Weakly tracially approximately representable actions

Abstract

We describe a weak tracial analog of approximate representability under the name "weak tracial approximate representability" for finite group actions. We then investigate the dual actions on the crossed products by this class of group actions. Namely, let G be a finite abelian group, let A be an infinite-dimensional simple unital C*-algebra, and let α : G → Aut(A) be an action of G on A which is pointwise outer. Then α has the weak tracial Rokhlin property if and only if the dual action α of the Pontryagin dual G on the crossed product C * (G, A, α) is weakly tracially approximately representable, and α is weakly tracially approximately representable if and only if the dual action α has the weak tracial Rokhlin property. This generalizes the results of Izumi in 2004 and Phillips in 2011 on the dual actions of finite abelian groups on unital simple C*-algebras.We further obtain some results on the Cuntz semigroup and the radius of comparison of the crossed product by a finite group action which has a weakened version of weak tracial approximate representability. Namely, let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*algebra, and let α : G → Aut(A) be a weakly tracially strictly approximately inner action of G on A. Let A α be the fixed point algebra. Then the radius of comparison satisfies rc(A) ≤ rc C * (G, A, α) and if C * (G, A, α) is simple, then rc(A) ≤ rc C * (G, A, α) ≤ rc(A α ). Further, the inclusion of A in C * (G, A, α) induces an isomorphism from the soft part of the Cuntz semigroup Cu(A) to its image in Cu(C * (G, A, α)). These results generalize our earlier results on the radius of comparison and the Cuntz semigroup of the crossed product by a tracially strictly approximately inner action.