Non-simple purely infinite C∗-algebras: the Hausdorff case

Blanchard, Etienne; Kirchberg, Eberhard

doi:10.1016/j.jfa.2003.06.008

Cited by 81 publications

(103 citation statements)

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“…6 We are indebted to E. Blanchard for pointing out the following result about tensor products of C 0 (X )-algebras to us (see [4,Proposition 3.1]-Blanchard states the result only for compact X , but the version below follows immediately by passing to the one-point compactification). As usual, we denote the minimal tensor product of A and B by A ⊗ B.…”

Section: Ifmentioning

confidence: 99%

$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

2007

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We study permanence properties of the classes of stable and so-called D-stable C * -algebras, respectively. More precisely, we show that a C 0 (X )-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for C * -algebras absorbing the Jiang-Su algebra Z tensorially). Furthermore, we prove that if D is a K 1 -injective strongly self-absorbing C * -algebra, then A absorbs D tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a C * -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of Z-absorbing C * -algebras. Mathematics Subject Classification (2000) 46L05 · 47L40

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Section: Ifmentioning

confidence: 99%

$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

2007

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“…The notion of approximate divisibility was introduced in [5]. We recall the definition: A unital C Ã -algebra A is approximately divisible if there is a sequence of unital Ã -homomorphisms j n : M 2 "M 3 -A such that j n ðxÞa À aj n ðxÞ-0 for all xAM 2 "M 3 and all aAA: Being approximately divisible implies being weakly divisible (at all projections p in A) (see [5]).…”

Section: Divisible C ã -Algebrasmentioning

confidence: 99%

“…We recall the definition: A unital C Ã -algebra A is approximately divisible if there is a sequence of unital Ã -homomorphisms j n : M 2 "M 3 -A such that j n ðxÞa À aj n ðxÞ-0 for all xAM 2 "M 3 and all aAA: Being approximately divisible implies being weakly divisible (at all projections p in A) (see [5]). The crucial difference between weak divisibility and approximate divisibility is the assumption of asymptotic centrality in the latter.…”

Section: Divisible C ã -Algebrasmentioning

confidence: 99%

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AF-embeddings into C∗-algebras of real rank zero

Perera

Rørdam

2004

Journal of Functional Analysis

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It is proved that every separable C Ã -algebra of real rank zero contains an AF-sub-C Ã -algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two C Ã -algebras and such that every projection in a matrix algebra over the large C Ã -algebra is equivalent to a projection in a matrix algebra over the AF-sub-C Ã -algebra. This result is proved at the level of monoids, using that the monoid of Murray-von Neumann equivalence classes of projections in a C Ã -algebra of real rank zero has the refinement property. As an application of our result, we show that given a unital C Ã -algebra A of real rank zero and a natural number n; then there is a unital Ã -homomorphism M n1 "?"M nr -A for some natural numbers r; n 1 ; y; n r with n j Xn for all j if and only if A has no representation of dimension less than n: r 2004 Elsevier Inc. All rights reserved.

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“…We restrict to exact C * -algebras to avoid considering quasitraces (using the results of Haagerup [16] and Blanchard-Kirchberg [3] that all quasitraces on these algebras are traces; cf. also [5]).…”

Section: Algebraic Simplicity and Tracesmentioning

confidence: 99%

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

Tikuisis

2013

Math. Ann.

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Abstract. The main result here is that a simple separable C * -algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [33,42] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C * -algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept. IntroductionThe program to classify unital, simple, separable, nuclear C * -algebras has recently seen a small paradigm shift [13]. In light of the fact that there exist such C * -algebras that cannot be classified by ordered K-theory and traces [34], the current trend is to try to identify, using regularity properties, the C * -algebras which are (or should be) classifiable. This idea is crystallized in a conjecture (cf. [36, Section 3.5]), due to Andrew Toms and Wilhelm Winter, that among these C * -algebras, the following properties are equivalent:(i) finite nuclear dimension (defined in [44]); (ii) Z-stability (being isomorphic to one's tensor product with the Jiang-Su algebra Z, introduced in [18]); and (iii) almost unperforated Cuntz semigroup. One might also add that, modulo the UCT, conditions (i)-(iii) should be equivalent to being classifiable (though in this form, such a statement is not well-formed because classifiability is a property of a class of C * -algebras, and not a property of a single C * -algebra). At the same time, certain examples of simple, stably projectionless C * -algebras have emerged -including certain crossed products of O 2 by R [21], a nuclear, separable, non-Z-stable example [31, Theorem 4.1], and others [12,17,37]. Certain tools, old and new, already allow one to understand some of the structure of these algebras under special hypotheses [6,27]. However, most of the theory on regularity properties for C * -algebras was developed only in the unital case, and leads one to ask what obstructions (if any) exist with nonunital algebras. Certainly, the unital case carries with it a number of simplifications: for instance, the simplex of traces on a C * -algebra which take the value 1 at the unit (i.e. which are states) plays an indispensible role in the theory, and it is far from obvious what the correct replacement is in the nonunital case. Nonetheless, one hopes that the simplifications in the unital case are superficial, and that ultimately, one can find and prove analogues or generalizations of the known unital results.2010 Mathematics Subject Classification. 46L35, 46L80, 46L05, 47L40, 46L85.

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Non-simple purely infinite C∗-algebras: the Hausdorff case

Cited by 81 publications

References 34 publications

$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

AF-embeddings into C∗-algebras of real rank zero

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

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