krieger algebras of infinite graphs and matrices

Raeburn

Szymański

2003

Trans. Amer. Math. Soc.

112

104

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Abstract. We show that the Cuntz-Krieger algebras of infinite graphs and infinite {0, 1}-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their K-theory. Since the finite approximating graphs have sinks, we have to calculate the K-theory of CuntzKrieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.The Cuntz-Krieger algebras O A were introduced by Cuntz and Krieger in 1980, and have been prominent in operator algebras ever since. At first the algebras O A were associated to a finite matrix A with entries in {0, 1}, but it was quickly realised that they could also be viewed as the C * -algebras of a finite directed graph [33]. Over the past few years, originally motivated by their appearance in the duality theory of compact groups [22], authors have considered analogues of the Cuntz-Krieger algebras for infinite graphs and matrices (see [21] the graph algebra C * (E) is the universal C * -algebra generated by a Cuntz-Krieger E-family {s e , p v }. The equations (0.1) make sense as they stand for row-finite graphs, in which the index set {e ∈ E 1 : s(e) = v} for the sum is always finite. If a vertex v emits infinitely many edges, the sum does not make sense in a C * -algebra, because infinite sums of projections cannot converge in norm. However, it was observed in [13] that the general theory of Cuntz-Krieger algebras carries over to arbitrary countable graphs if one simply removes the relations involving infinite sums from (0.1), and requires instead that the range projections S e S * e are mutually orthogonal and dominated by P s(e) . Exel and Laca have described a different generalisation of the Cuntz-Krieger algebras for infinite matrices A [10]. Their defining relations are complicated: loosely speaking, one has to include a Cuntz-Krieger relation whenever a row-operation

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Exact C*-Algebras, Tensor Products, and the Classification of Purely Infinite Algebras

Cited by 95 publications

References 24 publications

A family of 2-graphs arising from two-dimensional subshifts

A family of 2-graphs arising from two-dimensional subshifts

Unitary equivalence of automorphisms of separable C*-algebras

Cuntz-Krieger algebras of infinite graphs and matrices

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