Amenable unitary representations of locally compact groupoids

TL;DR

This paper extends the concept of amenability from groups to locally compact groupoids by defining amenable unitary representations, establishing key equivalences, and introducing a topological invariant mean for characterisation.

Contribution

It introduces a new notion of amenability for groupoid representations, generalizes Bekka's group results, and develops a topological invariant mean for characterisation.

Findings

G is amenable iff its left regular representation is amenable

Characterization of amenability via a topological invariant mean

Analysis of amenability in induced and properly amenable representations

Abstract

Let $G$ be a second countable locally compact groupoid equipped with a Haar system $\lambda$.In this work, we introduce and develop the notion of amenability for continuous unitary representations of $G$, formulated in terms of Hilbert bundles over the unit space $G^{0}$. We prove that $G$ is amenable if and only if its left regular representation is amenable, thereby extending Bekka's characterisation of amenable unitary representations from groups to groupoids. We further investigate the amenability of induced representations of $G$ and also study the representation of properly amenable groupoids. Finally, we define a topological invariant mean associated with a representation, constructed by utilising the theory of operator-valued vector measures on the unit space $G^{0}$, to characterise amenability.