Multilinear maps on products of operator algebras

Let A1,A2, . . . , Ar be C∗-algebras with second duals A 1 ,A 2 , . . . , A r, and let X be an arbitrary Banach space. Let Γ :A1 × A2 × ··· × Ar →X be a bounded r-linear map, and denote by Γ :A 1 ×A 2 ×···×A r →X the Johnson–Kadison–Ringrose extension (i.e., the separately weak∗ to weak∗ continuous r-linear extension) of Γ . The problem of characterising those Γ for which Γ 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.  2004 Elsevier Inc. All rights reserved