On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators

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...

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Main Author: Lajos Molnár
Format: Article
Language:English
Published: Wiley 2015-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2015/434020
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author Lajos Molnár
author_facet Lajos Molnár
author_sort Lajos Molnár
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description 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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spelling doaj-art-99d904401da94cb6acac3a784562a06e2025-02-03T01:22:54ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/434020434020On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite OperatorsLajos Molnár0 MTA-DE “Lendület” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, P.O. Box 12, Debrecen 4010, HungaryWe 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.http://dx.doi.org/10.1155/2015/434020
spellingShingle Lajos Molnár
On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators
Abstract and Applied Analysis
title On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators
title_full On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators
title_fullStr On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators
title_full_unstemmed On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators
title_short On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators
title_sort on the nonexistence of order isomorphisms between the sets of all self adjoint and all positive definite operators
url http://dx.doi.org/10.1155/2015/434020
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