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Every symmetric Kubo–Ando connection has the order-determining property

Part of: General theory of linear operators Selfadjoint operator algebras Special classes of linear operators

Published online by Cambridge University Press:  11 September 2023

Emmanuel Chetcuti  and
Curt Healey
Emmanuel Chetcuti
Affiliation:
Department of Mathematics, Faculty of Science, University of Malta, Msida MSD 2080, Malta e-mail: emanuel.chetcuti@um.edu.mt
Curt Healey*
Affiliation:
Department of Mathematics, Faculty of Science, University of Malta, Msida MSD 2080, Malta
*
e-mail: curt.c.healey.13@um.edu.mt
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Abstract

In this article, the question of whether the Löwner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo–Ando mean is studied. The question was affirmatively answered for certain classes of Kubo–Ando means, yet the general case was left as an open problem. We here give a complete answer to this question, by showing that the norm of every symmetric Kubo–Ando mean is order-determining, i.e., if $A,B\in \mathcal B(H)^{++}$ satisfy $\Vert A\sigma X\Vert \le \Vert B\sigma X\Vert $ for every $X\in \mathcal {A}^{{++}}$, where $\mathcal A$ is the C*-subalgebra generated by $B-A$ and I, then $A\le B$.

Keywords

Kubo–Ando connectionC*-algebrapositive definite coneorderpreservers

MSC classification

Primary: 47A64: Operator means, shorted operators, etc.
Secondary: 47B49: Transformers, preservers (operators on spaces of operators) 46L40: Automorphisms
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 10
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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