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TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

  • DAVID J. FOULIS (a1), SYLVIA PULMANNOVÁ (a2) and ELENA VINCEKOVÁ (a3)
Abstract
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 type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

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Copyright
Corresponding author
For correspondence; e-mail: pulmann@mat.savba.sk
Footnotes
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The second and third authors were supported by the Research and Development Support Agency under contract no. LPP-0199-07; grant VEGA 2/0032/09, Center of Excellence SAS–Quantum Technologies; and ERDF OP R & D Project CE QUTE ITMS 26240120009 and meta-QUTE ITMS 26240120022.

Footnotes
References
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[1] L. Beran , Orthomodular Lattices, An Algebraic Approach, Mathematics and its Applications, 18 (D. Reidel, Dordrecht, 1985).

[6] A. Dvurečenskij and S. Pulmannová , New Trends in Quantum Structures (Kluwer, Dordrecht, 2000).

[18] G. Kalmbach , Measures and Hilbert Lattices (World Scientific, Singapore, 1986).

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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