The authors define a Banach algebra \(A\) to be zero Lie product determined if, for every continuous bilinear functional \(\varphi\) satisfying \(\varphi(a,b)=0\), whenever \(a\) and \(b\) commute, then \(\varphi\) is of the form \(\varphi(a,b)=\tau ([a,b])\) for some \(\tau\in A^\ast\), the dual of \(A\). Zero Lie product determined algebras have occurred in quantum physics, for instance, in the consideration of canonical commutation relations to study the behaviour of bosons.
A zero Lie product determined \(A\) is said to satisfy condition B if, for any Banach space \(X\), every continuous bilinear mapping \(\varphi:A\times A\to X\) is such that \(\varphi(ab,c)=\varphi(a,bc)\) for all \(a,b,c\).
The authors define an approximate diagonal on \(A\) to be a bounded net \((u_\lambda)\) in the projective tensor product \(A\hat{\otimes} A\) such that \(\lim(au_\lambda -u_\lambda a)=0\) and \(\lim \pi(u _\lambda)a=a\) for each \(a \in A\), where \(\pi\) denotes the projection \(a\otimes b \mapsto ab\).
They then call \(A\) amenable if and only if it has an approximate diagonal. They show that any \(C^\ast\)-algebra is zero Lie product determined, cf.Proposition 2.1 in their article.
The main result of the paper is that an amenable Banach algebra satisfying property B is zero Lie product determined. As a consequence, \(L^1(G)\) is zero Lie product determined if \(G\) is a locally compact amenable group. The authors say that they do not know if the condition of amenability is really necessary for their result.