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arXiv · 1512.08976

Synaptic Algebras

D. J. Foulis
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Abstract

A synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective "synaptic," borrowed from biology, is meant to suggest that such an algebra coherently "ties together" the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice. Prototypic examples of synaptic algebras are the special Jordan algebra of all self-adjoint elements in a von Neumann algebra, the self-adjoint elements in a Rickart C*-algebra, the self-adjoint elements in an AW*-algebra, D. Topping's JW- and AJW-algebras, and the generalized Hermitian (GH-) algebras introduced and studied by the author and S. Pulmannová. All the foregoing examples are norm complete, but synaptic algebras are more general, and even a commutative synaptic algebra need not be norm complete.

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D. J. Foulis. 2015-12-05. Synaptic Algebras. https://arxiv.org/abs/1512.08976

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