Reduced Twisted Crossed Products over C*-Simple Groups

Rasmus Sylvester Bryder; Matthew Kennedy

doi:10.1093/imrn/rnw296

Reduced Twisted Crossed Products over C*-Simple Groups

Rasmus Sylvester Bryder, Matthew Kennedy

Institut for Matematiske Fag

5 Citationer (Scopus)

Abstract

We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We further show that the reduced crossed product satisfies an averaging property analogous to Powers’ averaging property. The dynamical systems are not required to be exact, and our results are new even in the non-twisted case.

Originalsprog	Engelsk
Tidsskrift	International Mathematics Research Notices
Vol/bind	2018
Udgave nummer	6
Sider (fra-til)	1638–1655
Antal sider	18
ISSN	1073-7928
DOI	https://doi.org/10.1093/imrn/rnw296
Status	Udgivet - 1 mar. 2018

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10.1093/imrn/rnw296

https://arxiv.org/pdf/1602.01533.pdfLicens: Andet

Citationsformater

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abstract = "We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We further show that the reduced crossed product satisfies an averaging property analogous to Powers{\textquoteright} averaging property. The dynamical systems are not required to be exact, and our results are new even in the non-twisted case.",

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N2 - We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We further show that the reduced crossed product satisfies an averaging property analogous to Powers’ averaging property. The dynamical systems are not required to be exact, and our results are new even in the non-twisted case.

AB - We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We further show that the reduced crossed product satisfies an averaging property analogous to Powers’ averaging property. The dynamical systems are not required to be exact, and our results are new even in the non-twisted case.

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DO - 10.1093/imrn/rnw296

M3 - Journal article

SN - 1073-7928

VL - 2018

SP - 1638

EP - 1655

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 6

ER -

Reduced Twisted Crossed Products over C*-Simple Groups

Abstract

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