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Order isomorphisms in $C^*$-algebras

Youssef El Khatiri

TL;DR

We provide a complete description of order isomorphisms between the self-adjoint parts of $C^*$-algebras and, more generally, between operator intervals in $AW^*$-algebras, showing such maps arise from Jordan $^*$-isomorphisms together with invertible elements in regular rings; explicitly, for $a\in A^+$ or $A^{++}$, $\Phi(a)=bJ(a)b$, and for $a\in A^{sa}$, $\Phi(a)=bJ(a)b+c$. The results extend Kadison's linear classification to nonlinear order-preserving maps, using regular monotone completions to reduce to monotone complete $C^*$-algebras and, in the AW^*-case, to the regular ring; Dye's theorem handles the effect-algebra case. The work provides a unified preserver-theoretic framework for operator intervals across noncommutative algebras and highlights a dichotomy between commutative and noncommutative AW^*-algebras. These findings deepen the algebraic understanding of quantum observables in noncommutative settings and suggest natural directions for further generalization.

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.

Order isomorphisms in $C^*$-algebras

TL;DR

We provide a complete description of order isomorphisms between the self-adjoint parts of -algebras and, more generally, between operator intervals in -algebras, showing such maps arise from Jordan -isomorphisms together with invertible elements in regular rings; explicitly, for or , , and for , . The results extend Kadison's linear classification to nonlinear order-preserving maps, using regular monotone completions to reduce to monotone complete -algebras and, in the AW^*-case, to the regular ring; Dye's theorem handles the effect-algebra case. The work provides a unified preserver-theoretic framework for operator intervals across noncommutative algebras and highlights a dichotomy between commutative and noncommutative AW^*-algebras. These findings deepen the algebraic understanding of quantum observables in noncommutative settings and suggest natural directions for further generalization.

Abstract

We provide a complete description of the order isomorphisms between the self-adjoint parts of -algebras. Furthermore, we characterize such isomorphisms between general operator intervals in -algebras. For the description, we use Jordan -isomorphisms and closed operators in the regular rings of -algebras. This work generalizes previous results on von Neumann algebras.
Paper Structure (10 sections, 21 theorems, 93 equations)

This paper contains 10 sections, 21 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator intervals in $C^*$-algebras
  4. Regular monotone completion of $C^*$-algebras
  5. Regular ring of finite $AW^*$-algebras
  6. The statement of main results
  7. Order isomorphisms between effect algebras
  8. Order isomorphisms between operator intervals
  9. The reduction of the problem
  10. Remarks and intriguing questions

Key Result

Theorem 1.1

(Kadison) Any unital linear order isomorphism between the self-adjoint parts of $C^*$-algebras extends uniquely to a Jordan $^*$-isomorphism between the corresponding $C^*$-algebras.

Theorems & Definitions (40)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Corollary 3.7
  • ...and 30 more