Let $\phi: G \rightarrow \mathbb{C}$ be a continuous function. We say that $\phi$ is positive type if $\sum_{i,j=1}^{n} c_i\bar{c_j}\phi(g_{j}^{-1}g_i)\geq 0$ for all $n \in N, c_i \in \mathbb{C}, g_i \in G$.
Let G be a topological group and H be an open subgroup of G and $\phi$ be the characteristic function on H. Can you help me prove that $\phi$ is positive type?
Note that $$\phi(g_{j}^{-1}g_i)=1 \Leftrightarrow g_{j}^{-1}g_i \in H$$
On $G$ define $g \equiv h \Leftrightarrow g^{-1}h \in H$.
Now, let $n \in N, c_i \in \mathbb{C}, g_i \in G$.
Split $g_1,.., g_n$ into left cosets. To make this clear, denote by $H_1,.., H_k$ the non-repeated left cosets $g_1H,..., g_nH$ (i.e. $H_i \neq H_j$ if $i \neq j$ and $\{ H_1,.., H_k\}= \{ g_1H,.., g_nH \}$).
Then $$ \sum_{i,j=1}^{n} c_i\bar{c_j}\phi(g_{j}^{-1}g_i)=( \sum_{l=1}^{k}\sum_{g_i,g_j \in H_l} c_i\bar{c_j}\phi(g_{j}^{-1}g_i))+ ( \sum_{1 \leq l \neq m\leq k}^{k}\sum_{g_i \in H_l, g_j \in H_m} c_i\bar{c_j}\phi(g_{j}^{-1}g_i)) \\ =( \sum_{l=1}^{k}\sum_{g_i,g_j \in H_l} c_i\bar{c_j})+ ( \sum_{1 \leq l \neq m\leq k}^{k}\sum_{g_i \in H_l, g_j \in H_m} c_i\bar{c_j} \cdot 0) = \sum_{l=1}^{k}\sum_{g_i,g_j \in H_l} c_i\bar{c_j} = \sum_{l=1}^{k}\left| \sum_{g_i \in H_l} c_i \right|^2 \geq 0 $$
