ARENS PRODUCTS AND AN IMBEDDING THEOREM

Let X be a separable Banach space, B(X) be the algebra of all bounded linear operators on X, and C be the algebra of all compact linear operators. This paper centers around the general question of giving a construction of B(X) as a Banach algebra starting from C.It is a result of Schatten and von Neumann that if H is a Hubert space, then there is an isometric imbedding of B(H) onto C*, where C* denotes the second dual of C Moreover, each of the two Arens products on C* coincides with the multiplication induced on C* by operator multiplication on B(H). The proofs of these results make strong use of the Hubert space structure.In this paper we generalize these results to a large class of uniformly convex spaces. Moreover, we show that even when B(X) is not equal to C* it is still possible to construct B(X) as a Banach algebra starting from C. © 1969 by Pacific Journal of Mathematics.