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Abstract and Applied Analysis
Volume 2015, Article ID 434020, 6 pages
http://dx.doi.org/10.1155/2015/434020
Research Article

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

MTA-DE “Lendület” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, P.O. Box 12, Debrecen 4010, Hungary

Received 20 October 2014; Accepted 8 December 2014

Academic Editor: Debora Amadori

Copyright © 2015 Lajos Molnár. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -algebras and present a proof in the finite dimensional case.