Example 41.3.11.label Let $T: M_{2}(\complex) \to M_{2}(\complex)$ be the transpose map, then $T$ is positive, but not $2$-positive.
Proof. Let
\[C = \begin{bmatrix}1&0&0&1 \\ 0&0&0&0 \\ 0&0&0&0 \\ 1&0&0&1\end{bmatrix} = (1, 0, 0, 1) \otimes (1, 0, 0, 1) \ge 0\]
then
\[T_{2}(C) = \begin{bmatrix}1&0&0&0 \\ 0&0&1&0 \\ 0&1&0&0 \\ 0&0&0&1\end{bmatrix}\]
is invertible, with negative determinant, so $T_{2}(C)$ is not positive.$\square$
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