Order isomorphisms in $C^*$-algebras

TL;DR

This paper characterizes order isomorphisms between self-adjoint parts of $C^*$-algebras and operator intervals in $AW^*$-algebras, extending prior results on von Neumann algebras using Jordan $^*$-isomorphisms.

Contribution

It provides a complete description of order isomorphisms in $C^*$-algebras and $AW^*$-algebras, generalizing previous work on von Neumann algebras.

Findings

Characterization of order isomorphisms using Jordan $^*$-isomorphisms.

Extension of results to general operator intervals in $AW^*$-algebras.

Generalization of known results from von Neumann algebras.

Abstract

We provide a complete description of the order isomorphisms between the self-adjoint parts of $C^*$-algebras. Furthermore, we characterize such isomorphisms between general operator intervals in $AW^*$-algebras. For the description, we use Jordan $^*$-isomorphisms and closed operators in the regular rings of $AW^{*}$-algebras. This work generalizes previous results on von Neumann algebras.