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, Volume 36, Issue 1, pp 1–17 | Cite as

Spectral Order on a Synaptic Algebra

  • David J. Foulis
  • Sylvia PulmannováEmail author
Article

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.

Keywords

Synaptic algebra Jordan algebra Order unit space Spectral ordering Effect algebra Spectral resolution Brower-Zadeh poset De Morgan properties 

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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