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 α\colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^α be the fixed point algebra. Then the radius of comparison satisfies rc (A^α) \leq rc (A) and rc ( C* (G, A, α) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^α in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^α) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, α) ) is isomorphic to this semigroup. We construct an example in which G is the two element group, A is a simple unital AH algebra, αhas the Rokhlin property, rc (A) > 0, rc (A^α) = rc (A), and rc (C* (G, A, α)) = (1/2) rc (A).

48 pages