A von Neumann algebra characterization of property (T) for 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) $.