{
  "id": "math/0611016",
  "version": "v2",
  "published": "2006-11-01T07:42:56.000Z",
  "updated": "2007-01-21T08:01:19.000Z",
  "title": "Achievement of continuity of $(φ,ψ)$-derivations without continuity",
  "authors": [
    "S. Hejazian",
    "A. R. Janfada",
    "M. Mirzavaziri",
    "M. S. Moslehian"
  ],
  "comment": "To appear in Bull. Belgian Math Soc",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "Suppose that $\\calak$ is a $C^*$-algebra acting on a Hilbert space $\\calhk$, and that $\\phi, \\psi$ are mappings from $\\calak$ into $B(\\calhk)$ which are not assumed to be necessarily linear or continuous. A $(\\phi, \\psi)$-derivation is a linear mapping $d: \\calak \\to B(\\calhk)$ such that $$d(ab)=\\phi(a)d(b)+d(a)\\psi(b)\\quad (a,b\\in \\calak).$$ We prove that if $\\phi$ is a multiplicative (not necessarily linear) $*$-mapping, then every $*$-$(\\phi,\\phi)$-derivation is automatically continuous. Using this fact, we show that every $*$-$(\\phi,\\psi)$-derivation $d$ from $\\calak$ into $B(\\calhk)$ is continuous if and only if the $*$-mappings $\\phi$ and $\\psi$ are left and right $d$-continuous, respectively.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-21T08:01:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L57",
      "46L05",
      "47B47"
    ],
    "keywords": [
      "continuity",
      "derivation",
      "achievement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11016H"
    }
  }
}