[Paper Review] Von Neumann Modules, Intertwiners and Self-Duality

Michael Skeide|ArXiv.org|Aug 25, 2003

Advanced Operator Algebra Research6 references22 citations

TL;DR

This paper presents a new, self-contained proof of the self-duality of von Neumann modules using the intertwiner space framework for representations of the commutant algebra. By identifying a von Neumann module $ E $ as the commutant of $ \mathcal{B}' $ in $ \mathscr{B}(G,H) $, and leveraging von Neumann's double commutant theorem and a technical lemma on operator norms, the authors establish that every bounded $ \mathcal{B} $-functional on $ E $ is represented by an element in $ E $, proving self-duality directly and simply.

ABSTRACT

We apply the ideas of Muhly, Skeide and Solel [MSS03] of considering von Neumann B-modules as intertwiner spaces for representations of B' to obtain a new, simple and self-contained proof for self-duality of von Neumann modules. This simplifies also the approach of [MSS03].

Motivation & Objective

• To provide a simplified, self-contained proof of the self-duality of von Neumann $ \mathcal{B} $-modules.
• To establish that every bounded right $ \mathcal{B} $-linear functional on a von Neumann module arises from an inner product with an element in the module.
• To show that the module $ E \subset \mathscr{B}(G,H) $ coincides with the commutant $ C_{\mathcal{B}'}(\mathscr{B}(G,H)) $, using the double commutant theorem.
• To clarify the duality between $ \mathcal{B}^\prime $-representations and von Neumann $ \mathcal{B} $-modules, enhancing accessibility of the theory.

Proposed method

• Represent the von Neumann $ \mathcal{B} $-module $ E $ as a strongly closed submodule of $ \mathscr{B}(G,H) $, where $ H = E \odot G $ is the interior tensor product.
• Define a normal unital representation $ \rho^\prime \colon \mathcal{B}^\prime \to \mathscr{B}(H) $ by $ \rho^\prime(b^\prime) = \text{id}_E \odot b^\prime $.
• Identify $ E $ as the space of intertwiners $ C_{\mathcal{B}'}(\mathscr{B}(G,H)) = \{ x \in \mathscr{B}(G,H) \mid \rho^\prime(b^\prime)x = x b^\prime \text{ for all } b^\prime \in \mathcal{B}^\prime \} $.
• Use the double commutant theorem to show $ E = C_{\mathcal{B}'}(\mathscr{B}(G,H)) $ by proving $ \mathcal{M}^{\prime\prime} = \mathcal{M} $ for the matrix algebra $ \mathcal{M} = \begin{pmatrix} \mathcal{B} & E^* \\ E & \mathscr{B}^a(E) \end{pmatrix} $.
• Establish self-duality by showing that every $ \mathcal{B} $-functional $ \Phi \in \mathcal{B}^r(E, \mathcal{B}) $ extends to a bounded operator in $ \mathscr{B}(H,G) $, whose adjoint gives the representing element in $ E $.
• Use a technical lemma to show $ \|L_\Phi\| = \|\Phi\| $, ensuring norm control over elementary tensors $ x \odot g $, which allows extension to $ H $.

Experimental results

Research questions

• RQ1 Can the self-duality of von Neumann modules be proven without relying on complete quasi orthonormal systems?
• RQ2 Is there a direct characterization of von Neumann $ \mathcal{B} $-modules as intertwiner spaces for $ \mathcal{B}^\prime $-representations?
• RQ3 Does the commutant $ C_{\mathcal{B}'}(\mathscr{B}(G,H)) $ coincide with the original module $ E $, and can this be shown via the double commutant theorem?
• RQ4 Can the self-duality of $ C_{\mathcal{B}'}(\mathscr{B}(G,H)) $ be established using only basic functional analysis and operator norm estimates?

Key findings

• The von Neumann $ \mathcal{B} $-module $ E $ is equal to the commutant $ C_{\mathcal{B}'}(\mathscr{B}(G,H)) $, i.e., $ E = \{ x \in \mathscr{B}(G,H) \mid \rho^\prime(b^\prime)x = x b^\prime \text{ for all } b^\prime \in \mathcal{B}^\prime \} $.
• The algebra of adjointable operators $ \mathscr{B}^a(E) $ coincides with the commutant of $ \rho^\prime(\mathcal{B}^\prime) $ in $ \mathscr{B}(H) $, i.e., $ \mathscr{B}^a(E) = \rho^\prime(\mathcal{B}^\prime)^\prime $.
• Every bounded $ \mathcal{B} $-functional $ \Phi \in \mathcal{B}^r(E, \mathcal{B}) $ is represented by an element $ y \in E $ such that $ \Phi(x) = \langle y, x \rangle $ for all $ x \in E $, proving self-duality.
• The norm of the operator $ L_\Phi \colon x \odot g \mapsto (\Phi x)g $ satisfies $ \|L_\Phi\| = \|\Phi\| $, which ensures the extension of $ L_\Phi $ to a bounded operator in $ \mathscr{B}(H,G) $.
• The proof reduces the self-duality of von Neumann modules to the Riesz representation theorem for Hilbert spaces, as the existence of adjoints is essential for the argument.