Simplest (completely) positive maps

Question

So, this question may be very simple, but I have an element $a$ in a C*-algebra, we can assume $a\in M_n(\mathbb{C})$ and therefore $$ a = b a b^{-1} $$ has the same spectrum. Therefore, $x\mapsto bxb^{-1}$ maps positive elements to positive elements. But on the other hand the simplest completely positive map is $x\mapsto cxc^*$ and I don't get now why this is a positive map or where is the mistake.

Answer 1

If $x$ is positive, we may decompose it as $x= vv^*$ and then $$cxc^* = cvv^*c^* = cv(cv)^*$$ is positive. Hence, $x \mapsto cxc^*$ is positive.

Answer 2

If $a$ is positive and $b$ is invertible, then $bab^{-1}$ is not necessarily positive. The problem is that for an element to be positive it does not suffice that its spectrum is contained in $[0,\infty)$, it additionally has to be self-adjoint. For example, the matrix $$ \begin{pmatrix}0&1\\0&0\end{pmatrix} $$ has spectrum $\{0\}$, but it is not positive (it's clearly not self-adjoint).