FACIAL SETS OF PROBABILITY MEASURES

Abstract: This is a discussion of probability measures in a noncommutative setting as
required by quantum mechanical probability theory. The concepts of a facial, orthostable and
orthofacial subset of probability measures on an orthomodular poset are introduced. They
provide a link between the poset and the boundary structure of convex sets of such
measures.

An orthomodular poset admitting a facial subset has interesting properties: e.g. it is a
complete lattice and every element in is a completely additive measure. We investigate
the connection between orthostability and the Jordan-Hahn decomposition of measures. It is
shown that the set of completely additive probability measures on the projection lattice of a
von Neumann algebra is orthofacial. Finally we use the notion of orthofaciality of a subset
of probability measures on an orthomodular poset to give a necessary and sufficient
condition for each bounded affine functional on to be the expectation functional of some
observable having finite spectrum.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -