n$-factorization Property of Bilinear Mappings

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:X\times Y\to Z$, depended on a natural number $n$ and a cardinal number $\kappa$; which is called $n$-factorization property of level $\kappa$. Then we study the relation between $n$-factorization property of level $\kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity and also strong Arens irregularity. These results may help us to prove some previous problems related to strong Arens irregularity more easier than old. These include some results proved by Neufang in ~\cite{neu1} and ~\cite{neu}. Some applications to certain bilinear mappings on convolution algebras, on a locally compact group, are also included. Finally, some solutions related to the Ghahramani-Lau conjecture is raised.