Sedigheh Barootkoob - Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box ۱۳۳۹, Bojnord, Iran.

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{neu۱} 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.

Bilinear map, Factorization property, Strongly Arens irregular, Automatically bounded and w^*-w^*-continuous