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 $\phi :A\times A\to \mathbb{C}$ satisfying $\phi (a,b)=0$ whenever $ab=ba$ is of the form $\phi (a,b)=\omega (ab-ba)$ for some $\omega \in A^{\ast }$. 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 $A^{\ast }$ 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.