{
  "id": "1008.2655",
  "version": "v1",
  "published": "2010-08-16T13:37:31.000Z",
  "updated": "2010-08-16T13:37:31.000Z",
  "title": "Topological centers of module actions and cohomological groups of Banach Algebras",
  "authors": [
    "Kazem Azem Haghnejad Azar"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\\ell_{B^{**}}(A^{**})$ and ${Z}^\\ell_{A^{**}}(B^{**})$ be the topological centers of the left module action $\\pi_\\ell:~A\\times B\\rightarrow B$ and the right module action $\\pi_r:~B\\times A\\rightarrow B$, respectively. We investigate some relationships between topological center of $A^{**}$, ${Z}_1({A^{**}})$ with respect to the first Arens product and topological centers of module actions ${Z}^\\ell_{B^{**}}(A^{**})$ and ${Z}^\\ell_{A^{**}}(B^{**})$. On the other hand, if $A$ has Mazure property and $B^{**}$ has the left $A^{**}-factorization$, then $Z^\\ell_{A^{**}}(B^{**})=B$, and so for a locally compact non-compact group $G$ with compact covering number $card(G)$, we have $Z^\\ell_{M(G)^{**}}{(L^1(G)^{**})}= {L^1(G)}$ and $Z^\\ell_{L^1(G)^{**}}{(M(G)^{**})}= {M(G)}$. By using the Arens regularity of module actions, we study some cohomological groups properties of Banach algebra and we extend some propositions from Dales, Ghahramani, Gr{\\o}nb{\\ae}k and others into general situations and we investigate the relationships between some cohomological groups of Banach algebra $A$. We obtain some results in Connes-amenability of Banach algebras, and so for every compact group $G$, we conclude that $H^1_{w^*}(L^\\infty(G)^*,L^\\infty(G)^{**})=0$. Suppose that $G$ is an amenable locally compact group. Then there is a Banach $L^1(G)-bimodule$ such as $(L^\\infty(G),.)$ such that $Z^1(L^1(G),L^\\infty(G))=\\{L_{f}:~f\\in L^\\infty(G)\\}$ where for every $g\\in L^1(G)$, we have $L_f(g)=f.g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-08-16T13:37:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L06",
      "46L07",
      "46L10",
      "47L25",
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "banach algebra",
      "topological center",
      "cohomological groups",
      "left module action",
      "arens regularity properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.2655A"
    }
  }
}