On crossed products with property 𝑇

Abstract

Let N N be a finite von Neumann algebra (with faithful normal normalized trace τ \tau ), G G a countable discrete group, and α \alpha a τ \tau -preserving action of G G on N N such that N ⋊ α G N{ \rtimes _\alpha }G is a factor. It is proved that if N ⋊ α G N{ \rtimes _\alpha }G has Property T {\text {T}} , then G G has Kazhdan’s Property T {\text {T}} .