A projection and an effect in a synaptic algebra

TL;DR

This paper explores the relationships between projections and effects within synaptic algebras, extending Halmos's projection theory and introducing new concepts for commutators in this mathematical framework.

Contribution

It generalizes Halmos's two projections theory to effects in synaptic algebras and proposes new candidates for commutators between projections and effects.

Findings

Extension of Halmos's CS-decomposition theorem to effects

Introduction of two new commutator candidates for p and e

Application of projection theory to synaptic algebras

Abstract

We study a pair p,e consisting of a projection p (an idempotent) and an effect e (an element between 0 and 1) in a synaptic algebra (a generalization of the self-adjoint part of a von Neumann algebra). We show that some of Halmos's theory of two projections (or two subspaces), including a version of his CS-decomposition theorem, applieas on this settinh, and we introduce and study two candidates for a commutator for p and e.