# Order isomorphisms between cones of JB-algebras

## TL;DR

This paper characterizes order isomorphisms between cones of atomic JBW-algebras, showing their structure and conditions under which they are linear, with implications for the algebraic and topological properties of JB-algebras.

## Contribution

It provides a complete description of order isomorphisms between cones of atomic JBW-algebras and establishes conditions for their linearity in general JB-algebras.

## Key findings

- Order isomorphisms between cones of atomic JBW-algebras are fully characterized.
- Every order isomorphism on the engaged part is linear.
- Non-linear order isomorphisms occur only in the disengaged part, which consists of copies of ℝ.

## Abstract

In this paper we completely describe the order isomorphisms between cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order isomorphism is linear and the disengaged part consists only of copies of $\mathbb{R}$. Furthermore, in the setting of general JB-algebras we prove the following. If either algebra does not contain an ideal of codimension one, then every order isomorphism between their cones is linear if and only if it extends to a homeomorphism, between the cones of the atomic part of their biduals, for a suitable weak topology.