{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.\nLet $A$ be a unital $*$-subalgebra of $B(H)$.\nLet $D$ be a densely defined [[unbounded|KRJX5RA0PW]] [[self-adjoint operator|0NEGN5CRSC]] on $H$ with domain $\\operatorname{dom} D$.\nThe triple $(A, H, D)$ is a *spectral triple* if:\n\n1. for every $a \\in A$, the operator $a$ maps $\\operatorname{dom} D$ into itself, and the commutator $[D, a] = Da - aD$ on $\\operatorname{dom} D$ extends to a bounded operator on $H$;\n2. the resolvent $(D - \\lambda I)^{-1}$ is a [[compact operator|M2NEP1K5AE]] for one, and then for every, $\\lambda \\in \\C \\setminus \\R$.\n\n## Remarks\n\nExample: let $A = C^\\infty(S^1)$ act on $H = L^2(S^1)$ by multiplication and let $D = -i \\, d/d\\theta$ on the Sobolev space of functions with square-integrable derivative.\nThe functions $e_n(\\theta) = e^{in\\theta}$, $n \\in \\Z$, form an orthonormal basis of eigenvectors with $D e_n = n e_n$, so $(D - i)^{-1}$ has eigenvalues $1/(n - i) \\to 0$ and is compact.\nFor $f \\in A$ the product rule gives $[D, f] = -i f'$, which is multiplication by a bounded function.\nMore generally, smooth functions on a compact Riemannian spin manifold, acting on square-integrable spinors, with the [[Dirac operator|3DXGJTEQZH]], form a spectral triple.\n\nNon-example: if $H$ is infinite-dimensional and $D = 0$, every commutator is $0$, so clause 1 holds.\nBut $(D - i I)^{-1} = i I$ is not compact, so clause 2 fails.\n\nOften $A$ is a dense $*$-subalgebra of a C*-algebra, and $A$ acts on $H$ through a faithful $*$-representation.\nA spectral triple is called even if there is a self-adjoint unitary $\\gamma$ on $H$ that commutes with $A$ and anticommutes with $D$.\nFor two states $\\varphi$ and $\\psi$ of the norm closure of $A$, Connes' distance formula is\n$$d(\\varphi, \\psi) = \\sup \\set{ \\abs{\\varphi(a) - \\psi(a)} : a \\in A, \\ \\|[D, a]\\| \\le 1 }.$$\nIt recovers the geodesic distance when $\\varphi$ and $\\psi$ are evaluations at two points of a manifold.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "FTQMQR3YS7",
    "kind": "definition",
    "msc": [
      "58B34",
      "46L87"
    ],
    "name": "Spectral triple",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spectral_triple"
      }
    ],
    "tag": "THX29SKKPB",
    "uses": [
      "HM2HHE3J51",
      "BMDG4JQPEH",
      "KRJX5RA0PW",
      "0NEGN5CRSC",
      "M2NEP1K5AE"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/THX29SKKPB/",
  "links": [
    {
      "field": "defines",
      "target": "FTQMQR3YS7"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "BMDG4JQPEH"
    },
    {
      "field": "uses",
      "target": "KRJX5RA0PW"
    },
    {
      "field": "uses",
      "target": "0NEGN5CRSC"
    },
    {
      "field": "uses",
      "target": "M2NEP1K5AE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/THX29SKKPB.md",
  "msc": [
    "58B34",
    "46L87"
  ],
  "name": "Spectral triple",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/THX/THX29SKKPB.md",
  "status": null,
  "tag": "THX29SKKPB",
  "uri": "m2://THX29SKKPB",
  "url": "https://math2.tamnd.com/wiki/spectral-triple-definition/",
  "writing": {
    "tag": "THX29SKKPB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}