{
  "id": "2405.10967",
  "version": "v1",
  "published": "2024-05-01T18:55:08.000Z",
  "updated": "2024-05-01T18:55:08.000Z",
  "title": "Toeplitz algebra and Symbol map via Berezin transform on $H^2(\\mathbb{D}^n)$",
  "authors": [
    "Mo Javed",
    "Amit Maji"
  ],
  "comment": "Preliminary version, 23 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "Let $\\mathscr{T}(L^{\\infty}(\\mathbb{T}))$ be the Toeplitz algebra, that is, the $C^*$-algebra generated by the set $\\{T_{\\phi} : \\phi\\in L^{\\infty}(\\mathbb{T})\\}$. Douglas's theorem on symbol map states that there exists a $C^*$-algebra homomorphism from $\\mathscr{T}(L^{\\infty}(\\mathbb{T}))$ onto $L^{\\infty}(\\mathbb{T})$ such that $T_{\\phi}\\mapsto \\phi$ and the kernel of the homomorphism coincides with commutator ideal in $\\mathscr{T}(L^{\\infty}(\\mathbb{T}))$. In this paper, we use the Berezin transform to study results akin to Douglas's theorem for operators on the Hardy space $H^2(\\mathbb{D}^n)$ over the open unit polydisc $\\mathbb{D}^n$ for $n\\geq 1$. We further obtain a class of bigger $C^*$-algebras than the Toeplitz algebra $\\mathscr{T}(L^{\\infty}(\\mathbb{T}^n))$ for which the analog of symbol map still holds true.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-01T18:55:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B35",
      "47A13",
      "32A65"
    ],
    "keywords": [
      "toeplitz algebra",
      "berezin transform",
      "douglass theorem",
      "symbol map states",
      "study results akin"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}