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On Realization of Partially Ordered Abelian Groups

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Abstract

The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators.

We show that there is a set of points separating ℝ-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space \(\mathcal{H}\) which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of ℝ. A similar characterization is derived for an order determining set of ℝ-maps and symmetric operators on a dense set in a complex Hilbert space \(\mathcal{H}\). We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.

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Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Ivan Chajda

  2. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jan Paseka

  3. Department of Mathematics, Harbin Institute of Technology, Harbin, 150006, People’s Republic of China

    Lei Qiang

Authors
  1. Ivan Chajda
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  2. Jan Paseka
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  3. Lei Qiang
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Corresponding author

Correspondence to Jan Paseka.

Additional information

The first two authors acknowledge the support by ESF Project CZ.1.07/2.3.00/20.0051 Algebraic methods in Quantum Logic of the Masaryk University. The third author acknowledges the support by the National Natural Science Foundation of China (Project 11101108).

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Chajda, I., Paseka, J. & Qiang, L. On Realization of Partially Ordered Abelian Groups. Int J Theor Phys 52, 2028–2037 (2013). https://doi.org/10.1007/s10773-012-1426-x

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  • DOI: https://doi.org/10.1007/s10773-012-1426-x

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