Characterization of Spaces of Countably Additive and Filtering States

It has been proved in [1] that any compact convex set can be the state space of an orthomodular lattice (OML). The characterization of the space of σ-additive states has been proved in one direction in [2] and completed by a necessary and sufficient condition in [3]. These spaces were found to be exactly s-semiexposed faces of compact convex sets, i.e., faces which can be expressed as intersections of level sets of some linear functionals. Inspired by [4], decompositions of states related to complete additivity were studied in [5]–[9]. The basic aim was to express any state as a convex combination of a completely additive state and a state which is far from completely additive. Several notions have been proposed for this purpose—weakly purely finitely additive states, filtering states, and weakly filtering states (see [7] for an overview of these notions). A state µ on an OML L is called filtering [9] if ∀a ∈ L \ {0} ∃b ∈ L \ {0} : µ(b) = 0. A state with this property is obviously weakly purely finitely additive, i.e., it cannot be obtained as a non-trivial convex combination of a completely additive state and any state. We study the convex structure of the space of filtering states. We prove that they can be any s-semiexposed faces of compact convex sets. Thus the space of filtering states has the same characterization as the space of completely additive states