I'm working through my Operator Algebra notes and I came across the notation $\ell^1(\Gamma)$ which is a Banach space when equipped with $\lVert \cdot \rVert_1$. Here, $\Gamma$ is a discrete group. It is not totally clear in the notes what the elements of the space look like.
It's written in the notes that $f \in \ell^1(\Gamma)$ can be written as $\sum_{g\in\Gamma} f(g) \delta_g$ where $\delta_g(g) =1$ and $0$ for elements of $\Gamma$ that are not $g$, and that this sum converges.
The problem is that the range nor the domain of $f$ is given in the notes. How should I think about such a space? Since $f$ is a function of formal sums of elements of $\Gamma$, is the norm $\lVert \cdot \rVert_1$ the operator norm defined in the usual way as $\textrm{sup} ( \lVert f(g) : \lVert g \rVert = 1, g \in \mathbb{C}[\Gamma] )$ (although this doesn't make sense at face value since we haven't defined a norm on the group ring)?
This is a space of functions defined on $\Gamma$. The codomain can vary, but generally people consider the codomain to be $\mathbb{C}$, in which case $\ell^1(\Gamma)=\ell^1(\Gamma,\mathbb{C})$ is a collection of functions $f: \Gamma\to \mathbb{C}$. The codomain in some situations could be $\mathbb{R}$ or another normed vector space, but I think it is safe to assume it is $\mathbb{C}$ unless otherwise stated.
In the case that $\Gamma =\mathbb{N}$ there is a natural order and so we identify such functions with sequences. e.g. the function $f:\mathbb{N}\to\mathbb{C}$ defined $f(n) = n^2$ is identified with the sequence $$(1,4,9,16,25,\dots)$$
Here, since an arbitrary discrete group may not come equipped with a natural order we don't have the benefit of thinking of the function $f:\Gamma\to\mathbb{C}$ as a sequence. Instead you should think of it as a collection of values, one for each element $g\in\Gamma$. Alternatively, we can express it in a very special basis given by the functions $\delta_g$. These are the indicator functions of a single element
$$\delta_g(x)=\begin{cases}1 & if\ x=g\\ 0 & if\ x\neq g\end{cases}$$
This is a basis for the collection of all functions $f:\Gamma\to \mathbb{C}$ (can you see why?) and so we can always express our function $f$ as $$ f(x) = \sum_{g\in\Gamma}c_g\delta_g(x)$$
for some coefficients $c_g\in\mathbb{C}$. Evaluating at $x=g$ shows us that $$f(g) = c_g$$ In words, the coefficients in this basis are exactly the values of the function at that point $g\in\Gamma$.
I should also say something about the norm. The most common way to define the norm is
$$\| f\|_1 = \sum_{g\in\Gamma} \vert f(g)\vert$$
Which means we sum the complex magnitude of each value of $f$ over all elements of $g$. I am not sure exactly what definition you are trying to use.