Supporting sequences of pure states on JB algebras

We show that any sequence ${\displaystyle \left({\phi }_n\right)}$ of mutually orthogonal pure states on a JB algebra A such that ${\displaystyle \left({\phi }_n\right)}$ forms an almost discrete sequence in the relative topology induced by the primitive ideal space of A admits a sequence ${\displaystyle \left(a_n\right)}$ consisting of positive, norm one, elements of A with pairwise orthogonal supports which is supporting for ${\displaystyle \left({\phi }_n\right)}$ in the sense of ${\displaystyle {\phi }_n\left(a_n\right)=1}$ for all n. Moreover, if A is separable then ${\displaystyle \left(a_n\right)}$ can be taken such that ${\displaystyle \left({\phi }_n\right)}$ is uniquely determined by the biorthogonality condition ${\displaystyle {\phi }_n\left(a_n\right)=1}$. Consequences of this result improving hitherto known extension theorems for C*-algebras and descriptions of dual JB algebras are given.