---
tag: JKANN4VBYH
kind: concept
name: Group C*-algebra
msc: ["22D25", "46L05", "22D15"]
sources:
  - url: https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_compact_group
---

## Summary

The group C*-algebra $C^*(G)$ of a [[locally compact group|AHV05ZBCYW]] $G$ is a completion of the [[group algebra|6GD25S53ND]] $L^1(G)$.
It uses the largest norm coming from [[unitary representations|74BPYJNZTM]] of $G$.
It is a C*-algebra whose nondegenerate representations correspond exactly to the unitary representations of $G$.
The reduced group C*-algebra $C^*_r(G)$ uses the left regular representation alone.
These algebras let one apply the theory of operator algebras to groups, and they are central in noncommutative geometry and the Baum–Connes conjecture.

## Explanations

### L1 human

The integrable functions on a group form an algebra, but its norm is not well suited to operator theory.
One fixes this by measuring each function through how large an operator it becomes in every unitary representation, and then filling in the gaps.
For the integers, this turns absolutely summable sequences into all continuous functions on the circle, via Fourier series.
So $C^*(\Z)$ is the algebra $C(\mathbb{T})$ of continuous functions on the circle.