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 noncommutativeC*-algebras and present a proof in the finite dimensional case.
Highlights
Introduction and ResultsIt is a trivial fact that the logarithmic function f(t) = log t, t > 0, is an order isomorphism from the set ]0, ∞[ of positive real numbers onto the real line ]−∞, ∞[; that is, it is a bijective map such that t ≤ s ⇐⇒ f (t) ≤ f (s) (1)holds for all t, s ∈ ]0, ∞[
We do not have a proof for our conjecture concerning general noncommutative C∗-algebras but in what follows we present a proof in the particular case of finite dimensional C∗-algebras
To make our paper self-contained, we present it with a short complete proof
Summary
MTA-DE “Lendulet” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, P.O. Box 12, Debrecen 4010, Hungary. 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
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