{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "9VA95Q3K6G"
    }
  ],
  "body": "\n## Summary\n\nAn approximate identity in a Banach algebra is a [[net|4DVJQJV0DG]] of elements that behaves more and more like a unit.\nMultiplying by it changes each element less and less.\nMany important algebras have no unit, such as the convolution algebra $L^1(\\R)$ and the algebra of continuous functions vanishing at infinity.\nIn these algebras an approximate identity often replaces the unit in proofs.\nEvery C*-algebra has one, and a bounded approximate identity is the hypothesis of the Cohen factorization theorem.\n\n## Explanations\n\n### L1 human\n\nThink of a sequence of bumps that equal $1$ on larger and larger intervals.\nMultiplying a function that dies out at infinity by such a bump changes it only far away, where it is already small.\nSo the bumps act almost like the constant function $1$, which itself is not in the algebra.\nIn $L^1(\\R)$ with convolution, taller and thinner spikes of area $1$ play the same role: convolving with them averages over tiny windows and changes a function very little.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46H05",
      "43A20"
    ],
    "name": "Approximate identity",
    "same_as": {
      "wikidata": "Q621743",
      "wikipedia": "Approximate identity"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Approximate_identity"
      }
    ],
    "tag": "CDYXTW3PJF"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/CDYXTW3PJF/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/CDYXTW3PJF.md",
  "msc": [
    "46H05",
    "43A20"
  ],
  "name": "Approximate identity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CDY/CDYXTW3PJF.md",
  "status": null,
  "tag": "CDYXTW3PJF",
  "uri": "m2://CDYXTW3PJF",
  "url": "https://math2.tamnd.com/wiki/approximate-identity/",
  "writing": {
    "tag": "CDYXTW3PJF",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}