A modern look at algebras of operators on $L^p$-spaces

TL;DR

This paper reviews recent advances in the study of operator algebras on $L^p$-spaces, highlighting new techniques that address longstanding questions and provide classifications of homomorphisms.

Contribution

It offers a modern overview of $L^p$-operator algebras, incorporating recent methods to solve classical problems and describe homomorphisms.

Findings

Description of all unital contractive homomorphisms between $p$-pseudofunctions of groups

Integration of new techniques with classical tools to resolve longstanding questions

Renewed interest and progress in $L^p$-operator algebra theory

Abstract

The study of operator algebras on Hilbert spaces, and C*-algebras in particular, is one of the most active areas within Functional Analysis. A natural generalization of these is to replace Hilbert spaces (which are $L^2$-spaces) with $L^p$-spaces, for $p\in [1,\infty)$. The study of such algebras of operators is notoriously more challenging, due to the lack of orthogonality in $L^p$-spaces. We give a modern overview of a research area whose beginnings can be traced back to the 50's, and that has seen renewed attention in the last decade through the infusion of new techniques. The combination of these new ideas with old tools was the key to answer some long standing questions. Among others, we provide a description of all unital contractive homomorphisms between algebras of $p$-pseudofunctions of groups.