Completely continuous multilinear operators on C(K) spaces

Given a k-linear operator T from a product of C(K) spaces into a Banach spa ce X, our main result proves the equivalence between T being completely con tinuous, T having an X-valued separately omega* - omega* continuous extensi on to the product of the biduals and T having a regular associated polymeas ure. It is well known that, in the linear case, these are also equivalent t o T being weakly compact, and that, for k > 1, T being weakly compact impli es the conditions above but the converse fails.