If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$.
The reason I am asking this because I want to find a continuous surjective map from $\hat{A}$ to $\hat{C}$.For this part I know that I can take $i:C\rightarrow A$ the inclusion map and take its dual $i^\ast:A^\ast\rightarrow C^\ast$, and restricting it to $\hat{A}$. My only problem is showing the surjection part in $\hat{C}$.
Yes, it follows from the following general theorem (see Dixmier "$C^*$-algebras and representations", 2.10.2):
Every irreducible representation $\rho$ of a $C^*$-subalgebra $C$ of a $C^*$-algebra $A$ can be continued to an irreducible representation $\pi$ of $A$ in a possibly larger Hilbert space.
You can also apply the Lemma 2.10.1 from Dixmier directly.