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

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.

31 pages

Cited by in corpus (7)

• Rigidity results for $L^p$-operator algebras and applications
• Topologically free actions and ideals in twisted Banach algebra crossed products
• Dynamical Complexity and $K$-Theory of $L^p$ Operator Crossed Products
• Multiplier algebras of $L^p$-operator algebras
• Persistence approximation property for $L^p$ operator algebras
• $L^p$ coarse Baum-Connes conjecture via $C_0$ coarse geometry
• C*-like modules and matrix $p$-operator norms