Group algebras acting on $L^p$-spaces

For $p\in [1,\infty)$ we study representations of a locally compact group $G$ on $L^p$-spaces and $QSL^p$-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 \ca{} 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 $1\leq p< q\leq 2$, there is a canonical map $\gamma_{p,q}\colon F^p(G)\to F^q(G)$ 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.