Property (TLΦ) and property (FLΦ) for Orlicz spaces LΦ

An Orlicz space LΦ(Ω) is a Banach function space defined by using a Young function Φ, which generalizes the Lp spaces. We show, for an Orlicz space LΦ([0,1]) which is not isomorphic to L∞([0,1]), if a locally compact second countable group has property (TLΦ([0,1])), which is a generalization of Kazhdan's property (T) for linear isometric representations on LΦ([0,1]), then it has Kazhdan's property (T). We also show, for a separable complex Orlicz space LΦ(Ω) with gauge norm, Ω=R,[0,1],N, if a locally compact second countable group has Kazhdan's property (T), then it has property (TLΦ(Ω)). We prove, for a finitely generated group Γ and a Banach space B whose modulus of convexity is sufficiently large, if Γ has Kazhdan's property (T), then it has property (FB), which is a fixed point property for affine isometric actions on B. Moreover, we see that, for a hyperbolic group Γ (which may have Kazhdan's property (T)) and an Orlicz sequence space ℓΦΨ with gauge norm such that the Young function Ψ sufficiently rapidly increases near 0, Γ doesn't have property (FℓΦΨ). These results are generalizations of the results for Lp-spaces.