Journal of Operator Theory

Volume 91, Issue 1, Winter 2024 pp. 27-54.

Symmetry for algebras associated to Fell bundles over groups and groupoids

Authors: Felipe Flores (1), Diego Jaure (2), Marius Mantoiu (3)
Author institution:(1) Departamento de Ingenieria Matematica, Universidad de Chile, Beauchef 851, Torre Norte, Oficina 436, Santiago, Chile
(2) Facultad de Ciencias, Departamento de Matematicas, Universidad de Chile, Las Palmeras 3425, Santiago, Chile
(3) Facultad de Ciencias, Departamento de Matematicas, Universidad de Chile, Las Palmeras 3425, Casilla 653, Santiago, Chile

Summary: To every Fell bundle $\mathscr C$ over a locally compact group $\sf G$ one associates a Banach $*$-algebra $L^1(G \vert \mathscr C)$. We prove that it is symmetric whenever $\sf G$ with the discrete topology is rigidly symmetric. A very general example is the Fell bundle associated to a twisted partial action of $\sf G$ on a $C^*$-algebra $\mathcal{A}$; 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 find in this case the right concept of rigid symmetry, involving representations in Hilbert bundles, and show its equivalence with an a priory stronger concept. Rigid symmetry of a discrete groupoid implies the symmetry of the transformation groupoids associated to any of its action. Symmetry is inherited from discrete groupoids to subgroupoids and transfers from a discrete groupoid to any of its epimorphic image.

DOI: http://dx.doi.org/10.7900/jot.2022jan27.2382
Keywords: Fell bundle, groupoid, partial group action, symmetric Banach algebra, cocycle, weight

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