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

TL;DR

This paper characterizes when $p$-convolution algebras can be represented on $L^q$-spaces, showing they are only operator algebras at $p=2$, and explores implications for $L^p$-crossed products in dynamical systems.

Contribution

It provides a complete characterization of $L^q$-representability of $p$-convolution algebras, revealing the limitations of $L^p$-representation theory for groups.

Findings

Banach algebras are operator algebras iff $p=2$

Representation on $L^q$-spaces occurs only under specific conditions

Isomorphism of $L^p$ and $L^q$ crossed products implies $rac{1}{p}+rac{1}{q}=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) $|\frac 1p - \frac 12|=|\frac 1q - \frac 12|$. 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.