Relations between effect algebras with Riesz decomposition properties and AF C*-algebras are studied. The well-known one-one correspondence between countable MV-algebras and unital AF C*-algebras whose Murray-von Neumann order is a lattice is extended to any unital AF C* algebras and some more general effect algebras having the Riesz decomposition property. One-one correspondence between tracial states on AF C*-algebras and states on the corresponding effect algebras is proved. In particular, pure (faithful) tracial states correspond to extremal (faithful) states on corresponding effect (...) algebras. (shrink)

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are (...) MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek's BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr. (shrink)

In the quantum logic approach, Bell inequalities in the sense of Pitowski are related with quasi hidden variables in the sense of Deliyannis. Some properties of hidden variables on effect algebras are discussed.

For any unit vector in an inner product space S, we define a mapping on the system of all ⊥-closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. (...) Finally, the result will be generalized to general inner product spaces. (shrink)

We relate so-called spin factors and generalized Hermitian (GH-) algebras, both of which are partially ordered special Jordan algebras. Our main theorem states that positive-definite spin factors of dimension greater than one are mathematically equivalent to generalized Hermitian algebras of rank two.

A synaptic algebra is a generalization of the self-adjoint part of a von Neumann algebra. In this article we extend to synaptic algebras the type-I/II/III decomposition of von Neumann algebras, AW∗-algebras, and JW-algebras.

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras.We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a (...) direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice. (shrink)

Two notions of grading of a quantum logic by a product of copies of the group ℤ 2 are introduced and used to define graded tensor products of quantum logics.

A notion of factorizability for vector-valued measures on a quantum logic L enables us to pass from abstract logics to Hilbert space logics and thereby to construct tensor products. A claim by Kruszynski that, in effect, every orthogonally scattered measure is factorizable is shown to be false. Some criteria for factorizability are found.

Two different characterizations of POV measures with commutative range are compared using a representation of some stochastic operators by (weak) Markov kernels. A representation by Choquet theorem is obtained as an integral over functions of a sharp observable appearing in one of the characterizations. A Naimark extension is constructed.

The theory of nearly orthosymmetric ortholattices generalizes the theory of orthosymmetric ortholattices defined by one of the authors in a previous paper. In this theory, some equations allow one to distinguish complex Hilbertian lattices from real and quaternion ones.

Two postulates concerning observables on a quantum logic are formulated. By Postulate 1 compatibility of observables is defined by the strong topology on the set of observables. Postulate 2 requires that the range of the sum of observables ought to be contained in the smallestC-closed sublogic generated by their ranges. It is shown that the Hilbert space logicL(H) satisfies the two postulates. A theorem on the connection between joint distributions of types 1 and 2 on the logic satisfying Postulate 2 (...) is proved. (shrink)

Starting with a quantum logic (a σ-orthomodular poset) L. a set of probabilistically motivated axioms is suggested to identify L with a standard quantum logic L(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space.

The notion of relative compatibility of observables is treated and its relation to the existence of joint distributions is obtained. The case of conventional quantum mechanics is studied and a generalization to the case of the quantum logic approach to quantum mechanics is given. It is shown that relative compatibility is equivalent to the existence of so-called “type 1” joint distributions.

Two different generalizations of Brouwer–Zadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from the so-called (...) special effects on a Hilbert space. It is shown that the standard Brouwer negation for effect operators produces a degenerate BZ poset with respect to the order induced from the partial sum operation. (shrink)