Geodesic
Every symmetric Kubo–Ando connection has the order-determining property
Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 279-288

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DOI

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 $A\le B$.
DOI : 10.4153/S0008439523000668
Mots-clés : Kubo–Ando connection, C*-algebra, positive definite cone, order, preservers 
Chetcuti, Emmanuel; Healey, Curt. Every symmetric Kubo–Ando connection has the order-determining property. Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 279-288. doi: 10.4153/S0008439523000668
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     title = {Every symmetric {Kubo{\textendash}Ando} connection has the order-determining property},
     journal = {Canadian mathematical bulletin},
     pages = {279--288},
     year = {2024},
     volume = {67},
     number = {2},
     doi = {10.4153/S0008439523000668},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439523000668/}
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