Extension of Multilinear Operators on Banach Spaces

These notes deal with the extension of multilinear operators on Banach spaces. The organization of the paper is as follows. In the first section we study the extension of the product on a Banach algebra to the bidual and some related structures including modules and derivations. Tha approach is elementary and uses the classical Arens' technique. Actually most of the results of section 1 can be easily derived from section 2. In section 2 we consider the problem of extending multilinear forms on a given Banach space to a larger space Y containing it as a closed subspace. In the third section we shall show that the extension operators of section 2 preserve the symmetry if (and only if) X is regular (that is, every linear operator X --> X' is weakly compact). Also, we give some applications to the (co)homology of Banach algebras. Given a multilinear operator T: X x ... x X --> Z, the (vector valued version of the) Aron-Berner extension provides us with a multilinear extension ab(T): X'' x ... x X'' --> Z'' which, in general, takes values in Z''. In section 4 we study some consequences of the fact that the range of ab(T) stays in the original space Z. We shall show that those operators whose Aron-Berner extensions are Z-valued play a similar role in the multilinear theory that weakly compact operators in the linear theory, thus obtaining multinear characterizations of some classical Banach space properties related to weak compactness in terms of operators having Z-valued Aron-Berner extensions. Finally in section 5 we give an application of the Aron-Berner extension to the representation of multilinear operators on spaces of continuous functions by polymeasures