{
  "id": "1307.5365",
  "version": "v1",
  "published": "2013-07-20T01:52:16.000Z",
  "updated": "2013-07-20T01:52:16.000Z",
  "title": "Spatiality of derivations on the algebra of $τ$-compact operators",
  "authors": [
    "Shavkat Ayupov",
    "Karimbergen Kudaybergenov"
  ],
  "comment": "15 pages. arXiv admin note: substantial text overlap with arXiv:1306.0251",
  "journal": "(2013) Integr. Equ. Oper. Theory",
  "doi": "10.1007/s00020-013-2095-8",
  "categories": [
    "math.OA"
  ],
  "abstract": "This paper is devoted to derivations on the algebra $S_0(M, \\tau)$ of all $\\tau$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $\\tau.$ The main result asserts that every $t_\\tau$-continuous derivation $D:S_0(M, \\tau)\\rightarrow S_0(M, \\tau)$ is spatial and implemented by a $\\tau$-measurable operator affiliated with $M$, where $t_\\tau$ denotes the measure topology on $S_0(M, \\tau)$. We also show the automatic $t_\\tau$-continuity of all derivations on $S_0(M, \\tau)$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_\\tau$-continuity of the derivation is redundant for its spatiality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-20T01:52:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47"
    ],
    "keywords": [
      "compact operators",
      "derivation",
      "spatiality",
      "properly infinite von neumann algebras",
      "main result asserts"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5365A"
    }
  }
}