Beschreibung
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 Birkhoffs problem. The Monograph contains a number of unsolved problems. Most of the results are carried over to projections from von Neumann algebras.
Autorenporträt
Marjan Matvejchuk Born 1947, Russia. -Graduated from Kazan State University, 1970, Russia. -PhD thesis, 1974. -Doctor of Sciences in 1991 from Institute of Mathematics Ukaine Academy of Science. -Professor, 1993. -At the present time he is a professor at Kazan National Research Technical University, (former Kazan Aviation University) Russia.
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