Zero Lie product determined Banach algebras, II

Abstract

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ satisfying $φ(a,b)=0$ whenever $ab=ba$ is of the form $φ(a,b)=ω(ab-ba)$ for some $ω\in A^*$. 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.