Commutativity from the Equality of two Heron-type Means in $C^*$-Algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917275513126912 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14123 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Commutativity from the Equality of two Heron-type Means in $C^*$-Algebras Zhang, Teng Functional Analysis Operator Algebras 46L05, 47A63, 47A64 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. |
| title | Commutativity from the Equality of two Heron-type Means in $C^*$-Algebras |
| topic | Functional Analysis Operator Algebras 46L05, 47A63, 47A64 |
| url | https://arxiv.org/abs/2602.14123 |