Effect algebras with state operator

State operators on convex effect algebras, in particular effect algebras of unital JC-algebras, JW-algebras and convex ?-MV-algebras are studied and their relations with conditional expectations in algebraic sense as well as in the sense of probability on MV-algebras are shown.

[1]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[2]  F. Jellett,et al.  PARTIALLY ORDERED ABELIAN GROUPS WITH INTERPOLATION (Mathematical Surveys and Monographs 20) , 1987 .

[3]  Tomás Kroupa,et al.  Conditional probability on MV-algebras , 2005, Fuzzy Sets Syst..

[4]  Stanley Gudder,et al.  What Is Fuzzy Probability Theory? , 2000 .

[5]  R. Kadison A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras , 1952 .

[6]  Z. Šidák On Relations Between Strict-Sense and Wide-Sense Conditional Expectations , 1957 .

[7]  Stanley Gudder,et al.  Lattice properties of quantum effects , 1996 .

[8]  D. Mundici Interpretation of AF -algebras in ukasiewicz sentential calculus , 1986 .

[9]  P. Busch,et al.  The quantum theory of measurement , 1991 .

[10]  R. Morrow,et al.  Foundations of Quantum Mechanics , 1968 .

[11]  Robert W. Spekkens,et al.  Foundations of Quantum Mechanics , 2007 .

[12]  E. Beltrametti,et al.  A classical extension of quantum mechanics , 1995 .

[14]  Beloslav Riečan,et al.  Probability on MV algebras , 1997 .

[15]  Anatolij Dvurecenskij,et al.  Loomis-Sikorski theorem and Stone duality for effect algebras with internal state , 2010, Fuzzy Sets Syst..

[16]  Sylvia Pulmannová,et al.  Orthomodular structures as quantum logics , 1991 .

[17]  D. Mundici,et al.  Algebraic Foundations of Many-Valued Reasoning , 1999 .

[18]  K. Kraus,et al.  States, effects, and operations : fundamental notions of quantum theory : lectures in mathematical physics at the University of Texas at Austin , 1983 .

[19]  Edwin Hewitt,et al.  Real And Abstract Analysis , 1967 .

[20]  Abner Shimony,et al.  The logic of quantum mechanics , 1981 .

[21]  A. Dvurecenskij Tensor product of difference posets and effect algebras , 1995 .

[22]  Daniele Mundici Tensor Products and the Loomis-Sikorski Theorem for MV-Algebras , 1999 .

[23]  J. Neumann Mathematische grundlagen der Quantenmechanik , 1935 .

[24]  H. Weber There Are Orthomodular Lattices Without Non-trivial Group-Valued States: A Computer-Based Construction , 1994 .

[25]  B. Riecan,et al.  Integral, Measure, and Ordering , 1997 .

[26]  Jan Kühr,et al.  De Finetti theorem and Borel states in [0, 1]-valued algebraic logic , 2007, Int. J. Approx. Reason..

[27]  Effect Algebras Are Not Adequate Models for Quantum Mechanics , 2010 .

[28]  Franco Montagna,et al.  MV-algebras with internal states and probabilistic fuzzy logics , 2009, Int. J. Approx. Reason..

[29]  DANIELE MUNDICI,et al.  Averaging the truth-value in Łukasiewicz logic , 1995, Stud Logica.

[30]  Anatolij Dvurečenskij,et al.  Loomis-sikorski theorem for σ-complete MV-algebras and ℓ-groups , 2000, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[31]  K. Goodearl Partially ordered abelian groups with interpolation , 1986 .

[32]  Roberto Giuntini,et al.  Toward a formal language for unsharp properties , 1989 .

[33]  Stanley Gudder Morphisms, tensor products and σ-effect algebras , 1998 .

[34]  Sylvia Pulmannová,et al.  New trends in quantum structures , 2000 .

[35]  Mirko Navara An orthomodular lattice admitting no group-valued measure , 1994 .

[36]  E. Effros,et al.  POSITIVE PROJECTIONS AND JORDAN STRUCTURE IN OPERATOR ALGEBRAS , 1979 .

[37]  Pekka Lahti,et al.  Partial order of quantum effects , 1995 .

[38]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

[39]  S. Gudder,et al.  Convex and linear effect algebras , 1999 .

[40]  D. Foulis,et al.  Effect algebras and unsharp quantum logics , 1994 .

[41]  Mirko Navara,et al.  A characterization of tribes with respect to the Łukasiewicz t-norm , 1997 .

[42]  Dan Butnariu,et al.  Triangular norm-based measures and their Markov kernel representation , 1991 .

[43]  H. Umegaki CONDITIONAL EXPECTATION IN AN OPERATOR ALGEBRA, II , 1954 .

[44]  A. Dvurecenskij Tensor product of difference posets , 1995 .

[45]  M. Hamana Injective envelopes of $C^{*}$-algebras , 1979 .

[46]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .

[47]  E. Effros,et al.  Jordan algebras of self-adjoint operators , 1967 .

[48]  G. Lüders Über die Zustandsänderung durch den Meßprozeß , 1950 .

[49]  Sylvia Pulmannová,et al.  Representation theorem for convex effect algebras , 1998 .

[50]  S. Gudder,et al.  Convex effect algebras, state ordered effect algebras, and ordered linear spaces , 2000 .

[51]  Anatolij Dvurecenskij,et al.  Conditional probability on sigma-MV-algebras , 2005, Fuzzy Sets Syst..

[52]  Tensor products of divisible effect algebras , 2003 .

[53]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.