Hilbert von Neumann modules
Panchugopal Bikram, Kunal Mukherjee, R. Srinivasan, V.S. Sunder

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 -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…
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Taxonomy
TopicsOrganic and Molecular Conductors Research · Graphene research and applications · Magnetism in coordination complexes
