{
  "id": "1207.0410",
  "version": "v1",
  "published": "2012-06-28T02:58:00.000Z",
  "updated": "2012-06-28T02:58:00.000Z",
  "title": "Dimension of spaces of polynomials on abelian topological semigroups",
  "authors": [
    "Bolis Basit",
    "A. J. Pryde"
  ],
  "comment": "9 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we study (continuous) polynomials $p: J\\to X$, where $J$ is an abelian topological semigroup and $X$ is a topological vector space. If $J$ is a subsemigroup with non-empty interior of a locally compact abelian group $G$ and $G=J-J$, then every polynomial $p$ on $J$ extends uniquely to a polynomial on $ G$. It is of particular interest to know when the spaces $P^n (J,X)$ of polynomials of order at most $n$ are finite dimensional. For example we show that for some semigroups the subspace $P^n_{R} (J,\\mathbf{C})$ of Riss polynomials (those generated by a finite number of homomorphisms $\\alpha: J\\to \\mathbf{R}$) is properly contained in $P^n (G,\\mathbf{C})$. However, if $P^1 (J,\\mathbf{C})$ is finite dimensional then $P^n_{R} (J,\\mathbf{C})= P^n (J,\\mathbf{C})$. Finally we exhibit a large family of groups for which $P^n (G,\\mathbf{C})$ is finite dimensional.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-28T02:58:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C08",
      "39A99",
      "22B05",
      "05A19"
    ],
    "keywords": [
      "abelian topological semigroup",
      "finite dimensional",
      "locally compact abelian group",
      "non-empty interior",
      "topological vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.0410B"
    }
  }
}