{
  "id": "2405.06453",
  "version": "v1",
  "published": "2024-05-10T13:02:32.000Z",
  "updated": "2024-05-10T13:02:32.000Z",
  "title": "Searching for linear structures in the failure of the Stone-Weierstrass theorem",
  "authors": [
    "Marc Caballer",
    "Sheldon Dantas",
    "Daniel L. Rodríguez-Vidanes"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We analyze the existence of vector spaces of large dimension inside the set $\\mathcal{C}(L, \\K) \\setminus \\overline{\\mathcal{A}}$, where $L$ is a compact Hausdorff space and $\\mathcal{A}$ is a self-adjoint subalgebra of $\\mathcal C(L, \\K)$ that vanishes nowhere on $L$ and does not necessarily separate the points of $L$. The results depend strongly on an equivalence relation that is defined on the algebra $\\mathcal{A}$, denoted by $\\sim_{\\mathcal{A}}$, and a cardinal number that depends on $\\sim_{\\mathcal{A}}$ which we call the order of $\\sim_{\\mathcal{A}}$. We then introduce two different cases, when the order of $\\sim_{\\mathcal{A}}$ is finite or infinite. In the finite case, we show that $\\mathcal{C}(L, \\K) \\setminus \\overline{\\mathcal{A}}$ is $n$-lineable but not $(n+1)$-lineable with $n$ being the order of $\\sim_{\\mathcal{A}}$. On the other hand, when the order of $\\sim_{\\mathcal{A}}$ is infinite, we obtain general results assuming, for instance, that the codimension of the closure of $\\mathcal{A}$ is infinite or when $L$ is sequentially compact. To be more precise, we introduce the notion of the Stone-Weiestrass character of $L$ which is closely related to the topological weight of $L$ and allows us to describe the lineability of $\\mathcal{C}(L, \\K) \\setminus \\overline{\\mathcal{A}}$ in terms of the Stone-Weierstrass character of subsets of $\\sim_{\\mathcal A}$. We also prove, in the classical case, that $(\\mathcal{C}(\\partial{D}, \\C) \\setminus \\overline{\\mbox{Pol}(\\partial{D})}) \\cup \\{0\\}$ (where $\\mbox{Pol}(\\partial{D})$ is the set of all complex polynomials in one variable restricted to the boundary of the unit disk) contains an isometric copy of $\\text{Hol}(\\partial{D})$ and is strongly $\\mathfrak c$-algebrable, extending previous results from the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-10T13:02:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stone-weierstrass theorem",
      "linear structures",
      "large dimension inside",
      "compact hausdorff space",
      "cardinal number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}