{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $n \\ge 2$ be an integer.\nThe *Cuntz algebra* $\\mathcal{O}_n$ is the universal C*-algebra generated by elements $s_1, \\ldots, s_n$ subject to the relations\n$$s_i^* s_i = 1 \\quad (1 \\le i \\le n), \\qquad \\sum_{i=1}^{n} s_i s_i^* = 1.$$\nSo the $s_i$ are isometries, and the projections $s_i s_i^*$ onto their ranges add up to $1$.\nThe relations imply $s_i^* s_j = 0$ for $i \\ne j$: the ranges are mutually orthogonal.\n\nConcrete check for $n = 2$: on the Hilbert space $\\ell^2(\\N)$ with orthonormal basis $(e_k)_{k \\ge 0}$, let $s_1 e_k = e_{2k}$ and $s_2 e_k = e_{2k+1}$.\nEach $s_i$ maps an orthonormal basis to an orthonormal family, so it is an isometry.\nThe range of $s_1$ is spanned by the even basis vectors and that of $s_2$ by the odd ones, so $s_1 s_1^* + s_2 s_2^* = 1$.\nThe C\\*-algebra generated by $s_1$ and $s_2$ is a copy of $\\mathcal{O}_2$.\n\n## Remarks\n\nCuntz showed that $\\mathcal{O}_n$ is simple, so any $n$ isometries with these relations generate an isomorphic C\\*-algebra.\nIt is also infinite: $s_1$ is an isometry with $s_1 s_1^* \\ne 1$.\nIts K-theory is $K_0(\\mathcal{O}_n) \\cong \\Z / (n-1)\\Z$ and $K_1(\\mathcal{O}_n) = 0$, so the algebras $\\mathcal{O}_n$ for different $n$ are not isomorphic.\nEach $\\mathcal{O}_n$ is separable and [[nuclear|D2M4VXJ7PA]].\n\n## Explanations\n\n### L1 human\n\nIn an infinite-dimensional space there is room to fit two full copies of the whole space side by side, as the even and odd positions of a sequence.\nThe Cuntz algebra is the algebra generated by such copying maps.\nIt is a basic example of a simple C\\*-algebra that is very far from the matrix algebras.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "example",
    "msc": [
      "46L05",
      "46L80"
    ],
    "name": "Cuntz algebra",
    "same_as": {
      "wikidata": "Q1144125",
      "wikipedia": "Cuntz algebra"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cuntz_algebra"
      }
    ],
    "tag": "36Q7XH9P7W"
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/36Q7XH9P7W/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/36Q7XH9P7W.md",
  "msc": [
    "46L05",
    "46L80"
  ],
  "name": "Cuntz algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/36Q/36Q7XH9P7W.md",
  "status": null,
  "tag": "36Q7XH9P7W",
  "uri": "m2://36Q7XH9P7W",
  "url": "https://math2.tamnd.com/wiki/cuntz-algebra/",
  "writing": {
    "tag": "36Q7XH9P7W",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or not_instance_of or illustrates out",
        "have": 0,
        "want": 1
      }
    ]
  }
}