Preservers of the p-power and the Wasserstein means on 2x2 matrices

Simon, Richárd and Virosztek, Dániel (2023) Preservers of the p-power and the Wasserstein means on 2x2 matrices. ELECTRONIC JOURNAL OF LINEAR ALGEBRA, 39. pp. 395-408. ISSN 1537-9582

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Abstract

In one of his recent papers, Moln & agrave;r showed that if A is a von Neumann algebra without I-1, I-2-type direct summands, then any function from the positive definite cone of A to the positive real numbers preserving the Kubo-Ando power mean, for some 0 ? p ? (-1, 1), is necessarily constant. It was shown in that paper that I-1-type algebras admit nontrivial p-power mean preserving functionals, and it was conjectured that I-2-type algebras admit only constant p-power mean preserving functionals. We confirm the latter. A similar result occurred in another recent paper of Moln & agrave;r concerning the Wasserstein mean. We prove the conjecture for I-2-type algebras in regard of the Wasserstein mean, too. We also give two conditions that characterise centrality in C-*-algebras.

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