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

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 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 rc (A^{\alpha}) \leq rc (A) and rc ( C* (G, A, \alpha) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^{\alpha} in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^{\alpha}) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, \alpha) ) is isomorphic to this semigroup. We construct an example in which G is the two element group, A is a simple unital AH algebra, \alpha has the Rokhlin property, rc (A) > 0, rc (A^{\alpha}) = rc (A), and rc (C* (G, A, \alpha)) = (1/2) rc (A).