I am trying to get my head around the left regular representation of a group, and I am not sure of the definition of the space $\ell^2(\Gamma)$ if $\Gamma$ is a discrete group.
To quote what I am working with:
Let $\Gamma$ be a discrete group and consider the Hilbert space $\ell^2(\Gamma)$ (whose canonical basis is denoted by $\{\delta_g\}_{g \in \Gamma}$. The right regular representation of $\Gamma$ is given by
$$\begin{array}{cccccccccc}
\rho : & \Gamma & \to & \mathcal{B}(\ell^2(\Gamma)) & & & & & & \\
& s & \mapsto & \rho(s): & \ell^2(\Gamma) & \to & \ell^2(\Gamma) & & & \\
& & & & \xi & \mapsto & \rho(s)(\xi): & \Gamma & \to \mathbb{C}\\
& & & & & & & t & \mapsto & \xi(ts).
\end{array}$$
I always thought that $\ell^2$ was the space of square-summable sequence, so I guess that's why I am confused. Is the correct definition given by $$\ell^2(\Gamma) = \left\{ f \colon \Gamma \to \mathbb{C} \mid \sum_{\gamma \in \Gamma} |f(\gamma)|^2 < +\infty\right\}?$$
And can someone give intuition on what the left representation of $\mathbb{Z}$ with the generators $S = \{\pm 1\}$ is?
Edit: What is the definition of $\ell^2(G)$ where $G$ is a group? this answered the first question, I did not see it before. I would still appreciate some insight on the second question :-).