Factorization property and Arens regularity

Abstract: In this paper, we study the Arens regularity properties of module actions and we extend some proposition from Baker, Dales, Lau and others into general situations. For Banach $A-bimodule$ $B$, let $Z_1(A^{**})$, ${Z}^\ell_{B^{}}(A^{})$ and ${Z}^\ell_{A^{}}(B^{})$ be the topological centers of second dual of Banach algebra $A$, left module action $\pi_\ell:~A\times B\rightarrow B$ and right module action $\pi_r:~B\times A\rightarrow B$, respectively. We establish some relationships between them and factorization properties of $A^*$ and $B^*$. We search some necessary and sufficient conditions for factorization of $A^*$, $B$ and $B^*$ with some results in group algebras. We extend the definitions of the left and right multiplier for module actions.