Published online by Cambridge University Press: 08 January 2020
A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if, for every Banach space $X$, every bilinear map $\unicode[STIX]{x1D711}:A\times A\rightarrow X$ satisfying $\unicode[STIX]{x1D711}(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba)$ for some continuous linear map $\unicode[STIX]{x1D70E}$. We show that all $C^{\ast }$-algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.
Communicated by A. Sims
The authors were supported by MINECO grant PGC2018-093794-B-I00. The first, the third and the fourth named authors were supported by Junta de Andalucía grant FQM-185. The second named author was supported by ARRS grant P1-0288.