A generalized Powers averaging property for commutative crossed products

We prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products $C(X) \rtimes_λG$, 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 $C(X) \rtimes_λG$ 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 $C(Y) \rtimes_λG$ simple, then any intermediate C*-algebra $A$ satisfying $C(Y) \rtimes_λG \subseteq A \subseteq C(X) \rtimes_λG$ is simple.

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