What is the continuous dual group of the multiplicative complex numbers?

Question

I was wondering how to compute the dual group of $\mathbb{C}_*=\mathbb{C}\backslash\{0\}$, where the dual group consists of continuous characters on $\mathbb{C}_*$ i.e. continuous homomorphisms from $\mathbb{C}_*$ into $\mathbb{T}$. I am having some difficulty figuring out what these might look like. Thanks!

Answer 1

$\Bbb C^*\cong\Bbb R^*_{>0}\times\Bbb T\cong\Bbb R_+\times\Bbb T$. The continuous characters of $\Bbb R_{>0}^*$ are $x\mapsto\exp(it\ln x)$ for $t\in\Bbb R$ and those of $\Bbb T$ are $z\mapsto z^n$ for $x\in \Bbb N$. So the general character of $\Bbb C^*$ is $$z\mapsto\exp(it\ln|z|)\left(\frac{z}{|z|}\right)^n.$$