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

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Tuesday, 18. April 2023, 10:30 - 23:45
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 $\mathcal{A}$ is a von Neumann algebra without  $I_1, I_2$-type direct summands,  then any function from the positive definite cone of $\mathcal{A}$  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 $I_1$-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 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 $I_2$-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.
 
The lecture will be in the Riesz lecture hall.

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