Symmetries in Synaptic Algebras

TL;DR

This paper explores symmetries within synaptic algebras, generalizing concepts from von Neumann algebras, and examines the properties of an induced equivalence relation on the projection lattice, especially in complete cases.

Contribution

It introduces and analyzes the properties of symmetry-induced equivalence relations in synaptic algebras, extending the understanding of their structure and parallels with dimension relations.

Findings

Equivalence relation has properties similar to dimension relations in complete lattices.

Symmetries are elements with squares equal to the unit, generalizing von Neumann algebra concepts.

Results apply to centrally orthocomplete projection lattices.

Abstract

A synaptic algebra is a generalization of the Jordan algebra of selfadjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.