Yuan Li, Mengqian Cui, Jiao Wu. Source: Annals of Functional Analysis, Volume 10, Number 1, 16--28.Abstract:
Let $K\mathcal{(H)}$ and ${\mathcal{B(H)}}$ be the sets of all compact operators and all bounded linear operators, respectively, on the Hilbert space $\mathcal{H}$ . In this article, we mainly show that if $\Phi\in\operatorname{CB}(K{\mathcal{(H)}}^{*},\mathcal{B}{\mathcal{(K)}})$ , then there exist $\Phi_{i}\in\operatorname{CP}(K{\mathcal{(H)}}^{*},{\mathcal{B(K)}})$ , for $i=1,2,3,4$ , such that $\Phi=(\Phi_{1}-\Phi_{2})+\sqrt{-1}(\Phi_{3}-\Phi_{4})$ . However, $\operatorname{CP}(K{\mathcal{(H)}}^{*},{\mathcal{B(K)}})\nsubseteq\operatorname{CB}(K{\mathcal{(H)}}^{*},\mathcal{B}{\mathcal{(K)}})$ , where $\operatorname{CB}(V,W)$ and $\operatorname{CP}(V,W)$ are the sets of all completely bounded maps and all completely positive maps from $V$ into $W$ , respectively.