Multilinear maps on products of operator algebras

Let A(1), A(2),..., A(r) be C*-algebras with second duals A"(1), A"(2),...,A"(r), and let X be an arbitrary trary Banach space. Let Gamma: A"(1) x A"(2) x (...) x A"(r) --> X" be a bounded r-linear map, and denote by Gamma": A"(1) x A"(2) x (...) x A"(r) --> X" the Johnson-Kadison-Rinrose extension (i.e., the separately weak* to weak* continuous r-linear extension) of Gamma. The problem of characterising those Gamma for which Gamma" takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C*-algebras, the 'obvious' extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C*-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra. (C) 2004 Elsevier Inc. All rights reserved.