arens regularity and factorization property

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. we establish some relationships between the topological centers of module actions and factorization properties of them with some results in group algebras. in 1951 arens shows that the second dual of banach algebra endowed with the either arens multiplications is a banach algebra, see [1]. the constructions of the two arens multiplications in lead us to definition of topological centers for with respect to both arens multiplications. the topological centers of banach algebras, module actions and applications of them were introduced and discussed in [3, 5, 6, 9, 15, 16, 17, 18, 19, 24, 25]. in this paper, we extend some problems from [3, 5, 6, 16, 22] to the general criterion on module actions with some applications in group algebras. baker, lau and pym in [3] proved that for banach algebra with bounded right approximate identity, is an ideal of right annihilators in and . in the following, for a banach , we study the similar discussion on the module actions and for banach , we show that