Group von Neumann algebra

Remarks

By the von Neumann bicommutant theorem, $L(G)$ is the closure in the strong operator topology of the linear span of the operators $\lambda(g)$, and it is a von Neumann algebra.

Example: $G = \Z$. Fourier series identify $\ell^2(\Z)$ with $L^2(\mathbb{T})$, and $\lambda(n)$ with multiplication by $z^n$. The commutant of these multiplications is the algebra of multiplications by functions in $L^\infty(\mathbb{T})$, which is its own commutant, so $L(\Z) \cong L^\infty(\mathbb{T})$.

Non-example: the reduced group C*-algebra $C^*_r(\Z) \cong C(\mathbb{T})$, acting by multiplication on $L^2(\mathbb{T})$, is not a von Neumann algebra. Its double commutant is $L^\infty(\mathbb{T})$, which is strictly larger than $C(\mathbb{T})$.

For a discrete group $G$, the vector $\delta_e$ gives a faithful normal trace $\tau(a) = \ip{a\delta_e}{\delta_e}$ on $L(G)$. If $G$ is a nontrivial discrete group and every conjugacy class of $G$ other than $\set{e}$ is infinite, then $L(G)$ is a factor of type $\mathrm{II}_1$.