Let $\phi$ : $A \rightarrow A$ be a positive map, where $A$ is a (unital) C* algebra. Suppose we are given that $\phi$ is n positive whenever n= $2^k$ for some $k \in \mathbb{N}$. Can we conclude that $\phi$ is completely postive?
1 Answer
Yes, an $n$ positive map is also $n1$ positive. Hence you map is $n$positive for all $n$, i.e. completely positive.
For a proof, you include $M_{n1}(A)$ as upper left block into $M_n(A)$.

4