Amenable unitary representations of locally compact groupoids
K. N. Sridharan, N. Shravan Kumar
TL;DR
The paper generalizes Bekka's operator-theoretic notion of amenability from locally compact groups to locally compact groupoids by introducing amenable continuous unitary representations defined on Hilbert bundles over the unit space $G^{0}$ with Haar system $\lambda$. It develops multiple equivalent characterizations using Hilbert-Schmidt and trace-class operators on the Hilbert $C_{0}(G^{0})$-module $C_{0}(G^{0},\mathcal{H}^{\pi})$, and introduces a topological invariant mean built from operator-valued measures on $G^{0}$ to characterize amenability. The main results establish that $G$ is amenable iff its left-regular representation is amenable, and extend amenability to induced representations and properly amenable groupoids, providing a cohesive operator-algebraic framework for groupoid harmonic analysis. This work unifies representation-theoretic amenability in the groupoid setting and offers tools for analyzing groupoid actions, their induced representations, and associated invariants.
Abstract
Let $G$ be a second countable locally compact groupoid equipped with a Haar system $λ$.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.