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

Abstract

In \cite{molnar} L.~Molnar studied 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. 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 $A\le B$.