On the nonexistence of order isomorphisms between the sets of all self-adjoint and all positive definite operators

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Abstract

We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutative C-algebras and present a proof in the finite dimensional case.

Original languageEnglish
Article number434020
JournalAbstract and Applied Analysis
Volume2015
DOIs
Publication statusPublished - 2015

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Hilbert spaces
Positive definite
Algebra
Nonexistence
Mathematical operators
Isomorphism
Bijective
Operator
Self-adjoint Operator
Assertion
C*-algebra
Hilbert space
Concepts

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutative C-algebras and present a proof in the finite dimensional case.",
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