Trace arithmetic--$κ_p$ inequality
Teng Zhang
TL;DR
The paper resolves an open question by showing that for any unital $C^*$-algebra with a faithful trace $τ$, the inequality $τ(A κ_p B) ≤ √(τ(A)τ(B)) ≤ τ(A ∇ B)$ holds for all $p>0$ and all $A,B≥0$, by embedding into finite von Neumann algebras and applying noncommutative Hölder. In the matrix setting, it proves a unitarily invariant-norm bound $|||A κ_p B||| ≤ |||A|||^{1/2} |||B|||^{1/2}$ for any unitarily invariant norm, which yields the trace inequality as a corollary. The paper also provides explicit $2×2$ counterexamples showing that the metric property of $d_p$ fails for $0<p<1$, addressing a second problem from KM24. Together, these results extend the reach of κ_p-type operator means in noncommutative spaces and clarify the metric structure of the associated distances, with implications for quantum distance measures such as Bures–Wasserstein distances.
Abstract
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}.