Group von Neumann algebra

Let $G$ be a locally compact group, choose a left Haar measure, and let $\lambda:G\to\mathcal U(L^2(G))$ be the left regular representation, $(\lambda_s\xi)(t)=\xi(s^{-1}t)$ . The group von Neumann algebra of $G$ is

\operatorname{VN}(G)=\{\lambda_s:s\in G\}''\subseteq\mathcal B(L^2(G)).

Equivalently, it is the weak-operator closure of the unital $*$ -algebra generated by the operators $\lambda_s$ , by the von Neumann bicommutant theorem. The notation $L(G)$ is especially common for discrete groups.

Basic structure

The commutant of $\operatorname{VN}(G)$ is generated by the appropriately normalized right regular representation. The algebra therefore records the regular representation rather than merely the abstract multiplication law. For general locally compact $G$ , its canonical noncommutative integral is the Plancherel weight; this need not be a finite trace.

Discrete and abelian cases

If $G$ is discrete, the vector $\delta_e$ defines the canonical faithful normal trace

\tau(x)=\langle x\delta_e,\delta_e\rangle.

In this case $L(G)$ is a factor exactly when every nonidentity conjugacy class of $G$ is infinite. If $G$ is abelian, Fourier transformation identifies $\operatorname{VN}(G)$ with the multiplication algebra $L^\infty(\widehat G)$ ; thus the group von Neumann algebra is commutative.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VII, §3 on the Plancherel weight and the von Neumann algebra of a locally compact group.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §7 on discrete group factors.