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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Rokhlin actions of finite groups on UHF-absorbing $\mathrm {C}^*$-algebras
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by Selçuk Barlak and Gábor Szabó PDF
Trans. Amer. Math. Soc. 369 (2017), 833-859 Request permission

Abstract:

This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on $\mathrm {C}^*$-algebras satisfying a certain UHF-absorption condition. We show that given any finite group $G$ and a separable, unital $\mathrm {C}^*$-algebra $A$ that absorbs $M_{|G|^\infty }$ tensorially, one can lift any group homomorphism $G\to \mathrm {Aut}(A)/_{{\approx _{\mathrm {u}}}}$ to an honest Rokhlin action $\gamma$ of $G$ on $A$. Unitality may be dropped in favour of stable rank one or being stable. If $A$ belongs to a certain class of $\mathrm {C}^*$-algebras that is classifiable by a suitable invariant (e.g. $K$-theory), then in fact every $G$-action on the invariant lifts to a Rokhlin action of $G$ on $A$. For the crossed product $\mathrm {C}^*$-algebra ${A\rtimes _\gamma G}$ of a Rokhlin action on a UHF-absorbing $\mathrm {C}^*$-algebra, an inductive limit decomposition is obtained in terms of $A$ and $\gamma$. If $G$ is assumed to be abelian, then the dual action $\hat {\gamma }$ is locally representable in a very strong sense. We then show how some well-known constructions of finite group actions with certain predescribed properties can be recovered and extended by the main results of this paper when paired with known classification theorems. Among these is Blackadar’s famous construction of symmetries on the CAR algebra whose fixed point algebras have non-trivial $K_1$-groups. Lastly, we use the results of this paper to reduce the UCT problem for separable, nuclear $\mathrm {C}^*$-algebras to a question about certain finite group actions on $\mathcal {O}_2$.
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Additional Information
  • Selçuk Barlak
  • Affiliation: Westfälische Wilhelms-Universität, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
  • Address at time of publication: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
  • Email: selcuk.barlak@uni-muenster.de, barlak@imada.sdu.dk
  • Gábor Szabó
  • Affiliation: Westfälische Wilhelms-Universität, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
  • MR Author ID: 1103496
  • ORCID: 0000-0001-7963-8493
  • Email: gabor.szabo@uni-muenster.de
  • Received by editor(s): August 12, 2014
  • Received by editor(s) in revised form: January 27, 2015
  • Published electronically: March 21, 2016
  • Additional Notes: The authors were supported by SFB 878 Groups, Geometry and Actions and GIF Grant 1137-30.6/2011
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 833-859
  • MSC (2010): Primary 46L55, 46L35
  • DOI: https://doi.org/10.1090/tran6697
  • MathSciNet review: 3572256