---
tag: 2N7CR03Q1X
kind: concept
name: Group von Neumann algebra
msc: ["22D25", "46L10"]
sources:
  - url: https://en.wikipedia.org/wiki/Von_Neumann_algebra
---

## Summary

The group von Neumann algebra $L(G)$ of a [[locally compact group|AHV05ZBCYW]] $G$ is the [[von Neumann algebra|PSFWQVZ0FS]] generated by the left translation operators on $L^2(G)$.
It is the weak closure of the group algebra acting on $L^2(G)$.
For discrete groups it is a main source of examples of factors.
When $G$ is a nontrivial discrete group in which every conjugacy class other than $\set{e}$ is infinite, $L(G)$ is a factor of type $\mathrm{II}_1$.

## Explanations

### L1 human

A group acts on functions on itself by shifting them, and each shift is a unitary operator on the Hilbert space $L^2(G)$.
Taking all these operators, their linear combinations and their limits in a weak sense gives the group von Neumann algebra.
For the integers, Fourier series turn shifting sequences into multiplying functions on the circle by $z^n$.
So $L(\Z)$ is the algebra of bounded measurable functions on the circle, acting by multiplication.
For noncommutative groups such as free groups, the algebra is much more mysterious, and its study is a large part of operator algebra theory.