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arXiv · 2301.06355

Every symmetric Kubo-Ando connection has the order-determining property on $\mathcal B(H)$

Emmanuel Chetcuti·Curt Healey
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Abstract

In \cite{molnar} L.~Molnar studied the question of whether the L\"owner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo-Ando mean. He affirmatively answered the question for certain classes of Kubo-Ando means and left as an open problem the general case. We here give an answer to this question, by showing that the norm of every symmetric Kubo-Ando mean $\sigma$ on $\mathcal B(H)$ is order-determining, i.e. if $A, B\in \mathcal B(H)^{{\sss{++}}}$ satisfy $\Vert A\sigma X\Vert \le \Vert B\sigma X\Vert$ for every $X\in \mathcal B(H)^{\sss{++}}$, then $A\le B$.

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Emmanuel Chetcuti, Curt Healey. 2023-01-16. Every symmetric Kubo-Ando connection has the order-determining property on $\mathcal B(H)$. https://arxiv.org/abs/2301.06355

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