{
  "id": "1007.3110",
  "version": "v1",
  "published": "2010-07-19T10:17:18.000Z",
  "updated": "2010-07-19T10:17:18.000Z",
  "title": "Arens Regularity And Factorization Property",
  "authors": [
    "Kazem Haghnejad Azar"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we will 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. In this paper, we will extend some problems from topological center of second dual of Banach algebra $A$, $Z_1(A^{**})$, into spaces ${Z}^\\ell_{B^{**}}(A^{**})$ and ${Z}^\\ell_{A^{**}}(B^{**})$. We investigate some relationships between ${Z}_1({A^{**}})$ and topological centers of module actions. For an unital Banach $A-module$ $B$ we show that ${Z}^\\ell_{A^{**}}(B^{**}){Z}_1({A^{**}})={Z}^\\ell_{A^{**}}(B^{**})$ and as results in group algebras, for locally compact group $G$, we have ${Z}^\\ell_{{L^1(G)}^{**}}(M(G)^{**})M(G)={Z}^\\ell_{{L^1(G)}^{**}}(M(G)^{**})$ and ${Z}^\\ell_{M(G)^{**}}({L^1(G)}^{**})M(G)={Z}^\\ell_{M(G)^{**}}({L^1(G)}^{**})$. For Banach $A-bimodule$ $B$, if we assume that $B^*B^{**}\\subseteq A^*$, then $~B^{**}{Z}_1(A^{**})\\subseteq {Z}^\\ell_{A^{**}}(B^{**})$ and moreover if $B$ is an unital as Banach $A-module$, then we conclude that $B^{**}{Z}_1({A^{**}})={Z}^\\ell_{A^{**}}(B^{**})$. Let ${Z}^\\ell_{A^{**}}(B^{**})A\\subseteq B$ and suppose that $B$ is $WSC$, so we conclude that ${Z}^\\ell_{A^{**}}(B^{**})=B$. If $\\overline{B^{*}A}\\neq B^*$ and $ B^{**}$ has a left unit $A^{**}-module$, then $Z^\\ell_{B^{**}}(A^{**})\\neq A^{**}$. We will also establish some relationships of Arens regularity of Banach algebras $A$, $B$ and Arens regularity of projective tensor product $A\\hat{\\otimes}B$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-19T10:17:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "F.2.2",
      "I.2.7"
    ],
    "keywords": [
      "factorization property",
      "topological center",
      "banach algebra",
      "left module action",
      "right module action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3110H"
    }
  }
}