Zero Lie product determined Banach algebras, II

TL;DR

This paper characterizes zero Lie product determined Banach algebras, showing they possess this property under conditions like weak amenability, cyclic derivation properties, or being matrix algebras over cyclically amenable Banach algebras.

Contribution

It establishes new sufficient conditions for Banach algebras to be zero Lie product determined, extending previous results and including matrix algebras over cyclically amenable algebras.

Findings

Banach algebras with property $ extbf{B}$ and bounded approximate identity are zero Lie product determined.

Cyclically amenable Banach algebras ensure the property via derivation conditions.

Matrix algebras over cyclically amenable Banach algebras are zero Lie product determined.

Abstract

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$ satisfying $\varphi(a,b)=0$ whenever $ab=ba$ is of the form $\varphi(a,b)=\omega(ab-ba)$ for some $\omega\in A^*$. We prove that $A$ has this property provided that any of the following three conditions holds: (i) $A$ is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all $n\times n$ matrices, where $n\ge 2$, over a cyclically amenable Banach algebra with a bounded approximate identity.