Local rigidity for actions of Kazhdan groups on non commutative $L_p$-spaces

TL;DR

This paper proves local rigidity of certain group actions on non-commutative Lp-spaces, showing small perturbations are conjugate to the original action under specific property (T) conditions.

Contribution

It establishes local rigidity results for actions of Kazhdan groups on non-commutative Lp-spaces, extending rigidity phenomena to a non-commutative setting.

Findings

Rigidity holds under Property (T) and ergodicity conditions.

Small perturbations of actions are conjugate to original actions.

Results apply to ICC Kazhdan groups acting on their von Neumann algebras.

Abstract

Given a discrete group $\Gamma$, a finite factor $\mathcal N$ and a real number $p\in [1, +\infty)$ with $p\neq 2,$ we are concerned with the rigidity of actions of $\Gamma$ by linear isometries on the $L_p$-spaces $L_p(\mathcal N)$ associated to $\mathcal N$. More precisely, we show that, when $\Gamma$ and $\mathcal N$ have both Property (T) and under some natural ergodicity condition, such an action $\pi$ is locally rigid in the group $G$ of linear isometries of $L_p(\mathcal N)$, that is, every sufficiently small perturbation of $\pi$ is conjugate to $\pi$ under $G$. As a consequence, when $\Gamma$ is an ICC Kazhdan group, the action of $\Gamma$ on its von Neumann algebra ${\mathcal N}(\Gamma)$, given by conjugation, is locally rigid in the isometry group of $L_p({\mathcal N}(\Gamma)).$