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?



Checking complete positivity of maps between C* algebras
Yes, an $n$ positive map is also $n-1$ positive. Hence you map is $n$-positive for all $n$, i.e. completely positive.


For a proof, you include $M\_{n-1}(A)$ as upper left block into $M\_n(A)$.


