Gleason Property and Extensions of States on Projection Logics
Jan Hamhalter · Bulletin of the London Mathematical Society · 1994
We prove that every state on the projection logic P(M) of a von Neumann algebra M not containing a direct summand of type I2 extends to a state of an arbitrary larger unital logic L. We also show that if a C*-algebra enjoys the Gleason property, and if it possesses sufficiently many projections, then an analogous result can be derived. Moreover, we prove that the extensions can be taken linear in a complete order unit norm space associated with L. (Results of this paper generalize results of [22] and may contribute to the noncommutative measure theory, convex theory of state spaces and foundations of quantum physics.)