Group Algebras Acting on $$L^p$$ L p -Spaces

Abstract

For $$p\in [1,\infty )$$ we study representations of a locally compact group $$G$$ on $$L^p$$ -spaces and $$\textit{QSL}^p$$ -spaces. The universal completions $$F^p(G)$$ and $$F^p_{\mathrm {QS}}(G)$$ 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 $$C^{*}$$ -algebra of $$G$$ (which is the case $$p=2$$ ). We study these completions of $$L^1(G)$$ in relation to the algebra $$F^p_\lambda (G)$$ 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 $$F^p_{\mathrm {QS}}(G)=F^p(G)=F^p_\lambda (G)$$ . One of our main results is that for $$1\le p< q\le 2$$ , there is a canonical map $$\gamma _{p,q}:F^p(G)\rightarrow F^q(G)$$ 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.