An order-theoretic characterization of JB-algebras

TL;DR

This paper characterizes JB-algebras using order-theoretic and geometric properties of complete order unit spaces, providing new insights into their structure and isomorphisms.

Contribution

It introduces an order-theoretic and geometric characterization of JB-algebras among complete order unit spaces, extending previous finite-dimensional and reflexive cases.

Findings

Characterization of JB-algebras via order-anti-automorphisms

Identification of the interior of the cone as a symmetric Banach--Finsler manifold

Isomorphism criterion via gauge-reversing bijections

Abstract

We give an order-theoretic characterization of the JB-algebras among the complete order unit spaces in terms of the existence of an order-anti-automorphism of the interior of the cone that is homogeneous of degree -1. More geometrically, we characterize JB-algebras as those complete order unit spaces for which the interior of the cone is a symmetric Banach--Finsler manifold under Thompson's metric. Furthermore, we show that two order unit spaces are isomorphic if there exists a gauge-reversing bijection between them, thus answering a question raised by Noll--Sch\"afer. These results have previously been established for finite-dimensional resp. reflexive order unit spaces by Walsh and Lemmens--R.--Wortel.