The Quasi-linearity problem for Jordan-Banach algebras: a topological characterization

TL;DR

This paper characterizes when local quasi-linear Jordan functionals on certain JB*-algebras are actually linear, based on their weak continuity properties on the unit ball.

Contribution

It provides a topological criterion for the linearity of local quasi-linear Jordan functionals in JB*-algebras without specific quotients.

Findings

A local quasi-linear Jordan functional is linear if and only if its restriction to the unit ball is uniformly weakly continuous.

The result applies to JB*-algebras with no quotients isomorphic to S_2(C).

The characterization links algebraic quasi-linearity to topological weak continuity.

Abstract

Let $\mathfrak{J}$ be a JB$^*$-algebra with no quotients isomorphic to $S_2(\mathbb{C})$. Let $\mu$ be a local quasi-linear Jordan functional on $\mathfrak{J}_{sa}$. We show that $\mu$ is a linear functional on $\mathfrak{J}_{sa}$ if and only if the restriction of $\mu$ to the closed unit ball of $\mathfrak{J}_{sa}$ is uniformly weakly continuous.