Trace arithmetic--$\hat{I}{º}_p$ inequality

Abstract

Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra equipped with a faithful tracial positive linear functional $\ddot{I}„$. Denote by ${\mathcal{A}}_{+}$ its positive cone. For $p>0$ and $A,B\in {\mathcal{A}}_{+}$, we consider the operations $$A\hat{I}{º}_{p}B:=(A^{p/4}{B}^{p/2}{A}^{p/4}{)}^{1/p},A\nabla B:=\frac{A+B}{2}.$$ We prove that, for all $p>0$ and all $A,B\in {\mathcal{A}}_{+}$, $$\ddot{I}„(A\hat{I}{º}_{p}B)\le \sqrt{\ddot{I}„(A)\ddot{I}„(B)}\le \ddot{I}„(A\nabla B),$$ 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 $2\times 2$ counterexamples showing that the triangle inequality for $d_p$ may fail when $0<p<1$ (already for $p=\frac{1}{2}$), giving a partial answer to \cite[Problem~2]{KM24}.

7 pages. All comments are welcome!