Completely Continuous Multilinear Operators on C(k) Spaces

Given a k-linear operator T from a product of C(K) spaces into a Banach space X, our main result proves the equivalence between T being completely continuous, T having an X-valued separately ω∗ − ω∗ continuous extension to the product of the biduals and T having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to T being weakly compact, and that, for k > 1, T being weakly compact implies the conditions above but the converse fails. The purpose of this paper is to present some results concerning vector-valued completely continuous operators from a product of C(K) spaces. First we will explain our notation: if K is a compact Hausdorff space, C(K) will be the space of scalar-valued continuous functions on K endowed with the supremun norm, Σ will denote the σ-algebra of the Borel sets of K, and B(Σ) will be the space of Σ-measurable functions on K which are the uniform limit of Σ-simple functions. X will denote a Banach space and X∗∗ its bidual; we will assume, when necessary, that X is embedded in X∗∗. We shall use the convention [i] . . . to mean that the i-th coordinate is not involved. If E1, . . . , Ek, X are Banach spaces, we will denote by Lk(E1, . . . , Ek; X) the Banach space of the continuous multilinear operators from E1 × · · · × Ek into X with the usual operator norm. As is well known, the Riesz representation theorem gives a representation of the operators on C(K) as integrals with respect to Radon measures, and this has been very fruitfully used in the study of the properties of the C(K) spaces and the operators defined on them. In a series of papers (see especially [9], [10]), Dobrakov developed a theory of polymeasures, functions defined on a product of σ-algebras which are measures on each variable separately, that can be used to obtain a Riesz-style representation theorem for multilinear operators defined on a product of C(K) spaces. We will denote the semivariation of a polymeasure γ by ‖γ‖ (for the general theory of polymeasures see [9] or [16]). It seems convenient to recall here that a polymeasure is called regular if it is separately regular and it is called countably additive if it is separately countably additive. We now state the previously announced general representation theorem which extends and completes previous results (see [6]). Received by the editors March 8, 1998 and, in revised form, April 24, 1998. 1991 Mathematics Subject Classification. Primary 46E15, 46B25.