Zero Jordan product determined Banach algebras

Abstract

A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $\phi :A\times A\to X$, 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.

References in corpus (2)

• Zero Lie product determined Banach algebras
• Matrix multiplication is determined by orthogonality and trace

Cited by in corpus (1)

• Strongly zero product determined Banach algebras