Amenability of Closed Subgroups and Orlicz Spaces

Abstract

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 $L^{\Phi }(G)$ for some ${\Delta }_2$-regular $N$-function $\Phi$ 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 $H^1(G,L^{\Phi }(G))$ is non-Hausdorff for some ${\Delta }_2$-regular $N$-function $\Phi$.

8 pages