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Complete orthogonality preservers between C$^*$-algebras

Jorge J. Garcés

Abstract

We introduce $n$-orthogonality (and completely orthogonality) preserving operators between C$^*$-algebras. Our main theorem states that every completely orthogonality preserving bounded linear mapping between C$^*$-algebras is a weighted TRO homomorphism. We also give several characterisations of TRO homomorphisms among triple homomorphisms.

Complete orthogonality preservers between C$^*$-algebras

Abstract

We introduce -orthogonality (and completely orthogonality) preserving operators between C-algebras. Our main theorem states that every completely orthogonality preserving bounded linear mapping between C-algebras is a weighted TRO homomorphism. We also give several characterisations of TRO homomorphisms among triple homomorphisms.
Paper Structure (5 sections, 20 theorems, 37 equations)

This paper contains 5 sections, 20 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A geometric characterisation of TRO homomorphisms
  4. Completely orthogonality preserving operators
  5. A generalisation of the order zero functional calculus

Key Result

Proposition 1

Every normal triple homomorphism between von Neumann algebras decomposes as the orthogonal sum of a TRO homomorphism and a TRO anti-homomorphism. $\Box$

Theorems & Definitions (33)

  • Proposition 1
  • Definition 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof
  • Proposition 7
  • ...and 23 more