Reduced Twisted Crossed Products over C*-Simple Groups

Bryder, Rasmus Sylvester; Kennedy, Matthew

doi:10.1093/imrn/rnw296

Cited by 12 publications

(16 citation statements)

References 15 publications

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“…A similar bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on A was proved by de la Harpe and Skandalis [6] in the case of Powers' groups and by Bryder and Kennedy [5] in the case of C * -simple groups .…”

Section: Remark 32supporting

confidence: 69%

“…The Powers' averaging property for the reduced crossed product r A of the action of a Powers' group on a unital C * -algebra A was proved by de la Harpe and Skandalis in [6]. Recent developments of the subject include independent results of Haagerup [8] and Kennedy [13], where they proved that the reduced C * -algebra C * r ( ) of any C *simple group has the Powers' averaging property; subsequently, Bryder and Kennedy [5] used similar techniques to prove the Powers' averaging property for the reduced (twisted) crossed product of C * -simple group actions. In this section we prove a more explicit version of the latter result which we need below to give a description of stationary states on the reduced crossed product C * -algebra.…”

Section: Powers' Averaging Property For Crossed Productsmentioning

confidence: 99%

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On simplicity of intermediate -algebras

Amrutam¹,

Kalantar²

2019

Ergod. Th. Dynam. Sys.

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We prove simplicity of all intermediate $C^{\ast }$ -algebras $C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$ in the case of minimal actions of $C^{\ast }$ -simple groups $\unicode[STIX]{x1D6E4}$ on compact spaces $X$ . For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary $C^{\ast }$ -dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ for any action $\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$ of a $C^{\ast }$ -simple group $\unicode[STIX]{x1D6E4}$ on a unital $C^{\ast }$ -algebra ${\mathcal{A}}$ , and use it to prove a one-to-one correspondence between stationary states on ${\mathcal{A}}$ and those on $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ .

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Section: Remark 32supporting

confidence: 69%

Section: Powers' Averaging Property For Crossed Productsmentioning

confidence: 99%

On simplicity of intermediate -algebras

Amrutam¹,

Kalantar²

2019

Ergod. Th. Dynam. Sys.

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“…This latter criterion has been recently used by Le Boudec and Matte Bon [40] to study the C*-simplicity of various groups of homeomorphisms and to show that Thompson's group V is C*-simple, while the C*-simplicity of T is equivalent to the non-amenability of F . Bryder and Kennedy [9] studied the ideal structure of (twisted) crossed products over C*-simple groups. In particular, they established a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra.…”

Section: Introductionmentioning

confidence: 99%

C*-simplicity and the unique trace property for discrete groups

et al. 2017

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A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.Comment: 40 pages; major restructuring; four new sections added, including a new characterization of C*-simplicity, a new proof of the Kalantar-Kennedy characterization of C*-simplicity and a discussion of the Connes-Sullivan propert

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“…After the first draft of this paper had been submitted for publication, R.S. Bryder and M. Kennedy have shown in [92] that if G is C * -simple, then C * r (Σ) satisfies a certain Powers type averaging property, related to property (DP). Using this, they are able to show that the bijection in item (4) above actually holds for any C * -simple group.…”

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confidence: 99%

Fourier theory andC∗-algebras

Bédos

Conti

2016

Journal of Geometry and Physics

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Twisted group C *-algebras Twisted C *-dynamical systems Twisted C *-crossed products Maximal ideals a b s t r a c t We discuss a number of results concerning the Fourier series of elements in reduced twisted group C *-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C *-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.

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Reduced Twisted Crossed Products over C*-Simple Groups

Cited by 12 publications

References 15 publications

On simplicity of intermediate -algebras

On simplicity of intermediate -algebras

C*-simplicity and the unique trace property for discrete groups

Fourier theory andC∗-algebras

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