Trace arithmetic--$\kappa_p$ inequality

Abstract

Let $\mathcal{A}$ be a unital $C^\ast$-algebra equipped with a faithful tracial positive linear functional $\tau$. Denote by $\mathcal{A}_+$ its positive cone. For $p>0$ and $A,B\in\mathcal{A}_+$, we consider the operations $$ A\kappa_p B := \bigl(A^{p/4} B^{p/2} A^{p/4}\bigr)^{1/p}, \qquad A\nabla B := \frac{A+B}{2}. $$ We prove that, for all $p>0$ and all $A,B\in\mathcal{A}_+$, $$ \tau(A\kappa_p B)\le \sqrt{\tau(A)\tau(B)}\le \tau(A\nabla B), $$ thereby answering \cite[Problem~1]{KM24}, posed by \'A.~Kom\'alovics and L.~Moln\'ar, 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}.