Bilinear operator multipliers into the trace class

Given Hilbert spaces H-1, H-2, H-3, we consider bilinear maps defined on the cartesian product S-2(H-2, H-3) x S-2 (H-1, H-2) of spaces of Hilbert-Schmidt operators and valued in either the space B(H-1, H-3) of bounded operators, or in the space S-1(H-1, H-3) of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras M-i subset of B(H-i), i = 1,2,3, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps u: S-2 (H-2, H-3) x S-2 (H-1, H-2) -> B(H-1, H-3) by the membership of a natural symbol of u to the von Neumann algebra tensor product M-1(circle times) over barM(2)(op)(circle times) over barM(3). In the case when M-2 is injective, we characterize completely bounded module maps u: S-2 (H-2, H-3) x S-2 (H-1, H-2) -> S-1(H-1, H-3) by a weak factorization property, which extends to the bilinear setting a famous description of bimodule linear mappings going back to Haagerup, Effros-Kishimoto, Smith and Blecher-Smith. We make crucial use of a theorem of Sinclair-Smith on completely bounded bilinear maps valued in an injective von Neumann algebra, and provide a new proof of it, based on Hilbert C*- modules. (C) 2020 Elsevier Inc. All rights reserved.