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

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