Maps on positive cones in operator algebras preserving power means (Q785647)

Let $\mathscr{A}, \mathscr{B}$ be unital $C^{*}$-algebras. The author treats two types of power means for $A,B\in \mathscr{A}^{++}$, positive definite elements in $\mathscr{A}$. The first power mean is defined by $$A \mathsf{m}_{p}B=\left(\frac{A^{p}+B^{p}}{2}\right)^{\frac{1}{p}}\quad\text{for}p\neq 0, p\in \mathbb{R}.$$ It is called the \textit{conventional $p$th power mean} The second power mean is defined in the sense of Kubo-Ando, i.e., $$ A\mathfrak{m}_{p} B=A^{-\frac{1}{2}}\left(\frac{I+(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{p}}{2} \right)^{\frac{1}{p}}A^{-\frac{1}{2}}\quad\text{for }p\in [-1,1],$$ where $I$ means the unit element in $\mathscr{A}$. In this paper, the author gives characterizations of (continuous) bijective map $\phi: \mathscr{A}^{++}\to \mathscr{B}^{++}$ satisfying the each of following condition: (i) $\phi (A\mathsf{m}_{p}B)=\phi(A)\mathsf{m}_{p}\phi(B)$ for a fixed $p\in \mathbb{R}$, $p\neq 0$ and all $A,B\in \mathscr{A}^{++}$, (ii) $\phi (A\mathfrak{m}_{p}B)=\phi(A)\mathfrak{m}_{p}\phi(B)$ for a fixed $p\in (-1,1)$ and all $A,B\in \mathscr{A}^{++}$. Moreover, the author gives a necessary condition of $\mathscr{A}$ and $\mathscr{B}$ such that there exists a continuous bijective map $\phi: \mathscr{A}^{++} \to\mathscr{B}^{++}$ satisfying $$ \phi (A\mathsf{m}_{p}B)=\phi(A) \mathfrak{m}_{p}\phi(B) $$ for a fixed $p\in (-1,1)$ and all $\mathscr{A}^{++}$. The author gives concrete characterizations of $\phi$ in the $B(H)$ setting, i.e., bounded linear operators on a complex Hilbert space $H$.