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)

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

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)

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

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 $\alpha(\phi)\in \ker \phi$. 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)

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

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)

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