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Modular operator multipliers into the trace class

Preprint, 2019

Given Hilbert spaces H_1,H_2,H_3, we consider bilinear maps defined on the cartesian product S_2(H_2,H_3)×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 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)×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\bar\otimesM^{op}_2\bar\otimesM_3. In the case when M_2 is injective, we characterize completely bounded module maps u:S_2(H_2,H_3)×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