Topological Centers and Factorization of Certain Module Actions

Abstract

Let $A$ be a Banach algebra and $X$ be a Banach $A$-bimodule with the left and right module actions $pi_ell: Atimes Xrightarrow X$ and $pi_r: Xtimes Arightarrow X$, respectively. In this paper, we  study  the topological centers of the left module action $pi_{ell_n}: Atimes X^{(n)}rightarrow X^{(n)}$ and the right module action $pi_{r_n}:X^{(n)}times Arightarrow X^{(n)}$,  which inherit from the module actions $pi_ell$ and $pi_r$,  and also the topological centers of their adjoints, from the factorization property point of view, and then, we investigate  conditions  under which these bilinear maps  are Arens regular or strongly Arens irregular.