Characterization of Spaces of Filtering States

Filtering states on orthomodular lattices have been introduced by G.T. Rüttimann as an “opposite” of completely additive states. He proved that they form a face of the state space. The question which faces can be the sets of filtering states remained open. Here we prove that, for any semi-exposed face F of a compact convex set C, there is an orthomodular lattice L and an affine homeomorphism φ of C onto the state space of L such that φ(F) is the space of filtering states.