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)