The Dual of the Haagerup Tensor Product

Abstract

The weak∗-Haagerup tensor product M ⊗w∗h N of two von Neumann algebras is related to the Haagerup tensor product M ⊗h Nin 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⊗hY)∗ 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.