An order-theoretic characterization of C*-algebras

TL;DR

This paper characterizes unital C*-algebras within operator systems using order-theoretic properties and JB-algebra structures, extending known results to real and complex cases.

Contribution

It provides an order-theoretic criterion for recognizing C*-algebras among operator systems, based on JB-algebra structures of matrix spaces.

Findings

A unital operator system is a C*-algebra iff each matrix space admits a compatible JB-algebra structure.

For n ≥ 4, the range of a unital n-positive projection on a unital real C*-algebra is a unital real C*-algebra.

The characterization extends to real and complex C*-algebras using order-theoretic properties.

Abstract

We give an order-theoretic characterization of the essential image of the forgetful functor from the category of real/complex unital C*-algebras to the category of real/complex unital operator systems. It is based on the characterization of JB-algebras among the order unit spaces in terms of the existence of gauge-reversing bijections obtained by M. Roelands and the author in arXiv:2507.09526. To this end, we show that a unital operator system is completely order isomorphic to a C*-algebra if and only if each of its matrix spaces admits a compatible JB-algebra structure. As an application, we prove that for $n\ge 4$ the range of a unital n-positive projection on a unital real C*-algebra is unitally n-order isomorphic to a unital real C*-algebra, which is the analogue of a result proven for complex C*-algebras by Choi--Effros.