Rigidity of twisted groupoid L^p-operator algebras

TL;DR

This paper investigates the rigidity of reduced twisted group and groupoid L^p-operator algebras, establishing conditions under which algebra isomorphisms imply group or groupoid isomorphisms for p not equal to 2.

Contribution

It introduces a classification of isometric isomorphisms of twisted L^p-operator algebras, linking algebraic isomorphisms to topological and cohomological properties of groups and groupoids.

Findings

Isometric isomorphisms correspond to group isomorphisms for groups with p≠2.

For groupoids, isomorphisms correspond to groupoid and twist isomorphisms under specified conditions.

The results extend rigidity phenomena known for C*-algebras to 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 $p\neq2$, 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 \'etale groupoid $\mathcal{G}$, we define the reduced twisted groupoid $L^p$-operator algebra $F^p_\lambda(\mathcal{G};\mathcal{E})$. In the main result of this paper, we show that for $p\neq 2$ if the groupoids are topologically principal, Hausdorff, \'etale 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.