Co-rigidity of groups, von Neumann algebras and Kac algebras

Abstract

In this paper, we consider a generalization of property T of Kazhdan for groups and property T of Connes for von Neumann algebras. We introduce another relative property T for groups corresponding to co-rigidity for von Neumann algebras, which is different from relative property T of Margulis. We investigate the connection between a pair of von Neumann algebras and a pair of their commutants with respect to co-rigidity. We define relative property T and co-rigidity for a pair of Kac algebras as the generalizations of relative property T and co-rigidity for groups. We show that the tensor product of two Kac algebras has property T if and only if two Kac algebras all have property T.