# Zero Jordan product determined Banach algebras

## TL;DR

This paper characterizes zero Jordan product determined Banach algebras, showing that all C*-algebras and L^1 groups of amenable groups possess this property, with implications for functional analysis.

## Contribution

It establishes that all C*-algebras and L^1(G) for amenable groups are zero Jordan product determined Banach algebras, expanding understanding of their structural properties.

## Key findings

- All C*-algebras are zero Jordan product determined.
- All group algebras L^1(G) of amenable groups have this property.
- Applications to functional analysis and algebraic structure.

## Abstract

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.