{
  "id": "1501.03986",
  "version": "v1",
  "published": "2015-01-16T14:12:55.000Z",
  "updated": "2015-01-16T14:12:55.000Z",
  "title": "Normed algebras of differentiable functions on compact plane sets",
  "authors": [
    "J. F. Feinstein",
    "H. G. Dales"
  ],
  "comment": "32 pages, 9 figures",
  "journal": "Indian Journal of Pure and Applied Mathematics (Platinum Jubilee special issue), 41 (2010), 153-187",
  "categories": [
    "math.FA"
  ],
  "abstract": "We investigate the completeness and completions of the normed algebras $D^{(1)}(X)$ for perfect, compact plane sets $X$. In particular, we construct a radially self-absorbing, compact plane set $X$ such that the normed algebra $D^{(1)}(X)$ is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets $X$ for which the completeness of $D^{(1)}(X)$ is equivalent to the pointwise regularity of $X$. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in $\\mathbb{C}$. In an earlier paper of Bland and Feinstein, the notion of an $\\mathcal{F}$-derivative of a function was introduced, where $\\mathcal{F}$ is a suitable set of rectifiable paths, and with it a new family of Banach algebras $D_{\\mathcal{F}}^{(1)}(X)$ corresponding to the normed algebras $D^{(1)}(X)$. In the present paper, we obtain stronger results concerning the questions when $D^{(1)}(X)$ and $D_{\\mathcal{F}}^{(1)}(X)$ are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever $X$ is '$\\mathcal{F}$-regular'. An example of Bishop shows that the completion of $D^{(1)}(X)$ need not be semisimple. We show that the completion of $D^{(1)}(X)$ is semisimple whenever the union of all the rectifiable Jordan arcs in $X$ is dense in $X$. We prove that the character space of $D^{(1)}(X)$ is equal to $X$ for all perfect, compact plane sets $X$, whether or not $D^{(1)}(X)$ is complete. In particular, characters on the normed algebras $D^{(1)}(X)$ are automatically continuous.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-16T14:12:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46H05",
      "46J10",
      "46E25"
    ],
    "keywords": [
      "compact plane set",
      "normed algebra",
      "differentiable functions",
      "jordan arcs",
      "completion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}