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

Abstract

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