Synaptic Algebras

TL;DR

This paper introduces synaptic algebras, a mathematical structure that unifies various algebraic and order-theoretic concepts inspired by operator algebras and quantum logic.

Contribution

It defines synaptic algebras as a new class of algebraic structures that generalize several known algebras used in quantum theory and operator algebras.

Findings

Synaptic algebras unify Jordan algebras, spectral spaces, and effect algebras.

Examples include self-adjoint parts of von Neumann, Rickart, and AW*-algebras.

Synaptic algebras extend beyond norm completeness, even in the commutative case.

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. Pulmannov\'a. All the foregoing examples are norm complete, but synaptic algebras are more general, and even a commutative synaptic algebra need not be norm complete.