Synaptic Algebras

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.