Let $A$ be a C\*-algebra. Suppose that $P:A \rightarrow A$ is a contractive completely positive projection. Does the range $P(A)$ is completely order isomorphic to a $C^\*$-algebra? 


In the case where the answer is false:
Is it true in supposing, in addition, that the range $P(A)$ is an operator system, $A$ unital and $P(1)=1$?



Range of completely positive projection
Yes. $P(A)$ is abstractly a C$^\ast$-algebra $B$ under the Choi-Effros product: $a\circ b := P(ab)$; and $P(A)$ is completely order isomorphic to $B$. See (the proof of) Theorem 3.1 in [M.-D. Choi and E. G. Effros; Injectivity and operator spaces. J. Functional Analysis 24 (1977), 156-–209]. Moreover, there is a surjective $\ast$-homomorphism from $C^\ast(P(A))$ onto the C$^\ast$-algebra $B$.


