Order isomorphisms on order intervals of atomic JBW-algebras

TL;DR

This paper characterizes order isomorphisms between effect algebras of atomic JBW-algebras, providing explicit formulas and extending results from type I factors to the entire algebra structure.

Contribution

It offers a complete description of order isomorphisms on effect algebras of atomic JBW-algebras, including explicit formulas for type I factors and their extension.

Findings

Derived a closed formula for order isomorphisms on type I factors

Proved the invariance of the invertible part of effect algebras

Extended formulas to the entire effect algebra

Abstract

In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I factors and order isomorphisms factor through the direct sum decomposition, this yields the desired description.