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Group amenability and actions on Z-stable C*-algebras

Gardella, Eusebio and Lupini, Martino (2021) Group amenability and actions on Z-stable C*-algebras. Advances in Mathematics, 389 . Art. No. 107931. ISSN 0001-8708. doi:10.1016/j.aim.2021.107931.

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We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if G contains a copy of F₂, then there exist uncountably many, non-cocycle conjugate strongly outer actions of G on any tracial, unital, separable C*-algebra that absorbs tensorially the Jiang-Su algebra. Similar conclusions hold for outer actions on McDuff II₁ factors. We moreover show that G is amenable if and only if the Bernoulli shift on any finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods are inspired by Jones' work [27], and consist in a careful study of weak containment for the Koopman representations of certain generalized Bernoulli actions.

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Lupini, Martino0000-0003-1588-7057
Additional Information:© 2021 Elsevier Inc. Received 16 March 2018, Revised 11 February 2021, Accepted 26 June 2021, Available online 23 July 2021. This work was initiated during a visit of the first named author to the second at the California Institute of Technology in January 2017, and was continued during a visit of both authors to the Centre de Recerca Matemàtica in March 2017 in occasion of the Intensive Research Programme on Operator Algebras. The authors gratefully acknowledge the hospitality and the financial support of both institutions. The first named author was partially funded by SFB 878 Groups, Geometry and Actions, and by a postdoctoral fellowship from the Humboldt Foundation, and the second named author was partially supported by the NSF Grant DMS-1600186, by a Research Establishment Grant from Victoria University of Wellington, and by the Marsden Fund Fast - Starting Grant, from the Royal Society of New Zealand.
Funding AgencyGrant Number
Centre de Recerca MatemàticaUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
Victoria University of WellingtonUNSPECIFIED
Marsden Fund of the Royal Society of New ZealandUNSPECIFIED
Subject Keywords:Amenable group; Free group; Bernoulli shift; GNS construction; Weak containment; Cocycle equivalence; Jiang-Su algebra; Hyperfinite II1 factor
Record Number:CaltechAUTHORS:20210817-165808970
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Official Citation:Eusebio Gardella, Martino Lupini, Group amenability and actions on Z-stable C*-algebras, Advances in Mathematics, Volume 389, 2021, 107931, ISSN 0001-8708, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110294
Deposited By: Tony Diaz
Deposited On:18 Aug 2021 18:27
Last Modified:18 Aug 2021 18:27

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