Leírás
In one of his recent papers L. Molnár, On dissimilarities of the conventional and Kubo-Ando power means in operator algebras, (J. Math. Anal. Appl., 504 (2021) 125356), Molnár showed that if A is a von Neumann algebra without I1, I2-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 nonzero p between -1 and 1 is necessarily constant. It was shown in that paper, that I1-type algebras admit nontrivial p-power mean preserving functionals, and it was conjectured, that I2-type algebras admit only constant p-power mean preserving functionals. We confirm the latter. A similar result occured in L. Molnár, Maps on positive definite cones of C*-algebras preserving the Wasserstein mean, Proc. Amer. Math. Soc. 150 (2022), 1209-1221., concerning the Wasserstein mean. We prove the conjecture for I2-type algebras in regard of the Wasserstein mean, too. We also give two conditions that characterise centrality in C*-algebras.
Joint work with Dániel Virosztek.