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

Abstract

Given a discrete group $\Gamma$, a finite factor $\mathcal{N}$ and a real number $p\in [1,+\infty )$ with $p\ne 2,$ 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 $L_p(\mathcal{N}(\Gamma )).$

20 pages

References in corpus (1)

• Hölder estimates for the noncommutative Mazur maps