Quotients of Banach algebras acting on $L^p$-spaces

TL;DR

This paper demonstrates that the class of Banach algebras representable on $L^p$-spaces (for $p eq 2$) is not closed under quotients, answering a long-standing open question.

Contribution

It provides the first example showing that such Banach algebras are not closed under quotients, using methods based on invertible isometries of $L^p$-spaces.

Findings

The class of Banach algebras on $L^p$-spaces is not closed under quotients.

The result answers a question posed by Le Merdy 20 years ago.

The methods involve analysis of Banach algebras generated by invertible isometries.

Abstract

We show that the class of Banach algebras that can be isometrically represented on an $L^p$-space, for $p\neq 2$, is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of $L^p$-spaces.