Property (T) with respect to non-commutative Lp-spaces

TL;DR

This paper extends Kazhdan's property (T) to non-commutative Lp-spaces associated with von Neumann algebras, demonstrating new rigidity properties in operator algebra contexts.

Contribution

It establishes that groups with property (T) also have property (T_B) for non-commutative Lp-spaces, a novel generalization in the theory of group properties.

Findings

Groups with property (T) have property (T_B) in non-commutative Lp-spaces.

The result applies to Haagerup's non-commutative Lp-spaces.

This generalizes classical property (T) to operator algebra frameworks.

Abstract

We show that a group with Kazhdan's property $(T)$ has property $(T_{B})$ for $B$ the Haagerup non-commutative $L_{p}(\mathcal{M})$-space associated with a von Neumann algebra $\mathcal{M}$, $1