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

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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
Publication statusPublished - 2015


ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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