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