Jordan weak amenability and orthogonal forms on JB*-algebras

TL;DR

This paper establishes isometric correspondences between symmetric and anti-symmetric orthogonal forms on JB*-algebras and certain classes of derivations, deepening understanding of their algebraic structure.

Contribution

It introduces new isometric correspondences linking orthogonal forms with Jordan and Lie Jordan derivations on JB*-algebras.

Findings

Isometric correspondence between symmetric orthogonal forms and Jordan derivations

Isometric correspondence between anti-symmetric orthogonal forms and Lie Jordan derivations

Enhances understanding of algebraic structures in JB*-algebras

Abstract

We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.