Hilbert von Neumann Modules versus Concrete von Neumann Modules

TL;DR

This paper clarifies the relationships between different types of von Neumann modules, introduces new insights into their tensor products, and corrects earlier literature, advancing the understanding of operator modules in noncommutative analysis.

Contribution

It demonstrates the equivalence of Hilbert-von Neumann modules and strongly full concrete von Neumann modules, and generalizes the tensor product of von Neumann correspondences.

Findings

Hilbert-von Neumann modules are equivalent to strongly full concrete von Neumann modules

Tensor product of von Neumann correspondences generalizes Connes correspondences

New arguments applicable to (pre-)Hilbert modules

Abstract

Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or $W^*$-objects among the Hilbert ($C^*$-)modules are around since the first papers by Paschke (1973) and Rieffel (1974) that lift Kaplansky's setting (1953) to modules over noncommutative $C^*$-algebras. While the formal definition of $W^*$-modules} is due to Baillet, Denizeau, and Havet (1988), the one of von Neumann modules as strongly closed operator spaces started with Skeide (2000). It has been paired with the definition of concrete von Neumann modules in Skeide (2006). It is well-known that (pre-)Hilbert modules can be viewed as ternary rings of operators and in their definition of Hilbert-von Neumann modules, Bikram, Mukherjee, Srinivasan, and Sunder (2012) take that point of view. We illustrate that a (suitably nondegenerate) Hilbert-von Neumann module is the same thing as a strongly full concrete von Neumann module. We also use this occasion to put some things in earlier papers right. We show that the tensor product of (concrete, or not) von Neumann correspondences is, indeed, (a generalization of) the tensor product of Connes correspondences (claimed in Skeide (2008)), viewed in a way quite different from Bikram et al.. We also present some new arguments that are useful even for (pre-)Hilbert modules.