Representations of $p$-convolution algebras on $L^q$-spaces

For a nontrivial locally compact group $G$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1,\infty )$, if any of these Banach algebras can be represented on an $L^q$-space, then one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $\left \vert\frac 1p - \frac 12\right \vert=\left \vert\frac 1q - \frac 12\right \vert$. This result can be interpreted as follows: for $p,q\in [1,\infty )$, the $L^p$- and $L^q$-representation theories of a group are incomparable, except in the trivial cases when they are equivalent.

As an application, we show that, for distinct $p,q\in [1,\infty )$, if the $L^p$- and $L^q$-crossed products of a topological dynamical system are isomorphic, then $\frac 1p + \frac 1q=1$. In order to prove this, we study the following relevant aspects of $L^p$-crossed products: existence of approximate identities, duality with respect to $p$, and existence of canonical isometric maps from group algebras into their multiplier algebras.