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Rokhlin actions of finite groups on UHF-absorbing $ \mathrm{C}^*$-algebras


Authors: Selçuk Barlak and Gábor Szabó
Journal: Trans. Amer. Math. Soc. 369 (2017), 833-859
MSC (2010): Primary 46L55, 46L35
DOI: https://doi.org/10.1090/tran6697
Published electronically: March 21, 2016
MathSciNet review: 3572256
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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_{\vert G\vert^\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
Email: gabor.szabo@uni-muenster.de

DOI: https://doi.org/10.1090/tran6697
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
Article copyright: © Copyright 2016 American Mathematical Society