Zero Jordan product determined Banach algebras

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $\phi \text{\hspace{0.17em}⁣}:A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $\phi (a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\phi (a,b)=\sigma (ab+ba)$ for some continuous linear map $\sigma$. 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.