The Rao-Reiter criterion for the amenability of homogeneous spaces
Let $G$ be a locally compact group $G$. The study of different actions of $G$ on various spaces (e.g. manifolds, graphs, etc.) is of fundamental importance, both for understanding the properties of $G$ and the structure of the spaces under consideration. Consider for example a closed subgroup $H\subseteq G$ of $G$ and let $M=G/H$ be the corresponding homogeneous space. Then, $M$ is called \textit{amenable} in the sense of \textit{P. Eymard} [Moyennes invariantes et représentations unitaires. Springer, Cham (1972; Zbl 0249.43004)] and \textit{F. P. Greenleaf} [J. Funct. Anal. 4, 295--315 (1969; Zbl 0195.42301)], if the space $L^{\infty}(M)$ supports an invariant mean. When $H=\{e\}$, this notion reduces to the classical concept of an amenable locally compact group, i.e. a locally compact group $G$ with a left-invariant Haar measure $m$ and a $G$-invariant mean on $L^{\infty}(G, m)$ (which is a positive linear functional $F$ on $L^{\infty}(G, m)$ such that $F(1_{G})=1$ and $F(\phi(g^{-1}x))=F(\phi)$ for any $\phi\in L^{\infty}(G, m)$ and $g\in G$). This is equivalent to say that every continuous action of $G$ via affine mappings on a non-empty compact convex subset $\mathcal{C}$ of a locally convex topological vector space has a fixed point. Similarly, given a closed subgroup $H\subseteq G$, if $G$ acts continuously via affine transformations on a non-trivial convex subset $\mathcal{C}$, such that $\mathcal{C}$ contains an $H$-fixed point, then $\mathcal{C}$ contains also a fixed point of $G$ and the coset space $G/H$ is amenable, and conversely. The author of the article treats the notion of amenable homogeneous spaces $G/H$ for the interesting case where $G$ and $H\subseteq G$ are not amenable and provides a criterion for amenability in terms of the Rao-Reiter condition. In particular, given a locally compact group $G$ and a closed subgroup $H$, the author shows that the homogeneous space $G/H$ is amenable if and only if the following holds: for every compact set $K\subset G$ and every $\epsilon>0$, there exists a non-negative function $f\in L^{\Phi}(G/H)$ with $\|f\|=1$, such that $\|\pi_{\Phi}(s)f-f\|<\epsilon$ for any $s\in K$, where $\|\cdot\|$ denotes the gause norm on the Orlicz space $L^{\Phi}(G/H)$ and $\pi_{\Phi}$ is an ``quasiregular'' action of $G$ on the unit sphere $S^{1}\big(L^{\Phi}(G/H)\big)$. In particular, $\Phi : \mathbb{R}\to\mathbb{R}$ is an $N$-function for which there exists some constant $c>2$, such that $\Phi(2x)\leq c\Phi(x)$ for all $x\geq 0$.
scientific article; zbMATH DE number 7035753
scientific article