{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $R$ be the [[hyperfinite type II1 factor|NZ4NY8SK1C]] with its faithful normal tracial state $\\tau$, and let $\\omega$ be a free [[ultrafilter|J64Y3JC0RE]] on $\\N$.\nLet $\\ell^\\infty(R)$ be the algebra of norm-bounded sequences in $R$, and let $I_\\omega$ be the ideal of sequences $(x_n)$ with $\\lim_{n \\to \\omega} \\tau(x_n^* x_n) = 0$.\nThe ultrapower $R^\\omega = \\ell^\\infty(R) / I_\\omega$ is a type $\\mathrm{II}_1$ factor with trace $\\tau_\\omega((x_n)) = \\lim_{n \\to \\omega} \\tau(x_n)$.\nThe problem asks: does every type $\\mathrm{II}_1$ factor $M$ with separable predual, with trace $\\tau_M$, admit an injective unital $*$-homomorphism $\\pi \\colon M \\to R^\\omega$ with $\\tau_\\omega \\circ \\pi = \\tau_M$?\n\n## Remarks\n\nConnes raised the question in his 1976 paper on the classification of injective factors.\nOver the following decades it was shown to be equivalent to several problems in other areas.\nAmong them are Kirchberg's QWEP conjecture on C*-algebras and Tsirelson's problem in quantum information theory.\nIn 2020, Ji, Natarajan, Vidick, Wright and Yuen announced a proof of the complexity result $\\mathrm{MIP}^* = \\mathrm{RE}$.\nIt implies a negative answer to Tsirelson's problem and hence to the Connes embedding problem.\nSo there exist $\\mathrm{II}_1$ factors with separable predual that do not embed into $R^\\omega$, but no explicit example is known.\nMany classes of factors are known to embed, for example the group von Neumann algebras of amenable and of free groups.\n\n## Explanations\n\n### L1 human\n\nThe hyperfinite factor $R$ is built from finite matrices, and its ultrapower collects all limits of sequences of such approximations.\nConnes asked whether every reasonable $\\mathrm{II}_1$ factor can be approximated by finite matrices in this way, at the level of traces of products.\nFor many years most experts expected the answer to be yes.\nThe problem was then linked to questions about quantum correlations, and a result in computational complexity theory showed that the answer is no.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "NZ4NY8SK1C"
    ],
    "history": {
      "first_stated": "1976",
      "posed_by": "Alain Connes"
    },
    "kind": "problem",
    "msc": [
      "46L10",
      "46L36"
    ],
    "name": "Connes embedding problem",
    "same_as": {
      "wikidata": "Q5161788",
      "wikipedia": "Connes embedding problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Connes_embedding_problem"
      }
    ],
    "tag": "YYP6QNRERY",
    "uses": [
      "NZ4NY8SK1C",
      "PSFWQVZ0FS",
      "J64Y3JC0RE"
    ]
  },
  "kind": "problem",
  "link": "https://math2.tamnd.com/node/YYP6QNRERY/",
  "links": [
    {
      "field": "about",
      "target": "NZ4NY8SK1C"
    },
    {
      "field": "uses",
      "target": "NZ4NY8SK1C"
    },
    {
      "field": "uses",
      "target": "PSFWQVZ0FS"
    },
    {
      "field": "uses",
      "target": "J64Y3JC0RE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YYP6QNRERY.md",
  "msc": [
    "46L10",
    "46L36"
  ],
  "name": "Connes embedding problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YYP/YYP6QNRERY.md",
  "status": null,
  "tag": "YYP6QNRERY",
  "uri": "m2://YYP6QNRERY",
  "url": "https://math2.tamnd.com/wiki/connes-embedding-problem/",
  "writing": {
    "tag": "YYP6QNRERY",
    "kind": "problem",
    "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
      }
    ]
  }
}