Non classical projections logics

Classical Quantum Logic consists of orthogonal projections of complex Hilbert space H. The famous Gleason theorem describes states (=probability measure) on the logic B(H) of all orthogonal projections of H. By Gleason theorem if dimH is equal to infinity then any (complex in general) measure is linear combination of states. In the Monograph we study the Quantum logics of projections P(H) in Krein space J(H) in Hilbert space with conjugation operator and S(E) of real-orthogonal projections in Euclidean space. An interesting case P(H) is when any projection is generated by a positive projection. We see that the logic S(E) consists of layers of isomorphic logics. The richest class of states exists on the logic S(E). If dimH is equal to infinity then on the logic J(H) every linear measure (that is the measure extending to a linear functional) is a complex-valued measure. The results of the monograph clarify one Birkhoff's problem. The Monograph contains a number of unsolved problems. Most of the results are carried over to projections from von Neumann algebras.