Symmetry for algebras associated to Fell bundles over groups and groupoids

Abstract

To every Fell bundle $\mathcal{C}$ over a locally compact group $\mathsf{G}$ one associates a Banach ${}^{\ast }$-algebra $L^1(\mathsf{G}|\mathcal{C})$. We prove that it is symmetric whenever $\mathsf{G}$ 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.

arXiv comment26 pages. Major changes with respect to the previous version. To appear in J. Operator Theory