POSITIVE BLOCK MATRICES

Abstract

‎Let $C$ and $D$ be positive operators such that $C\\leq D$ and $D$ be invertible‎. ‎We show that there exists a trace preserving unital completely positive map $\\Phi_{C,D}:\\mathbb{B}(\\mathcal{H})\\rightarrow \\mathbb{B}(\\mathcal{H})$ such that ‎the ‎block ‎operator ‎matrices<br><br>‎\\begin{equation*}‎<br><br>‎\\left(‎<br><br>‎\\begin{array}{cc}‎<br><br> ‎\\Phi_{C,D}(A) & C \\\\‎<br><br> ‎C & \\Phi_{C,D}(B) \\\\‎<br><br> ‎\\end{array}‎<br><br>‎\\right)‎<br><br>‎\\end{equation*}‎<br><br>are positive‎, ‎for all positive operators $A$ and $B$ such that $D=A\\sharp B$‎.