Rigidity of twisted groupoid L^p-operator algebras

Abstract

In this paper we will study the isomorphism problem for the reduced twisted group and groupoid $L^p$-operator algebras. For a locally compact group $G$ and a continuous 2-cocycle $\sigma$ we will define the reduced $\sigma$-twisted $L^p$-operator algebra $F_{\lambda }^p(G,\sigma )$. We will show that if $p\ne 2$, then two such algebras are isometrically isomorphic if and only if the groups are topologically isomorphic and the continuous 2-cocyles are cohomologous. For a twist $\mathcal{E}$ over an étale groupoid $\mathcal{G}$, we define the reduced twisted groupoid $L^p$-operator algebra $F_{\lambda }^p(\mathcal{G};\mathcal{E})$. In the main result of this paper, we show that for $p\ne 2$ if the groupoids are topologically principal, Hausdorff, étale and have a compact unit space, then two such algebras are isometrically isomorphic if and only if the groupoids are isomorphic and the twists are properly isomorphic.