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Generalized Hermitian Algebras

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Abstract

We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. In this paper we define and launch a study of a class of generalized Hermitian (GH) algebras. Among the examples of GH-algebras are ordered special Jordan algebras, JW-algebras, and AJW-algebras, but unlike these more restricted cases, a GH-algebra is not necessarily a Banach space and its lattice of projections is not necessarily complete. In this paper we develop the basic theory of GH-algebras, identify their unit intervals as effect algebras, and observe that their projection lattices are sigma-complete orthomodular lattices. We show that GH-algebras are spectral order-unit spaces and that they admit a substantial spectral theory.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, 1 Sutton Court, 01002, Amherst, MA, USA

    David J. Foulis

  2. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Sylvia Pulmannová

Authors
  1. David J. Foulis
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  2. Sylvia Pulmannová
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Corresponding author

Correspondence to Sylvia Pulmannová.

Additional information

The second author was supported by Research and Development Support Agency under the contract No. APVV-0071-06, grant VEGA 2/0032/09 and Center of Excellence SAS, CEPI I/2/2005.

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Foulis, D.J., Pulmannová, S. Generalized Hermitian Algebras. Int J Theor Phys 48, 1320–1333 (2009). https://doi.org/10.1007/s10773-008-9903-y

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  • DOI: https://doi.org/10.1007/s10773-008-9903-y

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