Gleason Property and Extensions of States on Projection Logics

We prove that every state on the projection logic P(M) of avon Neumann algebra M not containing a direct summand of typeI₂ 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 analogousresult can be derived. Moreover, we prove that the extensionscan be taken linear in a complete order unit norm space associatedwith L. (Results of this paper generalize results of 22] andmay contribute to the noncommutative measure theory, convextheory of state spaces and foundations of quantum physics.)