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

2023/01/16 by Chetcuti, Emmanuel, Healey, Curt
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2301.06355

Abstract

In \citemolnar 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 σ on \mathcal B(H) is order-determining, i.e. if A, B∈ \mathcal B(H)^\sss++ satisfy \Vert AσX\Vert ≤ \Vert BσX\Vert for every X∈ \mathcal B(H)^\sss++, then A≤ B.

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