I am trying to find sufficient conditions for an unital $C^*$-algebra homomorphism $\alpha:A \rightarrow B$ to be surjective on positive elements, that is, $\alpha(A_+)=\alpha(B_+)$. For example, if the range of $\alpha$ is a hereditary subalgebra of $B$, then one has that all elements $b \in B_+$ such that $b \leq 1$ are in $\alpha(A_+)$, however I don't know what to do with the rest of the elements.
For a particular case, let $\alpha: C(S^1) \rightarrow C(S^1)$ be the endomorphism $f(z) \mapsto f(z^2)$, I was trying to prove it is surjective on positive elements with no success.
Thanks in advance!
If $\alpha(A_+)=B_+$, then the range of $\alpha$ is contained in the subspace spanned by $B_+$ since $\alpha$ is a linear map. As every element in a $C^\ast$-algebra is a linear combination of four positive elements, the range of $\alpha$ must be $B$.
Conversely, assume that $\alpha$ is surjective. Every positive element of $B$ is of the form $b^\ast b$ with $b\in B$. By surjectivity, there exists $a\in A$ such that $\alpha(a)=b$. Hence $\alpha(a^\ast a)=b^\ast b$. Therefore $\alpha(A_+)=B_+$.