Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map $\Psi$, on this space, is a finite sum of such maps (Krauss form). Let $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow \mathcal{B}(\mathbb{C}^n)$ is a positive map (need not be cp). Let $C_\phi=\sum_{i,j}e_{i,j}\otimes\phi(e_{i,j})$, is the Choi matrix (using Choi–Jamiołkowski isomorphism). Using these notations, I am asking a few questions.
For any arbitrary $\phi$ and an arbitrary $A$, can we make the following statement. \begin{equation} C_{\phi\circ\chi_A}=\sum_i(A_i\otimes B_i)C_\phi(A_i\otimes B_i)^*, \end{equation} for some $\lbrace A_i\rbrace$ and $\lbrace B_i\rbrace$.
If $\phi_n:M_n(\mathcal{B}(\mathbb{C}^n))\rightarrow M_n(\mathcal{B}(\mathbb{C}^n))$ be the canonical extension, then can we conclude that, $(\phi\circ\chi_A)_n=\sum_i\chi_{A_i\otimes B_i}\circ\phi_n$.
I believe that the above statements are true (at least the first one), but I could not prove it (probably by making some stupid mistakes). Advanced thanks for all helps and suggestions.