Property $(T_{L^{\Phi}})$ and property $(F_{L^{\Phi}})$ for Orlicz spaces $L^{\Phi}$

Abstract

An Orlicz space $L^{\Phi}(\Omega)$ is a Banach function space defined by using a Young function $\Phi$, which generalizes the $L^p$ spaces. We show that, for a reflexive Orlicz space $L^{\Phi}([0,1])$, a locally compact second countable group has Kazhdan's property $(T)$ if and only if it has property $(T_{L^{\Phi}([0,1])})$, which is a generalization of Kazhdan's property $(T)$ for linear isometric representations on $L^{\Phi}([0,1])$. We also prove that, for a Banach space $B$ whose modulus of convexity is sufficiently large, if a locally compact second countable group has Kazhdan's property $(T)$, then it has property $(F_{B})$, which is a fixed point property for affine isometric actions on $B$. Moreover, we see that, for an Orlicz sequence space $\ell^{\Phi\Psi}$ such that the Young function $\Psi$ sufficiently rapidly increases near $0$, hyperbolic groups (with Kazhdan's property $(T)$) don't have property $(F_{\ell^{\Phi\Psi}})$. These results are generalizations of the results for $L^p$-spaces.