Banach algebras generated by an invertible isometry of an $L^p$-space

TL;DR

This paper characterizes Banach algebras generated by invertible isometries on $L^p$-spaces, introducing spectral configurations that determine their structure and showing these algebras are semisimple and often closed under functional calculus.

Contribution

It introduces spectral configurations as invariants for classifying Banach algebras generated by invertible isometries on $L^p$-spaces, providing a complete description and new insights into their structure.

Findings

Banach algebras generated by invertible isometries are semisimple.

Most such algebras are closed under continuous functional calculus.

Banach algebras acting on $L^1$-spaces are not closed under quotients.

Abstract

We provide a complete description of those Banach algebras that are generated by an invertible isometry of an $L^p$-space together with its inverse. Examples include the algebra $PF_p(\mathbb{Z})$ of $p$-pseudofunctions on $\mathbb{Z}$, the commutative $C^*$-algebra $C(S^1)$ and all of its quotients, as well as uncountably many `exotic' Banach algebras. We associate to each isometry of an $L^p$-space, a spectral invariant called `spectral configuration', which contains considerably more information than its spectrum as an operator. It is shown that the spectral configuration describes the isometric isomorphism type of the Banach algebra that the isometry generates together with its inverse. It follows from our analysis that these algebras are semisimple. With the exception of $PF_p(\mathbb{Z})$, they are all closed under continuous functional calculus, and their Gelfand transform is an isomorphism. As an application of our results, we show that Banach algebras that act on $L^1$-spaces are not closed under quotients. This answers the case $p=1$ of a question asked by Le Merdy 20 years ago.