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Transactions of the American Mathematical Society

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

 
 

 

Compact group actions with the Rokhlin property


Author: Eusebio Gardella
Journal: Trans. Amer. Math. Soc. 371 (2019), 2837-2874
MSC (2010): Primary 46L55; Secondary 46L35, 46L80
DOI: https://doi.org/10.1090/tran/7523
Published electronically: October 23, 2018
MathSciNet review: 3896099
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Abstract: We provide a systematic and in-depth study of compact group actions with the Rokhlin property. It is shown that the Rokhlin property is generic in some cases of interest; the case of totally disconnected groups is the most satisfactory one. One of our main results asserts that the inclusion of the fixed point algebra induces an order-embedding on $ K$-theory and that it has a splitting whenever it is restricted to finitely generated subgroups.

We develop new results in the context of equivariant semiprojectivity to study actions with the Rokhlin property. For example, we characterize when the translation action of a compact group on itself is equivariantly semiprojective. As an application, it is shown that every Rokhlin action of a compact Lie group of dimension at most one is a dual action. Similarly, for an action of a compact Lie group $ G$ on $ C(X)$, the Rokhlin property is equivalent to freeness together with triviality of the principal $ G$-bundle $ X\to X/G$.


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Additional Information

Eusebio Gardella
Affiliation: Fachbereich Mathematik, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
Email: gardella@uni-muenster.de

DOI: https://doi.org/10.1090/tran/7523
Keywords: Rokhlin property, crossed product, equivariant semiprojectivity, compact Lie group, $K$-theory, Cuntz semigroup
Received by editor(s): May 30, 2017
Received by editor(s) in revised form: January 22, 2018
Published electronically: October 23, 2018
Article copyright: © Copyright 2018 American Mathematical Society