Definition of group von Neumann algebra

Question

Given a discrete group $\Gamma$, in the definition of its group von Neumann algebra $L(\Gamma)$, the following map is considered, $$\lambda: \Gamma \to \mathcal{B}\left(l^2\left(\Gamma\right)\right),$$ where $\lambda(g)\delta_h:=\delta_{gh}$.

This might be a silly question. But what is $l^2(\Gamma)$? Is it the set of functions $l: \Gamma \to \mathbb{C}$ such that $\sum_{g\in\Gamma}\left|l(g)\right|^2<\infty$ (square summable)? But here should we worry about the cardinality of group $\Gamma$? Do we apply the usual definition of uncountable sum?

Also, what is the relationship between $L(\Gamma)$ and group algebra $\mathbb{C}[\Gamma]$? Or are they not related, but $l^2(\Gamma)$ and $\mathbb{C}[\Gamma]$ are related?

Thanks.

Answer 1

$\ell^2(\Gamma):=\{l:\Gamma\rightarrow\mathbb C \mid \sum_{g\in\Gamma}|l(g)|^2<\infty \}$ as you described.

There are a few different ways to define $\sum_{g\in\Gamma} |l(g)|$.

• We may insist $l$ has at most countable support, and then by a basic result in analysis, the sum doesn't depend on the order since $|l(g)|$ are all non-negative.
• We may also define the sum $\sum_{g\in\Gamma} |l(g)|^2$ as the limit of the filter or net of finite subsets of $\Gamma$ with includsion. In other was, we define $c=\sum_{g\in\Gamma} |l(g)|^2$ if for any $\epsilon>0$, there exists a finite subset $S\subset\Gamma$, such that for any finite subset $S\supset S'$, we have $|\sum_{g\in S'} |l(g)|-C|<\epsilon$.
• We can definie $\sum_{g\in\Gamma} |l(g)|^2$ as the Lebesgue integral of $|l(g)|^2$ for the counting measure on $\Gamma$ (the Haar measure). This definition can be used to define $\ell^2(\Gamma)$ when $\Gamma$ is not necessarily discrete but only locally compact Hausdorff.

Frankly, it may not be worth the effort to worry about groups that are uncountable in this scenario. After all, life is too short for non-separable Hilbert spaces, and "nonseparable Hilbert spaces are in some sense artifacts and can almost always be avoided". However, it's common to consider the uncountable groups with a nondiscrete topology, such as Lie groups for the study of group von Neumann algebras, hence the last perspective is really superior in many ways.

$\mathbb C[\Gamma]$ is a subalgebra of $L(\Gamma)$, and in a precise way, $L(\Gamma)$ is a topological closure of $\mathbb C[\Gamma]$ in $\mathcal B(\ell^2(\Gamma))$, see e.g. Wikipedia.

Answer 2

• For any family $(a_i)_{i\in I}$ of non-negative real numbers,$$\sum_{i\in I}a_i=\sup\{\sum_{i\in J}a_i\mid J\text{ finite }\subset I\}.$$
• $\Bbb C[\Gamma]$ is a dense subalgebra of $L[\Gamma]$ and a dense subspace of the Hilbert space $\ell^2(\Gamma).$