Trace arithmetic--$κ_p$ inequality
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908829770317824 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | Let $\mathcal{A}$ be a unital $C^\ast$-algebra equipped with a faithful tracial positive linear functional $τ$. Denote by $\mathcal{A}_+$ its positive cone. For $p>0$ and $A,B\in\mathcal{A}_+$, we consider the operations $$ Aκ_p B := \bigl(A^{p/4} B^{p/2} A^{p/4}\bigr)^{1/p}, \qquad A\nabla B := \frac{A+B}{2}. $$ We prove that, for all $p>0$ and all $A,B\in\mathcal{A}_+$, $$ τ(Aκ_p B)\le \sqrt{τ(A)τ(B)}\le τ(A\nabla B), $$ thereby answering \cite[Problem~1]{KM24}, posed by Á.~Komálovics and L.~Molnár, in the affirmative. We also record a unitarily invariant norm analogue of the key estimate in the matrix case, and we provide explicit $2\times2$ counterexamples showing that the triangle inequality for $d_p$ may fail when $0<p<1$ (already for $p=\tfrac12$), giving a partial answer to \cite[Problem~2]{KM24}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11922 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Trace arithmetic--$κ_p$ inequality Zhang, Teng Operator Algebras Functional Analysis 15A60, 46L52, 47A63 Let $\mathcal{A}$ be a unital $C^\ast$-algebra equipped with a faithful tracial positive linear functional $τ$. Denote by $\mathcal{A}_+$ its positive cone. For $p>0$ and $A,B\in\mathcal{A}_+$, we consider the operations $$ Aκ_p B := \bigl(A^{p/4} B^{p/2} A^{p/4}\bigr)^{1/p}, \qquad A\nabla B := \frac{A+B}{2}. $$ We prove that, for all $p>0$ and all $A,B\in\mathcal{A}_+$, $$ τ(Aκ_p B)\le \sqrt{τ(A)τ(B)}\le τ(A\nabla B), $$ thereby answering \cite[Problem~1]{KM24}, posed by Á.~Komálovics and L.~Molnár, in the affirmative. We also record a unitarily invariant norm analogue of the key estimate in the matrix case, and we provide explicit $2\times2$ counterexamples showing that the triangle inequality for $d_p$ may fail when $0<p<1$ (already for $p=\tfrac12$), giving a partial answer to \cite[Problem~2]{KM24}. |
| title | Trace arithmetic--$κ_p$ inequality |
| topic | Operator Algebras Functional Analysis 15A60, 46L52, 47A63 |
| url | https://arxiv.org/abs/2602.11922 |