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Projections in a Synaptic Algebra

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Abstract

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.

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Authors and Affiliations

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

    David J. Foulis

  2. Mathematical Institute, Slovak Academy of Sciences, Stefánikova 49, SK-814 73, Bratislava, Slovakia

    Sylvia Pulmannová

Authors
  1. David J. Foulis
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  2. Sylvia Pulmannová
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Correspondence to David J. Foulis.

Additional information

The second author was supported by Research and Development Support Agency under the contract No. APVV-0071-06, grant VEGA 2/6088/26 and Center of Excellence SAS, CEPI I/2/2005.

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Foulis, D.J., Pulmannová, S. Projections in a Synaptic Algebra. Order 27, 235–257 (2010). https://doi.org/10.1007/s11083-010-9148-2

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  • DOI: https://doi.org/10.1007/s11083-010-9148-2

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