| This thesis explores a possible operator space framework for the study of the second dual of a Banach algebra A. We prove some new characterizations for A to be Arens regular and we try to unify, for the Arens regularity problem, two of current approaches: by considering weakly almost periodic functionals on A and by considering the topological center of A**. Motivated by this study, we define two operator space tensor products, namely, the extended projective tensor product and the normal projective tensor product. We investigate the properties of these two products, and compare them with other operator space tensor products. It is shown that the extended projective tensor product is injective, and the normal projective tensor product can linearize a class of bilinear maps under the condition that the pair of operator spaces has certain type of Kaplansky density property. |
