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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Format: | Article |
Language: | English |
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Wiley
2015-01-01
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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 |
collection | DOAJ |
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. |
format | Article |
id | doaj-art-99d904401da94cb6acac3a784562a06e |
institution | Kabale University |
issn | 1085-3375 1687-0409 |
language | English |
publishDate | 2015-01-01 |
publisher | Wiley |
record_format | Article |
series | Abstract and Applied Analysis |
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 |
work_keys_str_mv | AT lajosmolnar onthenonexistenceoforderisomorphismsbetweenthesetsofallselfadjointandallpositivedefiniteoperators |