Powers averaging for actions on $C(X)$-algebras

Abstract

Given a unital $C(X)$-algebra $A$ discrete group $Γ$ and an action $α: Γ\to \text{aut}(A)$ which leaves $C(X)$ invariant and such that $C(X)\rtimes_{α,r} Γ$ is simple, and a $2$-cocycle $ω$, we obtain a bijective correspondence between maximal $Γ$-invariant ideals of $A$ and maximal ideals in $A\rtimes_{α,ω,r} Γ$. In particular, $A\rtimes_{α,ω,r} Γ$ is simple if and only if $A$ has no $Γ$-invariant ideals.