## Abstract and Applied Analysis

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

Lajos Molnár

#### 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 ${C}^{\ast}$-algebras and present a proof in the finite dimensional case.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015 (2015), Article ID 434020, 6 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.aaa/1429103748

Digital Object Identifier
doi:10.1155/2015/434020

Mathematical Reviews number (MathSciNet)
MR3312746

Zentralblatt MATH identifier
07095576

#### Citation

Molnár, Lajos. On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators. Abstr. Appl. Anal. 2015 (2015), Article ID 434020, 6 pages. doi:10.1155/2015/434020. https://projecteuclid.org/euclid.aaa/1429103748

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