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, projection $P$. 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.

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?