Preservers of the $p$-power and the Wasserstein means on $2 \times 2$ matrices

TL;DR

This paper proves that in certain von Neumann algebras, only trivial functions preserve specific matrix means, confirming conjectures about the behavior of these preservers in $I_2$-type algebras and providing new characterizations of central elements in $C^*$-algebras.

Contribution

It confirms that $I_2$-type algebras only admit constant preservers for the $p$-power and Wasserstein means, extending previous results and conjectures in operator algebra theory.

Findings

Only constant functions preserve $p$-power mean in $I_2$-type algebras.

Only constant functions preserve Wasserstein mean in $I_2$-type algebras.

Provides two new conditions characterizing centrality in $C^*$-algebras.

Abstract

In one of his recent papers \cite{ML1}, Moln\'ar 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 $0 \neq p \in (-1,1)$ 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\'ar \cite{ML2} 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.