{
  "id": "2403.16947",
  "version": "v2",
  "published": "2024-03-25T17:10:33.000Z",
  "updated": "2024-05-15T16:54:40.000Z",
  "title": "$M$-ideals in $H^\\infty(\\mathbb{D})$",
  "authors": [
    "Deepak K. D",
    "Jaydeb Sarkar",
    "Sreejith Siju"
  ],
  "comment": "Corrected and thoroughly revised. 34 pages",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "This article intends to initiate an investigation into the structure of $M$-ideals in $H^\\infty(\\mathbb{D})$, where $H^\\infty(\\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\\mathbb{D}$ in $\\mathbb{C}$. We introduce the notion of analytic primes and prove that $M$-ideals in $H^\\infty(\\mathbb{D})$ are analytic primes. From Hilbert function space perspective, we additionally prove that $M$-ideals in $H^\\infty(\\mathbb{D})$ are dense in the Hardy space. We show that outer functions play a key role in representing singly generated closed ideals in $H^\\infty(\\mathbb{D})$ that are $M$-ideals. This is also relevant to $M$-ideals in $H^\\infty(\\mathbb{D})$ that are finitely generated closed ideals in $H^\\infty(\\mathbb{D})$. We analyze $p$-sets of $H^\\infty(\\mathbb{D})$ and their connection to the \\v{S}ilov boundary of the maximal ideal space of $H^\\infty(\\mathbb{D})$. Some of our results apply to the polydisc. In addition to addressing questions regarding $M$-ideals, the results presented in this paper offer some new perspectives on bounded analytic functions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-05-15T16:54:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46H10",
      "30H10",
      "32A35",
      "47L20",
      "46J15",
      "58C10"
    ],
    "keywords": [
      "bounded analytic functions",
      "singly generated closed ideals",
      "analytic primes",
      "outer functions play",
      "open unit disc"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}