Amenability of Closed Subgroups and Orlicz Spaces

We prove that a closed subgroup $H$ of a second countable locally compact group $G$ is amenable if and only if its left regular representation on an Orlicz space $L^Φ(G)$ for some $Δ_2$-regular $N$-function $Φ$ almost has invariant vectors. We also show that a noncompact second countable locally compact group $G$ is amenable if and ony if the first cohomology space $H^1(G,L^Φ(G))$ is non-Hausdorff for some $Δ_2$-regular $N$-function $Φ$.

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