# showing that this specific functional is positive on $l^1(G)$ where $G$ is a discrete topological commutative group

Let $$G$$ be a discrete commutative group, and $$\phi:G \rightarrow \mathbb{T}$$ be a group homomorphism where $$\mathbb{T}$$ is the circle i.e. $$\mathbb{T} := \{z \in \mathbb{Z}:|z| = 1\}$$. Consider $$l^1(G)$$ with involution and convolution, and the define the function $$\varphi(f) := \sum_{x \in G}\phi(x) f(x)$$. I have shown that this is a linear functional on $$l^1(G)$$. I am having trouble showing that this functional is positive i.e. $$\varphi(f^* * f) \geq 0$$ for all $$f \in l^1(G)$$. I know it's not much, but this is what I have written so far $$\varphi(f^* * f) = \sum_{x \in G} \phi(x) (f^* * f)(x) = \sum_{x \in G} \phi(x) \sum_{y \in G} \overline{f(y^{-1})} f(y^{-1}x)$$

We have $$\sum_{x \in G} \phi(x) \sum_{y \in G} \overline{f(y^{-1})} f(y^{-1}x)\\ = \sum_{y \in G}\overline{\phi(y^{-1})} \overline{f(y^{-1})}\sum_{x \in G} \phi(y^{-1}x) f(y^{-1}x)\\ =\sum_{y \in G}\overline{\phi(y^{-1})} \overline{f(y^{-1})} \sum_{x \in G}{\phi(x)} {f(x)} \\ =\left |\sum_{x\in G} \phi(x)f(x)\right |^2$$
Remark Commutativity of $$G$$ is not essential.