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?



Positive type function on open subgroup
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
$$
\sum\_{i,j=1}^{n} c\_i\bar{c\_j}\phi(g\_{j}^{-1}g\_i)=( \sum\_{l=1}^{k}\sum\_{g\_i,g\_j \in H\_l} c\_i\bar{c\_j}\phi(g\_{j}^{-1}g\_i))+ ( \sum\_{1 \leq l \neq m\leq k}^{k}\sum\_{g\_i \in H\_l, g\_j \in H\_m} c\_i\bar{c\_j}\phi(g\_{j}^{-1}g\_i)) \\
=( \sum\_{l=1}^{k}\sum\_{g\_i,g\_j \in H\_l} c\_i\bar{c\_j})+ ( \sum\_{1 \leq l \neq m\leq k}^{k}\sum\_{g\_i \in H\_l, g\_j \in H\_m} c\_i\bar{c\_j} \cdot 0)
= \sum\_{l=1}^{k}\sum\_{g\_i,g\_j \in H\_l} c\_i\bar{c\_j} = \sum\_{l=1}^{k}\left| \sum\_{g\_i \in H\_l} c\_i \right|^2 \geq 0 
$$


