Symmetry for algebras associated to Fell bundles over groups and groupoids

TL;DR

This paper investigates the symmetry properties of Banach *-algebras associated with Fell bundles over groups and groupoids, generalizing known results and exploring conditions like rigidity and their implications for algebraic structures.

Contribution

It extends symmetry results to Fell bundles over groups and groupoids, introducing a generalized notion of rigidity and analyzing its impact on algebraic symmetry.

Findings

Banach *-algebra $L^1({ m G}ig| ext{bundle})$ is symmetric if ${ m G}$ is rigidly symmetric.

Generalization of symmetry concepts to Fell bundles over discrete groupoids.

Analysis of symmetry in transformation groupoids and permanence properties.

Abstract

To every Fell bundle $\mathscr C$ over a locally compact group ${\sf G}$ one associates a Banach $^*$-algebra $L^1({\sf G}\,\vert\,\mathscr C)$. We prove that it is symmetric whenever ${\sf G}$ with the discrete topology is rigidly symmetric. This generalizes the known case of a global action without a twist. There is also a weighted version as well as a treatment of some classes of associated integral kernels. We also deal with the case of Fell bundles over discrete groupoids. We formulate a generalization of rigid symmetry in this case and show its equivalence with an a priori stronger concept. We also study the symmetry of transformation groupoids and some permanence properties.