The set of completely bounded (CB) maps forms can be considered as a complex span of the set of completely positive (CP) maps. Can we find a basis for this complex linear space of CB maps such that every element of this is a CP map. What I am looking for is an explicit construction method for such a basis (if exists).
In same spirit we may consider the CB Hermiticity preserving maps which can be written as $\psi_1-\psi_2$, where $\psi_i\in$ CP maps and ask the similar question.
Even a partial answer is also helpful for me. (Example: Characterisations of all such maps between $\mathcal{B}(\mathbb{C}^n)\longrightarrow\mathcal{B}(\mathbb{C}^n)$, and similar partial cases). I hope, I explained my question correctly. I am ready to explain any ambiguity (if exists) and give further explanation if required (also re-edit my question to make it lucid and self-explanatory). Advanced thanks for all helps.
EDIT: As pointed by Tom Cooney, the above statement is not true for non-injective von Neumann algebras. Now the question is for injective $C^*$ algebras, does there exists a method by which we can actually construct a basis. For time being, we can consider only CB maps from $\mathcal{B(H)}$ to itself and ask for an explicit example of such basis.
Re-edit: Actually I tried to construct such basis for the maps from $\mathcal{B}(\mathbb{C}^n)$ to itself. However, structure of such maps, when CP, is well understood (Choi-Krauss representation), but does explicitly give a basis explicitly. At the best there is a result of Choi on the extremal points of such CP maps, but I failed to use it for the construction.