Completely positive map

Remarks

Equivalently, $\varphi \otimes \id_{M_n}$ is positive for every $n$. Every $*$-homomorphism $\pi$ is completely positive, since $\pi_n$ is a $*$-homomorphism and maps $x^* x$ to $\pi_n(x)^* \pi_n(x)$.

Example: for a fixed $m \times k$ complex matrix $V$, the map $\varphi \colon M_m(\C) \to M_k(\C)$, $\varphi(a) = V^* a V$, is completely positive. Indeed $\varphi_n(x) = W^* x W$ with $W = \operatorname{diag}(V, \ldots, V)$, and $W^* y^* y W = (y W)^* (y W) \ge 0$.

Non-example: the transpose $\theta(a) = a^T$ on $M_2(\C)$ is positive, since $a$ and $a^T$ have the same eigenvalues. It is not $2$-positive: the matrix $x = [E_{ij}]_{i,j=1}^2 \in M_2(M_2(\C))$, with $E_{ij}$ the matrix units, equals $v v^*$ with $v = e_1 \otimes e_1 + e_2 \otimes e_2$, so it is positive. But $\theta_2(x) = [E_{ji}]_{i,j}$ is the swap matrix, which has the eigenvalue $-1$.