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The Dual of the Haagerup Tensor Product
Author(s) -
Blecher David P.,
Smith Roger R.
Publication year - 1992
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-45.1.126
Subject(s) - tensor product of hilbert spaces , tensor product , tensor product of algebras , tensor product of modules , mathematics , multilinear map , pure mathematics , tensor contraction , tensor (intrinsic definition) , tensor density , product (mathematics) , bimodule , symmetric tensor , algebra over a field , tensor field , exact solutions in general relativity , mathematical analysis , geometry
The weak ∗ ‐Haagerup tensor product M ⊗ w∗h N of two von Neumann algebras is related to the Haagerup tensor product M ⊗ h N in the same way that the von Neumann algebra tensor product is related to the spatial tensor product. Many of the fundamental theorems about completely bounded multilinear maps may be deduced from elementary properties of the weak ∗ ‐Haagerup tensor product. We show that X ∗ ⊗ w∗h Y ∗ =( X⊗ h Y ) ∗ for all operator spaces X and Y . The weak ∗ ‐Haagerup tensor product has simple characterizations and behaviour with reference to slice map properties. The tensor product of two (not necessarily self‐adjoint) operator algebras is proven to have many strong commutant properties. All operator spaces possess a certain approximation property which is related to this tensor product. The connection between bimodule maps and commutants is explored.