Hilbert von Neumann modules

TL;DR

This paper presents a simplified operator-based approach to Hilbert von Neumann modules, establishing key theorems, developing tensor products, and analyzing bimodules from automorphisms, avoiding complex $C^*$-module machinery.

Contribution

It introduces a new, simpler framework for Hilbert von Neumann modules that bypasses $C^*$-modules and develops foundational theorems and tensor products.

Findings

Equivalent to Skeide's definition via Riesz lemma

Established Stinespring dilation theorem for bimodules

Connected Jones' basic construction to Stinespring dilation

Abstract

We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the Stinespring dilation theorem for Hilbert von Neumann bimodules, and we develop our version of `internal tensor products' which we refer to as Connes fusion for obvious reasons. In our discussion of examples, we examine the bimodules arising from automorphisms of von Neumann algebras, verify that fusion of bimodules corresponds to composition of automorphisms in this case, and that the isomorphism class of such a bimodule depends only on the inner conjugacy class of the automorphism. We also relate Jones' basic construction to the Stinespring dilation associated to the conditional expectation onto a finite-index inclusion (by invoking the uniqueness assertion regarding the latter).