Cone isomorphisms and expressions of some completely positive maps

Abstract

Let $B\mathcal{(H)}$, $K\mathcal{(H)}$ and $T\mathcal{(H)}$ be the set of all bounded linear operators, compact operators, and trace-class operators on the Hilbert space $\mathcal{H}$. The cone of all completely positive maps from $K\mathcal{(H)}$ into $T\mathcal{(K)}$ and all normal completely positive maps from $B\mathcal{(K)}$ into $T\mathcal{(H)}$ is denoted by $\mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$ and $\mathit{NCP}(B\mathcal{(K)},T\mathcal{(H)})$, respectively. In this note, the order structures of the positive cones $\mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$ and $\mathit{NCP}(B\mathcal{(K)},T\mathcal{(H)})$ are investigated. First, we show that $\mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$, $\mathit{NCP}(B\mathcal{(K)},T\mathcal{(H)})$, and $T{\mathcal{(K\otimes H)}}^{+}$ are cone-isomorphic. Then we give the operator sum representation for the map $\Phi \in \mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$.