Bilinear maps having Jordan product property

TL;DR

The paper analyzes symmetric bilinear maps on C$^*$-algebras that satisfy the square-zero property or the Jordan product property at a fixed element, showing that infinite simple von Neumann algebras are symmetrically square-zero determined and that finite direct sums and $c_0$-sums of such algebras inherit this structure. This yields representations of the form $V(a,b)=T(a\circ b)$ and enables a description of Jordan homomorphisms and derivations at a point, including decompositions through central multipliers for surjective preservers. The results unify and extend known outcomes on zero-product, Jordan maps, and derivations in von Neumann algebra direct sums, with concrete consequences for square-zero preservers and derivations at a point. Overall, the work provides a cohesive framework linking square-zero properties, Jordan products, and derivations in noncommutative operator algebra settings.

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.

Theorems & Definitions (29)

• Example 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof