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

TL;DR

This paper proves that the norm of any symmetric Kubo-Ando mean on bounded operators uniquely determines the L"owner order, resolving an open question for all such means.

Contribution

It establishes that the norm of every symmetric Kubo-Ando mean on (H) is order-determining, answering a longstanding open problem.

Findings

The norm of symmetric Kubo-Ando means determines the operator order.

Order relations can be inferred from mean norms in (H).

The result applies to all symmetric Kubo-Ando means.

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 $A\le B$.