A quick question, I am a bit confused about what this means here.
Let $G$ be a discrete commutative group i.e. a discrete topological commutative group, and $\phi \in \hat{G}$. Define $\varphi$ on $l^1(G)$ by $$\varphi(f) := \sum_{x \in G} \phi(x)f(x)$$The part that I am confused is $\phi \in \hat{G}$. Right now I am learning about Banach algebras and $C^*$-algebras. My professor writes $\hat{A}$ to refer to the set of characters of an unital Banach algebra. My algebra isn't that strong, but I was wondering if this is a group algebra(which I am not too familar with only heard the name) or is it some other algebraic term.If it's a group algebra is this a unital Banach algebra? Also I would like to mention the set of characters is a function from my algebra onto $\mathbb{C}$ that is a unital algebra homomorphism.