‎Let $A,B,$ and $C$ be positive operators in $\\\\\\\\\\\\\\\\mathbb{B}(\\\\\\\\\\\\\\\\mathscr{H})$‎. ‎The characterization of the operator geometric means by $2\\\\\\\\\\\\\\\\times 2$ positive block matrices states that‎, ‎if the block matrix‎ ‎\\\\\\\\\\\\\\\\begin{equation}\\\\\\\\\\\\\\\\label{111}‎ ‎\\\\\\\\\\\\\\\\left( \\\\\\\\\\\\\\\\begin{array}{cc}‎ ‎A & C \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‎ ‎C & B \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‎ ‎\\\\\\\\\\\\\\\\end{array} \\\\\\\\\\\\\\\\right)‎ ‎\\\\\\\\\\\\\\\\end{equation}‎ ‎is positive‎, ‎then $C\\\\\\\\\\\\\\\\leq A\\\\\\\\\\\\\\\\sharp B$‎. ‎For each $T\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathbb{B}(\\\\\\\\\\\\\\\\mathscr{H})$‎, ‎consider‎ ‎$$\\\\\\\\\\\\\\\\mathfrak{U}(T)=\\\\\\\\\\\\\\\\overline{{\\\\\\\\\\\\\\\\rm{conv}}}^{||.||}\\\\\\\\\\\\\\\\left\\\\\\\\\\\\\\\\{U^*TU \\\\\\\\\\\\\\\\ | \\\\\\\\\\\\\\\\ U\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathcal{U}(\\\\\\\\\\\\\\\\mathscr{H})\\\\\\\\\\\\\\\\right\\\\\\\\\\\\\\\\}.$$‎ ‎We show that if $C$ and $D$ are positive invertible operators such that $C\\\\\\\\\\\\\\\\leq D$‎, ‎then there exists a unital completely positive map $\\\\\\\\\\\\\\\\Phi_{C,D}:\\\\\\\\\\\\\\\\mathbb{B}(\\\\\\\\\\\\\\\\mathscr{H})\\\\\\\\\\\\\\\\rightarrow \\\\\\\\\\\\\\\\mathbb{B}(\\\\\\\\\\\\\\\\mathscr{H})$ such that $\\\\\\\\\\\\\\\\Phi_{C,D}(T)\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathfrak{U}(T)$ for each $T\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathbb{B}(\\\\\\\\\\\\\\\\mathscr{H})$ and‎ ‎\\\\\\\\\\\\\\\\begin{equation*}‎ ‎\\\\\\\\\\\\\\\\left(‎ ‎\\\\\\\\\\\\\\\\begin{array}{cc}‎ ‎\\\\\\\\\\\\\\\\Phi_{C,D}(A) & C \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‎ ‎C & \\\\\\\\\\\\\\\\Phi_{C,D}(B) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‎ ‎\\\\\\\\\\\\\\\\end{array}‎ ‎\\\\\\\\\\\\\\\\right)‎ ‎\\\\\\\\\\\\\\\\end{equation*}‎ ‎is positive for any pairs of positive operators $A$ and $B$ such that $D=A\\\\\\\\\\\\\\\\sharp B$‎. ‎In particular‎, ‎if $C\\\\\\\\\\\\\\\\leq A\\\\\\\\\\\\\\\\sharp B$‎, ‎then there exist positive operators $A_0\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathfrak{U}(A)$ and $B_0\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\mathfrak{U}(B)$ such that‎ ‎\\\\\\\\\\\\\\\\begin{equation*}‎ ‎\\\\\\\\\\\\\\\\left( \\\\\\\\\\\\\\\\begin{array}{cc}‎ ‎A_0 & C \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‎ ‎C & B_0 \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\‎ ‎\\\\\\\\\\\\\\\\end{array}‎ ‎\\\\\\\\\\\\\\\\right)\\\\\\\\\\\\\\\\geq 0‎. ‎\\\\\\\\\\\\\\\\end{equation*}‎

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