{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *factor* is a [[von Neumann algebra|PSFWQVZ0FS]] $M$ whose center $M \\cap M'$ is $\\C I$, where $M'$ is the [[commutant|1VC7EJ1ZBZ]].\nA *subfactor* of a factor $M$ is a von Neumann subalgebra $N \\subseteq M$ that is itself a factor and contains the identity of $M$.\nOne often writes the subfactor as the inclusion $N \\subseteq M$.\n\nIf $M$ is of type $\\mathrm{II}_1$ with faithful normal tracial state $\\tau$, let $L^2(M)$ be the Hilbert space completion of $M$ for $\\langle x, y \\rangle = \\tau(y^* x)$.\nThe *Jones index* of $N \\subseteq M$ is\n$$[M : N] = \\dim_N L^2(M) \\in [1, \\infty],$$\nthe Murray–von Neumann dimension of $L^2(M)$ as a module over $N$ acting by left multiplication.\n\n## Remarks\n\nExample: let $N$ be a $\\mathrm{II}_1$ factor and $k \\ge 1$.\nThen $N \\otimes I_k$, the matrices $\\operatorname{diag}(x, \\dots, x)$ with $x \\in N$, is a subfactor of $M = N \\otimes M_k(\\C)$, the $k \\times k$ matrices over $N$.\nBoth are factors with the same unit, and $L^2(M)$ is the direct sum of $k^2$ copies of $L^2(N)$ as an $N$-module, so $[M:N] = k^2$.\n\nNon-example: the diagonal matrices $D_2 \\subseteq M_2(\\C)$ form a unital von Neumann subalgebra of the factor $M_2(\\C)$.\nBut $D_2$ is commutative and two-dimensional, so its center is $D_2 \\ne \\C I$, and $D_2$ is not a factor.\n\nAnother example: if a finite group $G$ acts on a $\\mathrm{II}_1$ factor $M$ by outer automorphisms, the fixed-point algebra $M^G$ is a subfactor with $[M : M^G] = \\abs{G}$.\nThe index is $1$ exactly when $N = M$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "9RWS7TZHXH",
    "kind": "definition",
    "msc": [
      "46L37"
    ],
    "name": "Subfactor",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subfactor"
      }
    ],
    "tag": "YPPPPHE6TN",
    "uses": [
      "PSFWQVZ0FS",
      "1VC7EJ1ZBZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/YPPPPHE6TN/",
  "links": [
    {
      "field": "defines",
      "target": "9RWS7TZHXH"
    },
    {
      "field": "uses",
      "target": "PSFWQVZ0FS"
    },
    {
      "field": "uses",
      "target": "1VC7EJ1ZBZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YPPPPHE6TN.md",
  "msc": [
    "46L37"
  ],
  "name": "Subfactor",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YPP/YPPPPHE6TN.md",
  "status": null,
  "tag": "YPPPPHE6TN",
  "uri": "m2://YPPPPHE6TN",
  "url": "https://math2.tamnd.com/wiki/subfactor-definition/",
  "writing": {
    "tag": "YPPPPHE6TN",
    "kind": "definition",
    "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
      }
    ]
  }
}