A von Neumann algebra characterization of property (T) for groupoids

TL;DR

This paper characterizes property (T) for discrete probability-measure-preserving groupoids using their von Neumann algebras, extending to relative property (T) for subgroupoids via algebra inclusions.

Contribution

It provides a new von Neumann algebra-based characterization of property (T) for groupoids and their substructures, broadening the understanding of rigidity properties.

Findings

Characterization of property (T) via von Neumann algebras for groupoids

Extension to relative property (T) for subgroupoids

Framework applicable to arbitrary discrete probability-measure-preserving groupoids

Abstract

For an arbitrary discrete probability-measure-preserving groupoid $G$, we provide a characterization of property (T) for $G$ in terms of the groupoid von Neumann algebra $L(G)$. More generally, we obtain a characterization of relative property (T) for a subgroupoid $H\subset G$ in terms of the inclusion $L\left( H\right) \subset L\left( G\right) $.