Use Urysohn's Lemma to make integral non-zero

Question

Let $G$ be a compact abelian group and $H \leq G$ a closed subgroup. Let $\chi : H \rightarrow \mathbb{C}^{*}$ be a character of $H$.

Let $C(G)$ denote the ring of continuous functions on $G$ under convolution.

If $f : G \rightarrow \mathbb{C}$ is defined by $f(g) = \int_{H} \phi(gh) \chi(h)^{-1} \mathrm{d}h$, how can I use Urysohn's Lemma to construct a $\phi \in C(G)$ such that $f \ne 0$?

I think the difficult part is finding an $A,B \subseteq G$ closed and disjoint on which $\phi(A) = 1$ and $\phi(B) = 0$. I was think the $G/H$ cosets might be a candidate but I am not sure.

Any suggestions on how to use Urysohn's Lemma to make the integral non-zero?

Answer 1

I'm assumming your topology is Hausdorff. If it isn't, it's hard to imagine how to extend functions.

If you had $\phi(x)=\chi(g^{-1}x)$, then $f(g)=1$. The problem is that such $\phi$ is only defined on $gH$. Note that as $G$ is compact and Hausdorff, it is normal. Then, as $gH$ is closed, Tietze's Extension Theorem gives you a continuous function $\phi:G\to\mathbb C$ with $\phi|_{gH}=\chi(g^{-1}\cdot)$.