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Spectral Order on a Synaptic Algebra

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Abstract

We define and study an alternative partial order, called the spectral order, on a synaptic algebra—a generalization of the self-adjoint part of a von Neumann algebra. We prove that if the synaptic algebra A is norm complete (a Banach synaptic algebra), then under the spectral order, A is Dedekind σ-complete lattice, and the corresponding effect algebra E is a σ-complete lattice. Moreover, E can be organized into a Brouwer-Zadeh algebra in both the usual (synaptic) and spectral ordering; and if A is Banach, then E is a Brouwer-Zadeh lattice in the spectral ordering. If A is of finite type, then De Morgan laws hold on E in both the synaptic and spectral ordering.

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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, Štefánikova 49, 814 73, Bratislava, Slovakia

    Sylvia Pulmannová

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

Additional information

The second author was supported by the Research and Development Support Agency under the contract APVV-16-0073 and grant VEGA 2/0069/16.

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Foulis, D.J., Pulmannová, S. Spectral Order on a Synaptic Algebra. Order 36, 1–17 (2019). https://doi.org/10.1007/s11083-018-9451-x

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