Bilinear operator multipliers into the trace class

Abstract

Given Hilbert spaces H1,H2,H3, we consider bilinear maps defined on the cartesian product S2(H2,H3)×S2(H1,H2) of spaces of Hilbert-Schmidt operators and valued in either the space B(H1,H3) of bounded operators, or in the space S1(H1,H3) of trace class operators. We introduce modular properties of such maps with respect to the commutants of von Neumann algebras Mi⊂B(Hi), i=1,2,3, as well as an appropriate notion of complete boundedness for such maps. We characterize completely bounded module maps u:S2(H2,H3)×S2(H1,H2)→B(H1,H3) by the membership of a natural symbol of u to the von Neumann algebra tensor product M1⊗‾M2op⊗‾M3. In the case when M2 is injective, we characterize completely bounded module maps u:S2(H2,H3)×S2(H1,H2)→S1(H1,H3) 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.