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Every symmetric Kubo–Ando connection has the order-determining property
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 2 / June 2024
- Published online by Cambridge University Press:
- 11 September 2023, pp. 279-288
- Print publication:
- June 2024
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- Article
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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 
$A\le B$.
