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

Lajos Molnár

Abstract and Applied Analysis, 2015, vol. 2015, 1-6

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.

Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:434020

DOI: 10.1155/2015/434020

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