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

英文摘要：

An Orlicz space $L^Φ(Ω)$ is a Banach function space defined by using a Young function $Φ$, which generalizes the $L^p$ spaces. We show that, for a reflexive Orlicz space $L^Φ([0,1])$, a locally compact second countable group has Kazhdan's property $(T)$ if and only if it has property $(T_{L^Φ([0,1])})$, which is a generalization of Kazhdan's property $(T)$ for linear isometric representations on $L^Φ([0,1])$. 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 $\ell^{ΦΨ}$ such that the Young function $Ψ$ sufficiently rapidly increases near $0$, hyperbolic groups (with Kazhdan's property $(T)$) don't have property $(F_{\ell^{ΦΨ}})$. These results are generalizations of the results for $L^p$-spaces.