  For a nontrivial locally compact group $G$, and $p\in [1,\infty)$, 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
$q\in [1,\infty)$, these Banach algebras can be represented on an $L^q$-space
if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b)
$|\frac 1p - \frac 12|=|\frac 1q - \frac 12|$. 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.
