Type Decomposition of a Pseudoeffect Algebra

Abstract

Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how typedetermining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.2010 Mathematics subject classification: primary 06C15; secondary 17C65, 46L45.

Pseudo-D-lattices and separating points of measures

“…For basic properties of pseudo-effect algebras and pseudo-D-lattices we refer for instance to [7,8,16,18,19,22,30].…”

“…Moreover, pseudo-effect algebras have, of course, a larger number of models than effect algebras, in particular those which derive from noncommutative po-groups [18]. For a study see, for example, [15,16,18,19,25,22,26,30].…”

Decomposition of Pseudo-effect Algebras and the Hammer–Sobczyk Theorem

“…For basic properties of pseudo-effect algebras and pseudo-D-lattices we refer for instance to [4,5,13,14,16]. We now collect some rules.…”

“…Pseudo-D-lattices (D-lattices, respectively) are the same as lattice-ordered pseudoeffect algebras (lattice-ordered effect algebras, respectively). References on these structures are, for instance, [10,11,13,14,16,24,31].…”

The exocenter and type decomposition of a generalized pseudoeffect algebra

“…Our purpose in this article is to define and study extensions to generalized pseudoeffect algebras of the notions of the center, central orthocompleteness, central cover, type determining sets and type decompositions for an effect algebra, resp. for a pseudoeffect algebra (see [10,11,13,14,16,18]).…”