# Arens Regularity And Factorization Property

Research paper by **Kazem Haghnejad Azar**

Indexed on: **19 Jul '10**Published on: **19 Jul '10**Published in: **Mathematics - Functional Analysis**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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
$A\hat{\otimes}B$.