This is an announcement for the paper “Representations of $p$-convolution algebras on $L_q$-spaces” by Eusebio Gardellahttps://arxiv.org/find/math/1/au:+Gardella_E/0/1/0/all/0/1, Hannes Thielhttps://arxiv.org/find/math/1/au:+Thiel_H/0/1/0/all/0/1.

Abstract: 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) $|\frac{1}{p}-\frac{1}{2}|=|\frac{1}{q}-\frac{1}{2}|$. 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 $\frac{1}{p}+\frac{1}{q}=1$. 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.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.08612