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

Given a discrete group $\mathrm{\Gamma }$, a finite factor $\mathcal{N}$ and a real number $p\in [1,+\mathrm{\infty })$ with $p\mathrm{\ne }2,$ we are concerned with the rigidity of actions of $\mathrm{\Gamma }$ by linear isometries on the $L_p$-spaces $L_p(\mathcal N)$associatedto$\mathcal N$\mathrm{.}Moreprecisely,weshowthat,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 $\mathrm{\Gamma }$ is an ICC Kazhdan group, the action of $\mathrm{\Gamma }$ on its von Neumann algebra ${\mathcal N}(\Gamma)$,givenbyconjugation,islocallyrigidintheisometrygroupof$L_p({\mathcal N}(\Gamma)).$