By the word decomposable I mean a positive map $\phi:\mathcal{B(H)}\rightarrow \mathcal{B(K)}$; $\mathcal{H,K~}$ are some Hilbert spaces and $\phi=\psi_1+T\circ \psi_2$ where $T$ is the transpose operation and $\psi_i$ are completely positive maps. Transpose is not a completely bounded map. But is it possible that for some cases such decomposable maps are completely bounded? Is there a characterisation of such objects?
What if we only consider $\mathcal{H,K~}$ to be finite dimensional? Will it force the composition to be completely bounded? (Actually I am confused about the last one. While calculating the cb norm of transpose, we go for the norm of corresponding Choi matrix which turns out to be unitary and its norm becoms dependent on the dimension of $\mathcal{H}$ which say is $n$. What if we force $\mathcal{H}$ to be of finite dimension.) I think I am confusing myself here. Advanced thanks for any help suggestion etc.