---
tag: 6GD25S53ND
kind: concept
name: Group algebra of a locally compact group
msc: ["22D15", "43A20"]
sources:
  - url: https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_compact_group
same_as:
  wikidata: Q17019511
  wikipedia: Group algebra of a locally compact group
---

## Summary

The group algebra of a [[locally compact group|AHV05ZBCYW]] $G$ is the space $L^1(G)$ of integrable functions on $G$.
Its product is [[convolution|2Z29JN3VES]], and it has a natural involution.
It is a Banach $*$-algebra, and it plays for $G$ the role that the finite [[group algebra|MNP2HBWB5Q]] plays for a finite group.
Unitary representations of $G$ correspond to nondegenerate $*$-representations of $L^1(G)$, so the representation theory of $G$ becomes the representation theory of one algebra.

## Explanations

### L1 human

For a finite group one can add and multiply formal combinations of group elements.
For a group like the real line there are too many elements for finite sums, so one uses functions instead, with integrals in place of sums.
Multiplying two functions in this algebra means convolving them, and the point mass at a group element would play the role of that element.
For the integers, the group algebra consists of absolutely summable sequences, multiplied like the coefficients of power series in $z$ and $z^{-1}$.