A generalized Powers averaging property for commutative crossed products

TL;DR

This paper extends Powers' averaging property to characterize simplicity in reduced crossed products of commutative C*-algebras under minimal group actions, with implications for stationarity and intermediate algebra simplicity.

Contribution

It generalizes Powers' averaging property for crossed products, linking it to simplicity, stationarity, and intermediate algebra properties in a broad setting.

Findings

Generalized Powers' averaging property for commutative crossed products.

Characterization of simplicity of reduced crossed products via averaging.

Simplicity results for intermediate C*-algebras between crossed products.

Abstract

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products $C(X) \rtimes_\lambda G$, 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 $C(X) \rtimes_\lambda G$ and to Kawabe's generalized space of amenable subgroups $\operatorname{Sub}_a(X,G)$. 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 $C(Y) \rtimes_\lambda G$ simple, then any intermediate C*-algebra $A$ satisfying $C(Y) \rtimes_\lambda G \subseteq A \subseteq C(X) \rtimes_\lambda G$ is simple.