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Additive spectrum preserving mappings from von Neumann algebras

Martin Mathieu, Francois Schulz

TL;DR

The paper resolves Jafarian's 2009 conjecture by showing that any surjective additive spectrum-preserving map from a von Neumann algebra $A$ onto a semisimple Banach algebra is a Jordan isomorphism. Building on the theory of linear spectrum preservers, it develops a program to derive automatic continuity, idempotent preservation, and structural constraints for additive preserver mappings, and extends these ideas to C*-algebras with real rank zero. A key technical contribution is a constructive scheme using $R$, $\tilde{R}$ and $W$ to force linearity and obtain a unital Jordan homomorphism on real rank zero domains, ultimately yielding Jordan isomorphisms in several cases. In the von Neumann setting, a decomposition argument shows the additive spectrum-preserving map must be inner-automorphism equivalent to the identity, thus proving that $T$ is a Jordan isomorphism. Overall, the work advances preserver problems by establishing rigidity of spectrum-preserving maps and clarifying Jordan-structure preservation in operator algebras.

Abstract

We establish Jafarian's 2009 conjecture that every additive spectrum preserving mapping from a von Neumann algebra onto a semisimple Banach algebra is a Jordan isomorphism.

Additive spectrum preserving mappings from von Neumann algebras

TL;DR

The paper resolves Jafarian's 2009 conjecture by showing that any surjective additive spectrum-preserving map from a von Neumann algebra onto a semisimple Banach algebra is a Jordan isomorphism. Building on the theory of linear spectrum preservers, it develops a program to derive automatic continuity, idempotent preservation, and structural constraints for additive preserver mappings, and extends these ideas to C*-algebras with real rank zero. A key technical contribution is a constructive scheme using , and to force linearity and obtain a unital Jordan homomorphism on real rank zero domains, ultimately yielding Jordan isomorphisms in several cases. In the von Neumann setting, a decomposition argument shows the additive spectrum-preserving map must be inner-automorphism equivalent to the identity, thus proving that is a Jordan isomorphism. Overall, the work advances preserver problems by establishing rigidity of spectrum-preserving maps and clarifying Jordan-structure preservation in operator algebras.

Abstract

We establish Jafarian's 2009 conjecture that every additive spectrum preserving mapping from a von Neumann algebra onto a semisimple Banach algebra is a Jordan isomorphism.
Paper Structure (4 sections, 16 theorems, 43 equations)

This paper contains 4 sections, 16 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Some Properties of Additive Spectral Preservers
  3. Additive Spectrum Preserving Mappings On Certain C*-Algebras
  4. The von Neumann Algebra Case

Key Result

Proposition 2.1

Suppose that $A$ is semisimple and that $T\colon A\to B$ is a surjective map with the property that $r(x+y) = r(Tx+Ty)$ for all $x, y\in A$. Then $B$ is semisimple and $Z(B) = T(Z(A))$.

Theorems & Definitions (31)

  • Proposition 2.1
  • proof
  • Proposition 2.2: Askes2022
  • Corollary 2.3
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • ...and 21 more