Let $A$ be  a  simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
> Is there a continuous map $\alpha:A^{*}-\{0\}\to G(A)$ with $\alpha(\phi)\in \ker \phi$, in the other words: $\phi(\alpha_{\phi})=0$  where $\alpha_{\phi}=\alpha(\phi)$?
>
> If the answer is yes, is there an extension of $\phi$ to whole $A^{*}$?

Note that there is a (possible) non continuous $\phi$ as above.(As  a  consequence of the [Gleason Kahane Zelazko theorem](http://www.encyclopediaofmath.org/index.php/Gleason-Kahane-%C5%BBelazko_theorem)) 


What  about if  we  consider the norm topology for the dual space?