Zero Lie product determined Banach algebras

TL;DR

This paper characterizes zero Lie product determined Banach algebras, showing that all group algebras of amenable locally compact groups possess this property, thus advancing understanding of their algebraic structure.

Contribution

It introduces the class of zero Lie product determined Banach algebras and proves that all group algebras of amenable groups are included in this class.

Findings

All group algebras $L^1(G)$ for amenable groups are zero Lie product determined.

Provides general remarks on the class of zero Lie product determined Banach algebras.

Establishes the connection between amenability and the zero Lie product property.

Abstract

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$ with the property that $\varphi(a,b)=0$ whenever $a$ and $b$ commute is of the form $\varphi(a,b)=\tau(ab-ba)$ for some $\tau\in A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with $G$ an amenable locally compact group are zero Lie product determined.