Generalized Hermitian Algebras |
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Authors: | David J Foulis and Sylvia Pulmannová |
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Institution: | (1) Department of Mathematics and Statistics, University of Massachusetts, 1 Sutton Court, 01002 Amherst, MA, USA;(2) Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia |
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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.
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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Keywords: | GH-algebra Effect Projection Orthomodular lattice Carrier projection Comparability property Square root Absolute value Spectral resolution Spectrum |
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