Commutativity from the Equality of two Heron-type Means in $C^*$-Algebras

TL;DR

This paper proves that in unital $C^*$-algebras, the equality of two Heron-type means for positive invertible elements implies the elements commute, solving a previously posed problem without using tracial functionals.

Contribution

It establishes that equality of specific Heron-type and Wasserstein means in $C^*$-algebras implies commutativity, answering an open problem.

Findings

Equality of means implies commutativity of elements

Proof does not require tracial functional

Uses operator-valued triangle equality characterization

Abstract

Let $\mathcal{A}$ be a unital $C^*$-algebra, and let $\mathcal{A}^{++}$ denote the cone of positive invertible elements.We prove that for $A,B\in\mathcal{A}^{++}$, the equality between the conventional Heron-type mean $$ \Big(\frac{A^{1/2}+B^{1/2}}{2}\Big)^2 $$ and the Wasserstein mean $$ \frac14\big(A+B+A(A^{-1}\#B)+(A^{-1}\#B)A\big) $$ forces $A$ and $B$ to commute, thereby answering \cite[Problem~1]{MS24} posed by Moln\'ar and Simon.Our proof does not require any tracial functional; instead it relies on a characterization of the operator-valued triangle equality due to Ando and Hayashi.