{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "YGR48AJMFD"
    },
    {
      "field": "specializes",
      "target": "ZMY9H2XENE"
    }
  ],
  "body": "\n## Summary\n\nA reflexive operator algebra is an algebra of operators on a Hilbert space that is determined by its [[invariant subspaces|0ACX5QT8TE]].\nIt contains every operator that leaves all of them invariant.\nReflexivity links the algebraic structure of operator algebras with the lattice of their invariant subspaces.\nVon Neumann algebras and nest algebras are reflexive, and the question of which algebras are reflexive is close to the [[invariant subspace problem|5RF055GK4N]].\n\n## Explanations\n\n### L1 human\n\nEach set of operators has a family of subspaces that it maps into themselves.\nStarting from these subspaces, one can collect all operators that respect them, and this collection contains the original algebra.\nThe algebra is reflexive when nothing new appears.\nFor example, the upper triangular $2 \\times 2$ matrices are reflexive, while the matrices $\\begin{pmatrix} a \u0026 b \\\\ 0 \u0026 a \\end{pmatrix}$ are not, since they have the same invariant subspaces as all upper triangular matrices.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47L05",
      "47A15"
    ],
    "name": "Reflexive operator algebra",
    "same_as": {
      "wikidata": "Q7307359",
      "wikipedia": "Reflexive operator algebra"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Reflexive_operator_algebra"
      }
    ],
    "tag": "672HNBV803"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/672HNBV803/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/672HNBV803.md",
  "msc": [
    "47L05",
    "47A15"
  ],
  "name": "Reflexive operator algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/672/672HNBV803.md",
  "status": null,
  "tag": "672HNBV803",
  "uri": "m2://672HNBV803",
  "url": "https://math2.tamnd.com/wiki/reflexive-operator-algebra/",
  "writing": {
    "tag": "672HNBV803",
    "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
      }
    ]
  }
}