  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.
