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

Teng Zhang

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 Molnár 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.

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

Abstract

Let be a unital -algebra, and let denote the cone of positive invertible elements.We prove that for , the equality between the conventional Heron-type mean and the Wasserstein mean forces and to commute, thereby answering \cite[Problem~1]{MS24} posed by Molnár 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.
Paper Structure (2 sections, 2 theorems, 18 equations)

This paper contains 2 sections, 2 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. An affirmative answer to Problem \ref{['prob:MS']}

Key Result

Lemma 2.1

Let $H$ be a complex Hilbert space and let $X,Y\in B(H)$. If then there exists a partial isometry $U\in B(H)$ such that

Theorems & Definitions (3)

  • Lemma 2.1: Ando--Hayashi AH07
  • Theorem 2.2
  • proof