  A Banach algebra $A$ is said to be a zero Jordan product determined Banach
algebra if every continuous bilinear map $\varphi\colon A\times A\to X$, where
$X$ is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever
$a$, $b\in A$ are such that $ab+ba=0$, is of the form
$\varphi(a,b)=\sigma(ab+ba)$ for some continuous linear map $\sigma$. We show
that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally
compact groups have this property, and also discuss some applications.
