A question about discrete group

Question

There is a quotation below:

For a discrete group $\Gamma$ we let $\lambda:\Gamma\rightarrow B(l^{2}(\Gamma))$ denote the left regular representation: $\lambda_{s}(\delta_{t})=\delta_{st}$ for all $s, t\in \Gamma$, where $\{\delta_{t}: t\in \Gamma\}\subset l^{2}(\Gamma)$ is the canonical orthonormal basis.

I know seldom about algebraic topology, so I have some questions on this quotation:

1. What is the exact definition of discrete group?

2. What do the $l^{2}(\Gamma)$, $l^{\infty}(\Gamma)$ and $B(l^{2}(\Gamma))$ denote?

3. In the paragraph $\{\delta_{t}: t\in \Gamma\}\subset l^{2}(\Gamma)$ is the canonical orthonormal basis. What is the definition of the canonical orthonormal basis here?

Answer 1

This is more a question about functional analysis than algebraic topology:

1. A discrete group $\Gamma$ is a group equipped with the discrete topology. If $\Gamma$ has no topology (that is, we are in a purely algebraic setting), we can always equip it with the discrete topology, and we say that $\Gamma$ is discrete nonetheless.

2./3. Given a (discrete) set $X$ and a field $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (whichever you prefer), we denote, for $1\leq p<\infty$, $$\ell^p(X)=\left\{\alpha=(\alpha_x)_{x\in X}:\sum_{x\in X}|\alpha_x|^p<\infty\right\}$$ (the sum over elements of $X$ is simply the "limit of finite sums"). This is a $\mathbb{K}$-vector space with norm $\Vert\alpha\Vert_p=\left(\sum_x|\alpha_x|^p\right)^{1/p}$. We define $$\ell^\infty(X)=\left\{\alpha=(\alpha_x)_{x\in X}:\sup_{x\in X}|\alpha_x|<\infty\right\}.$$ This is a $\mathbb{K}$-vector space with norm $\Vert\alpha\Vert_\infty=\sup_{x\in X}|\alpha_x|$.
The definition is the same if $X=\Gamma$ is a group (This is studied in a Measure course). If $p=2$, we obtain a Hilbert space with inner product $\langle\alpha,\beta\rangle=\sum_{x\in X}\alpha_x\overline{\beta_x}$. $B(\ell^2(\Gamma))$ is simply the space of continuous linear operators from $\ell^2(\Gamma)$ to itself equipped with the operator norm.

Given $1\leq p\leq\infty$ and $t\in X$, $\delta_t\in\ell^p(X)$ is defined by $(\delta_t)_x=\begin{cases}1&,x=t\\0&,x\neq t\end{cases}$, and if $p=2$, we obtain an orthonormal basis.

Answer 2

1. A group with the discrete topology (one may say that a discrete group is a group with no topology at all).

2. $\ell_2(\Gamma)$ is the Hilbert space of square-summable functions on $\Gamma$, $\ell_\infty(\Gamma)$ the space of all bounded functions on $\Gamma$ with the supremum norm, $B(\ell_2(\Gamma))$ the algebra of all bounded operators on the Hilbert space $\ell_2(\Gamma)$. Note that both $\ell_\infty(\Gamma)$ and $B(\ell_2(\Gamma))$ have a natural structure of a C*-algebra (these are also von Neumann algebras).

3. The canonical basis consists of functions which are zero everywhere except some point $t$ on which they take value 1.