On The Factorization Property and Arens Regularity of Banach Algebras

In this paper, we establish some relationships between topological centers of module actions and factorization properties and give some results in group algebras case. We also consider sufficient and necessary conditions under which the Banach algebra $A\widehat{\otimes}B$ is Arens regular. Let $A$ and $B$ be Banach algebras and $B$ is unital and suppose that $B$ is a Banach $A$-bimodule. In this case, if $A\widehat{\otimes}B$ is Arens regular, then $A$ is Arens regular. In another case, assume that $B$ is non-trivial on $A$ and $B$ is a unitary Banach $A$-bimodule. Then $A$ and $B$ are Arens regular if and only if $A\widehat{\otimes}B$ is Arens regular.