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Permanence properties for crossed products and fixed point algebras of finite groups


Authors: Cornel Pasnicu and N. Christopher Phillips
Journal: Trans. Amer. Math. Soc. 366 (2014), 4625-4648
MSC (2010): Primary 46L55; Secondary 46L35, 46L40
Published electronically: April 25, 2014
MathSciNet review: 3217695
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Abstract: Let $ \alpha \colon G \to \operatorname {Aut} (A)$ be an action of a finite group $ G$ on a C*-algebra $ A.$ We present some conditions under which properties of $ A$ pass to the crossed product $ C^* (G, A, \alpha )$ or the fixed point algebra $ A^{\alpha }.$ We mostly consider the ideal property, the projection property, topological dimension zero, and pure infiniteness. In many of our results, additional conditions are necessary on the group, the algebra, or the action. Sometimes the action must be strongly pointwise outer, and in a few results it must have the Rokhlin property. When $ G$ is finite abelian, we prove that crossed products and fixed point algebras by $ G$ preserve topological dimension zero with no condition on the action.

We give an example to show that the ideal property and the projection property do not pass to fixed point algebras (even when the group is $ {\mathbb{Z}}_{2}$). The construction also gives an example of a C*-algebra $ B$ which does not have the ideal property but such that $ M_{2} (B)$ does have the ideal property; in fact, $ M_{2} (B)$ has the projection property.


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Additional Information

Cornel Pasnicu
Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
Email: Cornel.Pasnicu@utsa.edu

N. Christopher Phillips
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: ncp@darkwing.uoregon.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06036-4
Received by editor(s): August 27, 2012
Published electronically: April 25, 2014
Additional Notes: Some of this material is based upon work of the second author, supported by the US National Science Foundation under Grants DMS-0302401, DMS-0701076, and DMS-1101742.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.