On the order determining property of the norm of a Kubo-Ando mean in operator algebras (Q2230510)

\textit{F. Kubo} and \textit{T. Ando} means [Math. Ann. 246, 205--224 (1980; Zbl 0412.47013)] are non-commutative extensions of scalar means. It is known that every Kubo-Ando mean $\sigma$ has a monotone property in the sense that, if $A\leq X$ and $B\leq Y$, then $A\sigma B\leq X\sigma Y$. In the present paper, the author investigates the conditions under which the norm of a Kubo-Ando mean $\sigma$ corresponds to the order of positive definite cones of an operator algebra. In particular, the author studies the conditions under which $$A\leq B \Longleftrightarrow \|A\sigma X\|\leq \|B\sigma X \| \text{ for any positive definite }X$$ holds for any pair $A,B$ of positive definite elements of a general $C^*$-algebra as well as the $C^*$-algebra $\mathbb{B}(\mathcal{H})$ of all bounded linear operators on a Hilbert space $\mathcal{H}$.