What are the Pontryagin duals of additive and multiplicative group of complex number?

Question

What are the Pontryagin duals of additive and multiplicative group of complex number?

So basically what are all characters of $(\mathbb{C},+$) and $(\mathbb{C^*},.)$?

Answer 1

For Pontryagin duality, the characters are the continuous homomorphisms to $S^1$, not to all of $\mathbb{C}^\ast$, as characters are defined in some other circumstances.

We have $\widehat{G\times H} = \widehat{G}\oplus \widehat{H}$, and it is probably known that $\widehat{\mathbb{R}} \cong \mathbb{R}$ and $\widehat{S^1} \cong \mathbb{Z}$.

Writing $(\mathbb{C},+)$ and $(\mathbb{C}^\ast,\cdot)$ as products of simpler groups then gives the answer.