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

Let $\mathfrak{J}$ be a JB${}^{\ast }$-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.