Continuous representations of groupoids

TL;DR

This paper develops a framework for continuous groupoid representations on Hilbert spaces, exploring their properties, standard constructions, and connections to operator algebras, with a focus on the regular representation and Banach *-categories.

Contribution

It introduces the concept of unitary continuous groupoid representations on Hilbert spaces and analyzes their properties and associated operator algebras, extending classical representation theory.

Findings

Characterization of continuous groupoid representations

Analysis of the regular representation's role

Relationship between groupoid and Banach *-category representations

Abstract

We introduce unitary representations of continuous groupoids on continuous fields of Hilbert spaces. We investigate some properties of these objects and discuss some of the standard constructions from representation theory in this particular context. An important r\^{ole} is played by the regular representation. We conclude by discussing some operator algebra associated to continuous representations of groupoids; in particular, we analyse the relationship of continuous representations of $G$ and continuous representations of the Banach $*$-category $\hat{L}^{1}(G)$.