The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

Abstract

AbstractLetGbe a finite group, letAbe an infinite-dimensional stably finite simple unital C*-algebra, and let$\alpha \colon G \to {\operatorname {Aut}} (A)$be an action ofGonAwhich has the weak tracial Rokhlin property. Let$A^{\alpha }$be the fixed point algebra. Then the radius of comparison satisfies${\operatorname {rc}} (A^{\alpha }) \leq {\operatorname {rc}} (A)$and${\operatorname {rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{{{\operatorname{card}}} (G))} \cdot {\operatorname {rc}} (A)$. The inclusion of$A^{\alpha }$inAinduces an isomorphism from the purely positive part of the Cuntz semigroup${\operatorname {Cu}} (A^{\alpha })$to the fixed points of the purely positive part of${\operatorname {Cu}} (A)$, and the purely positive part of${\operatorname {Cu}} ( C^* (G, A, \alpha ) )$is isomorphic to this semigroup. We construct an example in which$G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$,Ais a simple unital AH algebra,$\alpha $has the Rokhlin property,${\operatorname {rc}} (A)> 0$,${\operatorname {rc}} (A^{\alpha }) = {\operatorname {rc}} (A)$, and${\operatorname {rc}} ({C^* (G, A, \alpha)} ) = ({1}/{2}) {\operatorname {rc}} (A)$.