# Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is

$$\text{Tr}(P(A)) = \text{Tr}(A)$$ $$P\otimes I_{k\times k} \geq0 \quad \forall k$$ $$P(A^*) = P(A)^*$$ $$P^2 = P$$

I'm interested in results that give information about the range of $P$, in particular if $P(M_n(\mathbb{C}))$ is a subalgebra of $M_n(\mathbb{C})$. I did explicit examples and this always seems the case. Any information will be appreciated.

PS: My projection in general is not unital, so i can't use that to conclude $\text{Ran}(P)$ is a subalgebra. Is it possible to do this dropping the unital condition?

• @ChrisHeunen I've already saw that, i'm aware you can abstractly view $\text{Ran}(P)$ as a W* algebra with Choi-Effros product. But i don't think that is relevant for my question, i look results with the same product that inherits from $M_n(\mathbb{C})$. I know that if you change the trace preserving condition by unital, the conclusion i look for is false. But i can't produce counterexamples for the case trace preserving (unital AND trace preserving guarantee the conclusion). Mar 16, 2015 at 16:08
• Take a positive operator $B$ that is not a multiple of a projection with $\textrm{Tr}(B)=1$ and define $P(A)=\textrm{Tr}(A)B.$ Then the range is 1 dimensional but $B$ doesn't generate a 1 dimensional C*-algebra. Mar 16, 2015 at 16:16