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

TL;DR

This paper investigates how certain structural properties of C*-algebras are preserved under crossed products and fixed point algebras when acted upon by finite groups, identifying conditions for preservation and providing counterexamples.

Contribution

It establishes new conditions under which properties like ideal property, projection property, and topological dimension zero are preserved in crossed products and fixed point algebras for finite group actions.

Findings

Topological dimension zero is preserved for finite abelian groups without additional conditions.

The ideal and projection properties do not always pass to fixed point algebras, even for simple groups.

Counterexamples demonstrate the failure of property preservation and the existence of C*-algebras with specific property behaviors.

Abstract

For an action of a finite group on a C*-algebra, we present some conditions under which properties of the C*-algebra pass to the crossed product or the fixed point algebra. 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 the group is finite abelian, we prove that crossed products and fixed point algebras 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 for the two element group). The construction also gives an example of a C*-algebra which does not have the ideal property but such that the algebra of 2 by 2 matrices over it does have the ideal property; in fact, this matrix algebra has the projection property.