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Isomorphisms of Birkhoff-James orthogonality on finite-dimensional $C^*$-algebra

Bojan Kuzma, Srdjan Stefanović

TL;DR

This work classifies bijective maps that strongly preserve Birkhoff-James orthogonality on finite-dimensional complex $C^*$-algebras. It proves that such BJ isomorphisms are, up to a unimodular scalar and a central positive multiplier, real-linear isometries, and on simple summands they reduce to conjugate-linear isometries combined with block-unitary similarities. Extending to general finite-dimensional algebras without abelian summands, the authors obtain a blockwise description $\\Phi(A)=\\Gamma(A)A$ with $\\Gamma(A)=\\gamma(A)P(A)$, where $P(A)$ lies in the center and is positive, and $\\gamma(A)$ is unimodular. The results delineate when the neat simple-case form holds and explain the complications introduced by abelian summands, which can yield weighted or nonlocal structures. Overall, the paper sharpens the understanding of BJ-orthogonality preservers in finite-dimensional $C^*$-algebras and clarifies how algebraic structure governs the allowable preservers.

Abstract

We classify bijective maps which strongly preserve Birkhoff-James orthogonality on a finite-dimensional complex $C^*$-algebra. It is shown that those maps are close to being real-linear isometries whose structure is also determined.

Isomorphisms of Birkhoff-James orthogonality on finite-dimensional $C^*$-algebra

TL;DR

This work classifies bijective maps that strongly preserve Birkhoff-James orthogonality on finite-dimensional complex -algebras. It proves that such BJ isomorphisms are, up to a unimodular scalar and a central positive multiplier, real-linear isometries, and on simple summands they reduce to conjugate-linear isometries combined with block-unitary similarities. Extending to general finite-dimensional algebras without abelian summands, the authors obtain a blockwise description with , where lies in the center and is positive, and is unimodular. The results delineate when the neat simple-case form holds and explain the complications introduced by abelian summands, which can yield weighted or nonlocal structures. Overall, the paper sharpens the understanding of BJ-orthogonality preservers in finite-dimensional -algebras and clarifies how algebraic structure governs the allowable preservers.

Abstract

We classify bijective maps which strongly preserve Birkhoff-James orthogonality on a finite-dimensional complex -algebra. It is shown that those maps are close to being real-linear isometries whose structure is also determined.

Paper Structure

This paper contains 7 sections, 28 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal subspaces of relative left-symmetry
  4. BJ isomorphisms on TEXT
  5. BJ isomorphisms on TEXT
  6. Proof of the main theorem
  7. Concluding remarks

Key Result

Theorem 1.1

Let $\Phi\colon\mathcal{A}\to\mathcal{B}$ be a BJ isomorphism between complex $C^\ast$-algebras $\mathcal{A}$ and $\mathcal{B}$. If $\mathcal{A}$ is finite-dimensional and has no abelian summand, then $\mathcal{A}$ and $\mathcal{B}$ are isomorphic $C^\ast$-algebras and there exists a real-linear sur

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 2.1: Kuzm-Sush-abelian, Lemma 2.1
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 51 more