map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$? Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^*)=f(a)^*$ for all $a\in A$, $f(1_A)=1_B$ and $f(a)f(b)=0$ for all $a,b\in A_{sa}$ with $ab=0$. Follows $f(a^2)=f(a)^2$ for all $a\in A$?
I have tried to proof this first for self-adjoint elements $a\in A$ using the continuous functional calculus but I'm stuck. Do you know a proof or a reference? Greetings
 A: Yemon Choi pointed out in a comment to this related question that we can conclude $f$ is a *-homomorphism under the stronger assumption that $ab = 0$ implies $f(a)f(b) = 0$ for arbitrary (not just self-adjoint) $a,b \in A$. This folows from a result stated in the abstract of the paper Alaminos, Brešar, Extremera, and Villena, Maps preserving zero products, Studia Math. 193 (2009), 131-159 by taking $\phi(a,b) = f(a)f(b)$ and setting $b = 1$ in the conclusion.
A: We can reduce to the case where $A$ and $B$ are commutative subalgebras, e.g. taking the closed star-algebra generated by $a$.  Then $A = C(X)$ and $B = C(Y)$ for compact Hausdorff spaces $X$ and $Y$, and we have a continuous linear map $f :   A \to B$.  From this we get a continuous map $f^*$ between the duals $C(Y)^*$ and $C(X)^*$ in the weak* topology.  
We also have natural embeddings $X \to C(X)^*$ and $Y \to C(Y)^*$ coming from point evaluations (i.e. delta-masses, in terms of measures).  You can recognize the measures supported at a single point by the property that, if $\phi(ab) = 0$, then $\phi(a)=0$ or $\phi(b)=0$.  By the condition $f(a)f(b)=0$ whenever $ab=0$, it follows that $f^*$ sends delta masses to multiples of delta masses; and from $f(1)=1$ we get that $f^*$ sends $Y$ continuously to $X$.  
Thus $f: A \to B$ is a morphism of star-algebras, induced by a continuous map from $Y$ to $X$.  In particular $f(a^2) = f(a)^2$.
