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 1


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 $M_n(A)$.

  • 4
    $\begingroup$ Of course! Stupid question on my part. $\endgroup$
    – voldemort
    Dec 20, 2013 at 7:34

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