Decompositions of completely bounded maps into completely positive maps involving trace class operators

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 $\Phi \in CB(K{\mathcal{(H)}}^{\ast },\mathcal{B}\mathcal{(K)})$, then there exist ${\Phi }_i\in CP(K{\mathcal{(H)}}^{\ast },\mathcal{B(K)})$, for $i=1,2,3,4$, such that $\Phi =({\Phi }_1-{\Phi }_2)+\sqrt{-1}({\Phi }_3-{\Phi }_4)$. However, $CP(K{\mathcal{(H)}}^{\ast },\mathcal{B(K)})⊈CB(K{\mathcal{(H)}}^{\ast },\mathcal{B}\mathcal{(K)})$, where $CB(V,W)$ and $CP(V,W)$ are the sets of all completely bounded maps and all completely positive maps from $V$ into $W$, respectively.