On the Properties of the Arens Regularity of Bounded Bilinear Mappings

  Let ,  and be Banach spaces and  be a bilinear mapping. In 1951 Arens found two extension for  as  and  from  into .  The mapping  is the unique extension of  such that  from  into  is  continuous for every , but the mapping  is not in general  continuous from  into  unless .  Thus for all  the mapping is  continuous if and only if  is Arens regular. Regarding  as a Banach , the operation  extends to  and  defined on . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space  becomes a Banach algebra.

   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 some manuscripts. It is known that the multiplication map of every non-reflexive,  -algebra is Arens regular.  In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras.

We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping  is Arens regular if and only if the linear map  with  is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity.

The following conclusions were drawn from this research.if and only if the bilinear mapping   is Arens regular.A bounded bilinear mapping  is Arens regular if and only if the linear map  with  is weakly compact. if and only if the bilinear mapping   is Arens regular.Assume that  has approximate identity. Then   is Arens regular if and only if  is reflexive.