Unit in the image of a cp map

Question

This is another question which looks non-trivial to me. Suppose that we have a completely positive map $f\colon M_n \to M_m$ such that $f(a) = I_m$, the identity matrix on $M_m$. Is there a positive element $b\in M_n$ such that $f(b)=I_m$?

Answer 1

The answer is no, in general.

Let $n=m=2$. Define, for $x\in M_2(\mathbb C)$, $$ f(x)=x_{11}\,\begin{bmatrix}2&0\\0&1\end{bmatrix}+x_{22}\,\begin{bmatrix}1&0\\0&0\end{bmatrix}. $$ This map is linear. It is also positive, because if $x\geq0$ then $x_{11},x_{22}\geq0$ and the two matrices are positive. And it is completely positive, because its range lies within an abelian subalgebra of $M_2(\mathbb C)$ (the diagonal algebra).

We have $$ f\left(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\right)=I_2. $$ And no positive matrix is sent to the identity: if $f(x)=I_2$, then necessarily $$ 2x_{11}+x_{22}=1,\ \ x_{11}=1. $$ So the only solution is $x_{11}=1$, $x_{22}=-1$.