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) _λ 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) _λ G and to Kawabe’s generalized space of amenable subgroups 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 ) ⊆ C(X) is an inclusion of unital commutative G-C*-algebras with X minimal and C(Y )_λG simple, then any intermediate C*-algebra A satisfying C(Y ) _λ G ⊆ A ⊆ C(X) _λ G is simple.