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

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.