Arens Regularity And Factorization Property

In this paper, we will study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let $Z_{B^{\ast \ast }}^{\mathrm{\ell }}(A^{\ast \ast })$ and $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$ be the topological centers of the left module action ${\pi }_{\mathrm{\ell }}:A\times B\to B$ and the right module action ${\pi }_r:B\times A\to B$, respectively. In this paper, we will extend some problems from topological center of second dual of Banach algebra $A$, $Z_1(A^{\ast \ast })$, into spaces $Z_{B^{\ast \ast }}^{\mathrm{\ell }}(A^{\ast \ast })$ and $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$. We investigate some relationships between $Z_1(A^{\ast \ast })$ and topological centers of module actions. For an unital Banach $A-module$ $B$ we show that $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })Z_1(A^{\ast \ast })=Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$ and as results in group algebras, for locally compact group $G$, we have $Z_{{L^1(G)}^{\ast \ast }}^{\mathrm{\ell }}(M(G{)}^{\ast \ast })M(G)=Z_{{L^1(G)}^{\ast \ast }}^{\mathrm{\ell }}(M(G{)}^{\ast \ast })$ and $Z_{M(G{)}^{\ast \ast }}^{\mathrm{\ell }}({L^1(G)}^{\ast \ast })M(G)=Z_{M(G{)}^{\ast \ast }}^{\mathrm{\ell }}({L^1(G)}^{\ast \ast })$. For Banach $A-bimodule$ $B$, if we assume that $B^{\ast }{B}^{\ast \ast }\subseteq A^{\ast }$, then $B^{\ast \ast }{Z}_1(A^{\ast \ast })\subseteq Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$ and moreover if $B$ is an unital as Banach $A-module$, then we conclude that $B^{\ast \ast }{Z}_1(A^{\ast \ast })=Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })$. Let $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })A\subseteq B$ and suppose that $B$ is $WSC$, so we conclude that $Z_{A^{\ast \ast }}^{\mathrm{\ell }}(B^{\ast \ast })=B$. If $\overline{B^{\ast }A}\mathrm{\ne }{B}^{\ast }$ and $B^{\ast \ast }$ has a left unit $A^{\ast \ast }-module$, 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 $A\widehat{\otimes }B$.