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.

中文翻译：

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双线性运算符乘数进入迹类

给定 Hilbert 空间 H1,H2,H3，我们考虑定义在 Hilbert-Schmidt 算子空间的笛卡尔积 S2(H2,H3)×S2(H1,H2) 上并在空间 B(H1,H3) 中取值的双线性映射有界算子，或在迹类算子的空间 S1(H1,H3) 中。我们介绍了此类映射关于冯诺依曼代数 Mi⊂B(Hi), i=1,2,3 的换算子的模性质，以及此类映射的完全有界性的适当概念。我们通过 u 的自然符号对冯诺依曼代数张量积 M1⊗‾M2op⊗‾的隶属度来表征完全有界模块映射 u:S2(H2,H3)×S2(H1,H2)→B(H1,H3) M3。在 M2 是单射的情况下，我们通过弱分解性质表征完全有界模块映射 u:S2(H2,H3)×S2(H1,H2)→S1(H1,H3)，它扩展到双线性设置，这是对双模线性映射的著名描述，可以追溯到 Haagerup、Effros-Kishimoto、Smith 和 Blecher-Smith。我们在单射冯诺依曼代数中对完全有界双线性映射上的辛克莱-史密斯定理至关重要，并提供了基于希尔伯特 C⁎-模的新证明。