I am trying to determine all the characters on the group $\mathbb C^\times$ of non-zero complex numbers under multiplication. I believe that all the characters are given by: $$\chi : z=re^{ix} \to {r\over r}e^{ikx}=e^{ikx} \quad for \ k \in \mathbb R$$ But I'm not sure if this is correct or if this is all of the characters for $\mathbb C^\times$.
Note: The definition of character I am using is: A character of a locally compact abelian group A is a continuous group homomorphism: $$\chi :A \to \mathbb T$$ where $\mathbb T$ is the circle group i.e. the multiplicative group of all complex numbers of absolute value 1.