{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "NVFF58NEKV"
    }
  ],
  "body": "\n## Summary\n\nThe crossed product $A \\rtimes_\\alpha G$ is a C*-algebra built from a C\\*-algebra $A$ and a group $G$.\nThe group acts on $A$ by automorphisms through a [[group action|0WCY42EV9V]] $\\alpha$.\nWhen $A$ is unital and $G$ is discrete, it contains $A$ and unitaries $u_g$ for $g \\in G$.\nIn it the action becomes conjugation: $u_g a u_g^* = \\alpha_g(a)$.\nIt turns a dynamical system into a single algebra, and many examples, such as the noncommutative tori, arise this way.\nThere is a parallel construction for von Neumann algebras.\n\n## Explanations\n\n### L1 human\n\nSuppose a group moves the points of a space around.\nThe crossed product puts the functions on the space and the group elements into one algebra.\nMoving a function by the group then becomes the algebraic operation $a \\mapsto u_g a u_g^{*}$.\nFor the group $\\Z$ rotating a circle by an irrational angle, the crossed product is the noncommutative torus.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "46L55",
      "46L05",
      "47L65"
    ],
    "name": "Crossed product",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Crossed_product"
      }
    ],
    "tag": "8HG8TS46A6"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/8HG8TS46A6/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/8HG8TS46A6.md",
  "msc": [
    "46L55",
    "46L05",
    "47L65"
  ],
  "name": "Crossed product",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8HG/8HG8TS46A6.md",
  "status": null,
  "tag": "8HG8TS46A6",
  "uri": "m2://8HG8TS46A6",
  "url": "https://math2.tamnd.com/wiki/crossed-product/",
  "writing": {
    "tag": "8HG8TS46A6",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}