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TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

Part of: Selfadjoint operator algebras Modular lattices, complemented lattices

Published online by Cambridge University Press:  28 February 2011

DAVID J. FOULIS ,
SYLVIA PULMANNOVÁ  and
ELENA VINCEKOVÁ
DAVID J. FOULIS
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA (email: foulis@math.umass.edu)
SYLVIA PULMANNOVÁ*
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Stefánikova 49, SK-814 73 Bratislava, Slovakia (email: pulmann@mat.savba.sk)
ELENA VINCEKOVÁ
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Stefánikova 49, SK-814 73 Bratislava, Slovakia (email: vincek@mat.savba.sk)
*
✉For correspondence; e-mail: pulmann@mat.savba.sk
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Abstract

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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.

Keywords

(pseudo)effect algebra(pseudo)MV algebraorthomodular posetorthomodular latticeboolean algebravon Neumann algebraJW algebraLoomis dimension latticetypes IIIand III

MSC classification

Secondary: 06C15: Complemented lattices, orthocomplemented lattices and posets 46L45: Decomposition theory for $C^*$-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 3 , December 2010 , pp. 335 - 358
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

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.

References

[1]Beran, L., Orthomodular Lattices, An Algebraic Approach, Mathematics and its Applications, 18 (D. Reidel, Dordrecht, 1985).Google Scholar
[2]Carrega, J. C., Chevalier, G. and Mayet, R., ‘Direct decompositions of orthomodular lattices’, Algebra Universalis 27 (1990), 480–496.CrossRefGoogle Scholar
[3]Chang, C. C., ‘Algebraic analysis of many-valued logic’, Trans. Amer. Math. Soc. 88 (1958), 467–490.CrossRefGoogle Scholar
[4]Dixmier, J., Les algèbres d’opérateurs dans l’espace hilbertien (Gauthier-Villars, Paris, 1957).Google Scholar
[5]Dvurečenskij, A., ‘Central elements and Cantor–Bernstein’s theorem for pseudo-effect algebras’, J. Aust. Math. Soc. 74 (2003), 121–143.CrossRefGoogle Scholar
[6]Dvurečenskij, A. and Pulmannová, S., New Trends in Quantum Structures (Kluwer, Dordrecht, 2000).CrossRefGoogle Scholar
[7]Dvurečenskij, A. and Vetterlein, T., ‘Pseudoeffect algebras. I. Basic properties’, Internat. J. Theoret. Phys. 40 (2001), 685–701.CrossRefGoogle Scholar
[8]Dvurečenskij, A. and Vetterlein, T., ‘Pseudoeffect algebras. II. Group representations’, Internat. J. Theoret. Phys. 40 (2001), 703–726.CrossRefGoogle Scholar
[9]Dvurečenskij, A. and Vetterlein, T., ‘Non-commutative algebras and quantum structures’, Internat. J. Theoret. Phys. 43(7/8) (2004), 1599–1612.CrossRefGoogle Scholar
[10]Foulis, D. J. and Bennett, M. K., ‘Effect algebras and unsharp quantum logics’, Found. Phys. 24(10) (1994), 1331–1352.CrossRefGoogle Scholar
[11]Foulis, D. J. and Pulmannová, S., ‘Centrally orthocomplete effect algebras’, Algebra Universalis, DOI 10.1007/S00012-010-0100-5.Google Scholar
[12]Foulis, D. J. and Pulmannová, S., ‘Hull mappings and dimension effect algebras’, Math. Slovaca, to appear.Google Scholar
[13]Foulis, D. J. and Pulmannová, S., ‘Type decomposition of an effect algebra’, Found. Phys. 40 (2010), 1543–1565.CrossRefGoogle Scholar
[14]Georgescu, G. and Iorgulescu, A., ‘Pseudo-MV algebras: a non-commutative extension of MV algebras’, in: Proc. Fourth Inter. Symp. on Econ. Inform., May 6–9, Bucharest (eds. Sumeurenau, I.et al.) (INFOREC Printing House, Bucharest, 1999), pp. 961–968.Google Scholar
[15]Goodearl, K. R. and Wehrung, F., ‘The complete dimension theory of partially ordered systems with equivalence and orthogonality’, Mem. Amer. Math. Soc. 176(831) (2005).Google Scholar
[16]Greechie, R. J., Foulis, D. J. and Pulmannová, S., ‘The center of an effect algebra’, Order 12 (1995), 91–106.CrossRefGoogle Scholar
[17]Kalmbach, G., Orthomodular Lattices (Academic Press, New York, 1983).Google Scholar
[18]Kalmbach, G., Measures and Hilbert Lattices (World Scientific, Singapore, 1986).CrossRefGoogle Scholar
[19]Kaplansky, I., ‘Projections in Banach algebras’, Ann. of Math. (2) 53(2) (1951), 235–249.CrossRefGoogle Scholar
[20]Kaplansky, I., Rings of Operators (W.A. Benjamin, New York and Amsterdam, 1968).Google Scholar
[21]Loomis, L. H., The Lattice Theoretic Background of the Dimension Theory of Operator Algebras, Memoirs of the American Mathematical Society, 18 (American Mathematical Society, Providence, RI, 1955).CrossRefGoogle Scholar
[22]Maeda, S., ‘Dimension functions on certain general lattices’, J. Sci. Hiroshima Univ. A19 (1955), 211–237.Google Scholar
[23]von Neumann, J., ‘On rings of operators’, J. von Neumann Collected Works, Vol. III (Pergamon Press, Oxford, 1961).Google Scholar
[24]Rachůnek, J., ‘A non-commutative generalization of MV algebras’, Czechoslovak Math. J. 52 (2002), 255–273.CrossRefGoogle Scholar
[25]Ramsay, A., ‘Dimension theory in complete weakly modular orthocomplemented lattices’, Trans. Amer. Math. Soc. 116 (1965), 9–31.CrossRefGoogle Scholar
[26]Sikorski, R., Boolean Algebras, 2nd edn (Academic Press, New York and Springer, Berlin, 1964).Google Scholar
[27]Topping, D. M., Jordan Algebras of Self-adjoint Operators, Memoirs of the American Mathematical Society, 53 (American Mathematical Society, Providence, RI, 1965).Google Scholar
[28]Xie, Y. and Li, Y., ‘Central elements in pseudo-effect algebras’, Math. Slovaca, to appear.Google Scholar