Bilinear maps having Jordan product property

TL;DR

This paper investigates symmetric bilinear maps with the Jordan product property on certain C*-algebras, showing they have a specific form and characterizing related algebraic structures.

Contribution

It establishes that such bilinear maps are of a particular form and characterizes Jordan homomorphisms and derivations at a fixed element in specific C*-algebras.

Findings

Maps have the square-zero property on certain sums of von Neumann algebras.

Such maps can be expressed as compositions with a bounded linear map.

Characterization of Jordan homomorphisms and derivations at a fixed element.

Abstract

We study symmetric continuous bilinear maps $V$ on a C$^*$-algebra $A$ that have the Jordan product property at a fixed element $z\in A$. We show that, whenever $A$ is a finite direct sum or a $c_0$-sum of infinite simple von Neumann algebras, such a map $V$ has the square-zero property. Then, it is proved that $V(a,b)=T(a\circ b)$ for some bounded linear map $T$ on $A$. As a consequence, Jordan homomorphisms and derivations at $z\in A$ are characterized.