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On the Order Determining Property of the Norm of a Kubo–Ando Mean in Operator Algebras

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Abstract

In this paper we deal with the question of when the norm of a Kubo–Ando mean \(\sigma \) determines the order on the positive definite cone of an operator algebra. More precisely, we investigate the following problem: when, for any given pair A, B of positive definite elements of the algebra, we have that \(A\le B\) holds if and only if \(\Vert A\sigma X\Vert \le \Vert B\sigma X\Vert \) is valid for all positive definite elements X. We present several results and formulate related open problems.

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Acknowledgements

The author expresses his sincere thanks to the anonymous referee for the very careful reading of the paper, for his/her remarks and comments that helped to improve the presentation.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Lajos Molnár

  2. Institute of Mathematics, Budapest University of Technology and Economics, Budapest, 1521, Hungary

    Lajos Molnár

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  1. Lajos Molnár
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Correspondence to Lajos Molnár.

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The author acknowledges supports from the Ministry for Innovation and Technology, Hungary, Grant NKFIH-1279-2/2020 and from the National Research, Development and Innovation Office of Hungary, NKFIH, Grant No. K134944.

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Molnár, L. On the Order Determining Property of the Norm of a Kubo–Ando Mean in Operator Algebras. Integr. Equ. Oper. Theory 93, 53 (2021). https://doi.org/10.1007/s00020-021-02666-0

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  • DOI: https://doi.org/10.1007/s00020-021-02666-0

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