DECOMPOSITIONS OF COMPLETELY BOUNDED MAPS INTO COMPLETELY POSITIVE MAPS INVOLVING TRACE CLASS OPERATORS

Let K(H) and B(H) be the sets of all compact operators and all bounded linear operators, respectively, on the Hilbert space H. In this article, we mainly show that if Phi is an element of CB(K(H)*, B(K)), then there exist Phi(i) is an element of CP(K(H)*, B(K)), for i = 1, 2, 3, 4, such that Phi = (Phi(1) - Phi(2)) + root-1(Phi(3)-Phi(4)). However, CP(K(H)*, B(K)) not subset of CB(K(H)*, B(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.