A VON NEUMANN ALGEBRA CHARACTERIZATION OF PROPERTY (T) FOR GROUPOIDS

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 inclusions L(H)\subset L(G).