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Symmetry for algebras associated to Fell bundles over groups and groupoids

Felipe Flores, Diego Jauré, Marius Mantoiu

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.

Symmetry for algebras associated to Fell bundles over groups and groupoids

Abstract

To every Fell bundle over a locally compact group one associates a Banach -algebra . We prove that it is symmetric whenever 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.

Paper Structure

This paper contains 14 sections, 22 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. The case of discrete groupoids
  3. The case of locally compact groups
  4. The case of discrete groupoids
  5. Fell bundles over discrete groupoids and symmetry issues
  6. Actions of discrete groupoids on continuous fields of $C^*$-algebras
  7. Behavior of symmetry with respect to epimorphisms and monomorphisms
  8. The case of locally compact groups
  9. The main results for locally compact groups
  10. Discretization of representations
  11. Proof of Theorem \ref{['cruciala']}
  12. Twisted partial group actions and crossed products
  13. A weighted symmetry result
  14. Kernels for Fell bundles over locally compact groups

Key Result

Theorem 2.2

For a discrete groupoid, rigid symmetry and hypersymmetry coincide.

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Corollary 2.5
  • proof
  • ...and 43 more