{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7PY20WMZDZ"
    },
    {
      "field": "uses",
      "target": "A625F93ANV"
    },
    {
      "field": "defines",
      "target": "DHYS1W3Y4N"
    }
  ],
  "body": "\n## Summary\n\nThe group algebra of a [[locally compact group|AHV05ZBCYW]] $G$ is the space $L^1(G)$ of integrable functions on $G$.\nIts product is [[convolution|2Z29JN3VES]], and it has a natural involution.\nIt is a Banach $*$-algebra, and it plays for $G$ the role that the finite [[group algebra|MNP2HBWB5Q]] plays for a finite group.\nUnitary representations of $G$ correspond to nondegenerate $*$-representations of $L^1(G)$, so the representation theory of $G$ becomes the representation theory of one algebra.\n\n## Explanations\n\n### L1 human\n\nFor a finite group one can add and multiply formal combinations of group elements.\nFor 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.\nMultiplying two functions in this algebra means convolving them, and the point mass at a group element would play the role of that element.\nFor the integers, the group algebra consists of absolutely summable sequences, multiplied like the coefficients of power series in $z$ and $z^{-1}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22D15",
      "43A20"
    ],
    "name": "Group algebra of a locally compact group",
    "same_as": {
      "wikidata": "Q17019511",
      "wikipedia": "Group algebra of a locally compact group"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_compact_group"
      }
    ],
    "tag": "6GD25S53ND"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/6GD25S53ND/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/6GD25S53ND.md",
  "msc": [
    "22D15",
    "43A20"
  ],
  "name": "Group algebra of a locally compact group",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6GD/6GD25S53ND.md",
  "status": null,
  "tag": "6GD25S53ND",
  "uri": "m2://6GD25S53ND",
  "url": "https://tamnd.github.io/math2/wiki/group-algebra-of-a-locally-compact-group/",
  "writing": {
    "tag": "6GD25S53ND",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}