Reversible AJW-algebras

TL;DR

This paper proves that special AJW-algebras can be decomposed into parts that are reversible or totally nonreversible, providing a structural classification based on central projections.

Contribution

It introduces a decomposition of special AJW-algebras into reversible and nonreversible components using central projections.

Findings

Existence of central projections dividing the algebra into three parts.

Characterization of reversible parts via ideals and duals.

Identification of a totally nonreversible component.

Abstract

In this article it is proved that for every special AJW-algebra $A$ there exist central projections $e$, $f$, $g\in A$, $e+f+g=1$ such that (1) $eA$ is reversible and there exists a norm-closed two sided ideal $I$ of $C^*(eA)$ such that $eA={{}^\perp}(^\perp(I_{sa})_+)_+$; (2) $fA$ is reversible and $R^*(fA)\cap iR^*(fA)=\{0\}$; (3) $gA$ is a totally nonreversible AJW-algebra.