Universal state space embeddability of Jordan-Banach algebras

We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra A admits the universal extension property if and only if the Gleason theorem is valid for A. As a consequence we get that any positive Stone algebra-valued measure on projection lattice of a quotient of a JBW algebra without type I-2 direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice.