Group algebras acting on $L^p$-spaces

Abstract

For $p\in [1,\infty )$ we study representations of a locally compact group $G$ on $L^p$-spaces and $QS{L}^p$-spaces. The universal completions $F^p(G)$ and $F_{\mathrm{QS}}^p(G)$ of $L^1(G)$ with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group \ca{} of $G$ (which is the case $p=2$). We study these completions of $L^1(G)$ in relation to the algebra $F_{\lambda }^p(G)$ of $p$-pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, $G$ is amenable if and only if $F_{\mathrm{QS}}^p(G)=F^p(G)=F_{\lambda }^p(G)$. One of our main results is that for $1\le p<q\le 2$, there is a canonical map ${\gamma }_{p,q}:F^p(G)\to F^q(G)$ which is contractive and has dense range. When $G$ is amenable, ${\gamma }_{p,q}$ is injective, and it is never surjective unless $G$ is finite. We use the maps ${\gamma }_{p,q}$ to show that when $G$ is discrete, all (or one) of the universal completions of $L^1(G)$ are amenable as a Banach algebras if and only if $G$ is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to $L^p$-operator crossed products of topological spaces.

Version 1: 27 pages. Version 2: lots of minor corrections, and we got rid of the second-countability assumption on the groups. 31 pages

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