{
  "id": "1911.09440",
  "version": "v1",
  "published": "2019-11-21T12:36:51.000Z",
  "updated": "2019-11-21T12:36:51.000Z",
  "title": "Classification of integro-differential $C^*$-algebras",
  "authors": [
    "Anton A. Kutsenko"
  ],
  "categories": [
    "math.OA",
    "math.FA"
  ],
  "abstract": "The integro-differential algebra $\\mathscr{F}_{N,M}$ is the $C^*$-algebra generated by the following operators acting on $L^2([0,1)^N\\to\\mathbb{C}^M)$: 1) operators of multiplication by bounded matrix-valued functions, 2) finite differential operators, 3) integral operators. We give a complete characterization of $\\mathscr{F}_{N,M}$ in terms of its Bratteli diagram. In particular, we show that $\\mathscr{F}_{N,M}$ does not depend on $M$ but depends on $N$. At the same time, it is known that differential algebras $\\mathscr{H}_{N,M}$, generated by the operators 1) and 2), do not depend on both dimensions $N$ and $M$, they are all $*$-isomorphic to the universal UHF-algebra. We explicitly compute the Glimm-Bratteli symbols (for $\\mathscr{H}_{N,M}$ it was already computed earlier) $$ \\mathfrak{n}(\\mathscr{F}_{N,M})=\\prod_{n=1}^{\\infty}\\begin{pmatrix} n & 0 n-1 & 1 \\end{pmatrix}^{\\otimes N}\\begin{pmatrix}1 1 \\end{pmatrix}^{\\otimes N},\\ \\ \\ \\ \\mathfrak{n}(\\mathscr{H}_{N,M})=\\prod_{n=1}^{\\infty}n, $$ which characterize completely the corresponding AF-algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-21T12:36:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classification",
      "finite differential operators",
      "glimm-bratteli symbols",
      "integro-differential algebra",
      "integral operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}