arXiv subjects
Publications and source records attributed to Sylvia Pulmannova.
Dimension effect algebras were introduced in (A. Jencova, S. Pulmannova, Rep. Math. Phys. 62 (2008), 205-218), and it was proved that they are unit intervals in dimension groups. We prove that the effect algebra tensor product of dimension effect algebras is a dimension effect algebra, which is the unit interval in the unital abelian po-groups tensor product of the corresponding dimension groups.
We investigate the orthoalgebras of certain non-Boolean models which have a classical realization. Our particular concern will be the partition logics arising from the investigation of the empirical propositional structure of Moore and Mealy type automata.
We define and study an alternative partial order, called the spectral order, on a synaptic algebra-a generalization of the self-adjoint part of a von Neumann algebra. We prove that if the synaptic algebra A is norm complete (a Banach synaptic algebra), then under the spectral order, A is Dedekind sigma-complete lattice, and the corresponding effect algebra E is a sigma-complete lattice. Moreover, E can be organized into a Brouwer-Zadeh algebra in both the usual (synaptic) and spectral ordering; and if A is Banach, then E is a Brouwer-Zadeh lattice in the spectral ordering. If A is of finite type, then De Morgan laws hold on E in both the synaptic and spectral ordering.
We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor iff A is an antilattice. We also generalize several other results of R. Kadison pertaining to infima and suprema in operator algebras.
Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Stormer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C*-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW*-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.
A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $ξ_a$ on the $σ$-orthomodular lattice of projections in $A$ and that $ξ_a$ determines the spectral resolution of $a$. Also, if $f$ is a continuous function defined on the spectrum of $a$, we formulate a definition of $f(a)$, thus obtaining a continuous functional calculus for $A$.
Different versions of the notion of a state have been formulated for various so-called quantum structures. In this paper, we investigate the interplay among states on synaptic algebras and on its sub-structures. A synaptic algebra is a generalization of the partially ordered Jordan algebra of all bounded self-adjoint operators on a Hilbert space. The paper culminates with a characterization of extremal states on a commutative generalized Hermitian algebra, a special kind of synaptic algebra.
We study a pair p,e consisting of a projection p (an idempotent) and an effect e (an element between 0 and 1) in a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra). We show that some of Halmos's theory of two projections (or two subspaces), including a version of his CS-decomposition theorem, applieas on this settinh, and we introduce and study two candidates for a commutator for p and e.
A synaptic algebra $A$ is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace $V$ of $A$ in regard to the question of when $V$ is a vector lattice. Our main theorem states that if $V$ contains the identity element of $A$ and is closed under the formation of both the absolute value and the carrier of its elements, then $V$ is a vector lattice if and only if the elements of $V$ commute pairwise.
A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW*-algebra. In this paper we prove that a synaptic algebra A has the monotone square property, i.e., if a and b are positive elements, then if a is less or equal than b, then the square root of a is less or equal than the square root of b.
We investigate P. Halmos' two projections theorem, (or two subspaces theorem) in the context of a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra).
A generalized pseudo effect algebra (GPEA) is a partially ordered partial algebraic structure with a smallest element 0, but not necessarily with a unit (i.e, a largest element). If a GPEA admits a so-called unitizing automorphism, then it can be embedded as an order ideal in its so-called unitization, which does have a unit. We study unitizations of GPEAs with respect to a unitizing automorphism, paying special attention to the behavior of congruences and ideals in this setting.
State operators on convex effect algebras, in particular effect algebras of unital JC-algebras, JW-algebras and convex sigma-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.
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.
A synaptic algebra is a generalization of the Jordan algebra of selfadjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.
We extend to a generalized pseudoeffect algebra (GPEA) the notion of the exocenter of a generalized effect algebra (GEA) and show that elements of the exocenter are in one-to-one correspondence with direct decompositions of the GPEA; thus the exocenter is a generalization of the center of a pseudoeffect algebra (PEA). The exocenter forms a boolean algebra and the central elements of the GPEA correspond to elements of a sublattice of the exocenter which forms a generalized boolean algebra. We extend to GPEAs the notion of central orthocompleteness, prove that the exocenter of a centrally orthocomplete GPEA (COGPEA) is a complete boolean algebra and show that the sublattice corresponding to the center is a complete boolean subalgebra. We also show that in a COGPEA, every element admits an exocentral cover and that the family of all exocentral covers, the so-called exocentral cover system, has the properties of a hull system on a generalized effect algebra. We extend the notion of type determining (TD) sets, originally introduced for effect algebras and then extended to GEAs and PEAs, to GPEAs, and prove a type-decomposition theorem, analogous to the type decomposition of von Neumann algebras.
In this paper we define and study dimension generalized effect algebras (DGEAs), i.e., Dedekind orthocomplete and centrally orthocomplete generalized effect algebras equipped with a dimension equivalence relation. Our theory is a bona fide generalization of the theory of dimension effect algebras (DEAs), i.e., it is formulated so that, if a DGEA happens to be an effect algebra (i.e., it has a unit element), then it is a DEA. We prove that a DGEA decomposes into type I, II, and III DGEAS in a manner analogous to the type I/II/III decomposition of a DEA.
Pseudo-effect algebras are partial algebraic structures, that were introduced as a non-commutative generalization of effect algebras. In the present paper, lattice ordered pseudo-effect algebras are considered as possible algebraic non-commutative analogs of non-commutative non-standard reasoning. To this aim, the interplay among conjunction, implication and negation connectives is studied. It turns out that in the non-commutative reasoning, all these connectives are doubled. In particular, there are two negations and two pairs consisting of conjunction and implication, related by residuation laws. The main result of the paper is a characterization of lattice pseudo-effect algebras in terms of so-called pseudo Sasaki algebras. We also show that all pseudo-effect algebras can be characterized in terms of certain partially defined double residuated structures.