Zero Lie product determined Banach algebras
Volume 239 / 2017
Studia Mathematica 239 (2017), 189-199
MSC: 43A20, 46H05, 46L05, 47B48.
DOI: 10.4064/sm8734-4-2017
Published online: 7 July 2017
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 $\varphi (a,b)=0$ 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.
