A generalized Powers averaging property for commutative crossed products

Abstract

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products $C(X){⋊}_{\lambda }G$, where $G$ is a countable discrete group, and $X$ is a compact Hausdorff space which $G$ acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of $C(X){⋊}_{\lambda }G$ and to Kawabe's generalized space of amenable subgroups ${\mathrm{Sub}}_a(X,G)$. This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if $C(Y)\subseteq C(X)$ is an inclusion of unital commutative $G$-C*-algebras with $X$ minimal and $C(Y){⋊}_{\lambda }G$ simple, then any intermediate C*-algebra $A$ satisfying $C(Y){⋊}_{\lambda }G\subseteq A\subseteq C(X){⋊}_{\lambda }G$ is simple.

24 pages; To appear in Transactions of the American Mathematical Society. This is the accepted manuscript, which includes all changes requested by the referee