{
  "id": "1007.3106",
  "version": "v1",
  "published": "2010-07-19T10:11:39.000Z",
  "updated": "2010-07-19T10:11:39.000Z",
  "title": "Derivations And Cohomological Groups Of Banach Algebras",
  "authors": [
    "Kazem Haghnejad Azar"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $B$ be a Banach $A-bimodule$ and let $n\\geq 0$. We investigate the relationships between some cohomological groups of $A$, that is, if the topological center of the left module action $\\pi_\\ell:A\\times B\\rightarrow B$ of $A^{(2n)}$ on $B^{(2n)}$ is $B^{(2n)}$ and $H^1(A^{(2n+2)},B^{(2n+2)})=0$, then we have $H^1(A,B^{(2n)})=0$, and we find the relationships between cohomological groups such as $H^1(A,B^{(n+2)})$ and $H^1(A,B^{(n)})$, spacial $H^1(A,B^*)$ and $H^1(A,B^{(2n+1)})$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H^1_{w^*}(L^\\infty(G)^*,L^\\infty(G)^{**})=0$. Let $G$ be an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\\infty(G),.)$ such that $Z^1(L^1(G),L^\\infty(G))=\\{L_{f}:~f\\in L^\\infty(G)\\}.$ We also obtain some conclusions in the Arens regularity of module actions and weak amenability of Banach algebras. We introduce some new concepts as $left-weak^*-to-weak$ convergence property [$=Lw^*wc-$property] and $right-weak^*-to-weak$ convergence property [$=Rw^*wc-$property] with respect to $A$ and we show that if $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property and $A^{**}$ is weakly amenable, then $A$ is weakly amenable. We also show to relations between a derivation $D:A\\rightarrow A^*$ and this new concepts.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-19T10:11:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L06",
      "46L07",
      "46L10",
      "47L25",
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "banach algebras",
      "cohomological groups",
      "derivation",
      "convergence property",
      "left module action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3106H"
    }
  }
}