The notion of Hilbert module - some call it Hilbert $C^*$-module - is a straightforward generalization of the notion of Hilbert space, where the scalar field $\bf C$ is replaced with a $C^*$ or von Neumann algebra. Consequently, the inner product takes values in that algebra. Some properties we are used to from Hilbert spaces survive to Hilbert modules (for instance, Cauchy-Schwartz inequality and its consequences), others do not (for instance, self-duality and its consequences). Hilbert modules, today, are ubiquitous in all contexts where operator algebras play a role; but not always their presence is noticed - and the price for not noticing is, frequently, having unnecessarily complicated proofs. (Just two examples: Stinespring construction {\it versus} GNS-construction for CP-maps; Connes' tensor product of Connes correspondences {\it versus} tensor product of von Neumann correspondences.) If you work with Hilbert modules, it is good to have a picture in mind to guide your intuition - but that picture depends on the context and on the problem you wish to resolve. We cannot discuss all these pictures in a single lecture. In this lecture, we will concentrate on the well-known fact that Hilbert modules may be understood as an off-diagonal corner $p{\cal A}({\bf 1}-p)$ in a $C^*$-algebra $\cal A$ in the same way as a Hilbert space $H$ is the $21$-corner in the algebra ${\cal B}({\bf C}\oplus H)$. Requiring that $\cal A$ is a von Neumann algebra, naturally leads to the notion of von Neumann module. Von Neumann modules turn out to be self-dual. They are a good compromise conserving all the algebraic power of Hilbert modules {\bf and} the consequences of self-duality.

A Brief on Hilbert Modules and von Neumann Modules

SKEIDE, Michael
2013

Abstract

The notion of Hilbert module - some call it Hilbert $C^*$-module - is a straightforward generalization of the notion of Hilbert space, where the scalar field $\bf C$ is replaced with a $C^*$ or von Neumann algebra. Consequently, the inner product takes values in that algebra. Some properties we are used to from Hilbert spaces survive to Hilbert modules (for instance, Cauchy-Schwartz inequality and its consequences), others do not (for instance, self-duality and its consequences). Hilbert modules, today, are ubiquitous in all contexts where operator algebras play a role; but not always their presence is noticed - and the price for not noticing is, frequently, having unnecessarily complicated proofs. (Just two examples: Stinespring construction {\it versus} GNS-construction for CP-maps; Connes' tensor product of Connes correspondences {\it versus} tensor product of von Neumann correspondences.) If you work with Hilbert modules, it is good to have a picture in mind to guide your intuition - but that picture depends on the context and on the problem you wish to resolve. We cannot discuss all these pictures in a single lecture. In this lecture, we will concentrate on the well-known fact that Hilbert modules may be understood as an off-diagonal corner $p{\cal A}({\bf 1}-p)$ in a $C^*$-algebra $\cal A$ in the same way as a Hilbert space $H$ is the $21$-corner in the algebra ${\cal B}({\bf C}\oplus H)$. Requiring that $\cal A$ is a von Neumann algebra, naturally leads to the notion of von Neumann module. Von Neumann modules turn out to be self-dual. They are a good compromise conserving all the algebraic power of Hilbert modules {\bf and} the consequences of self-duality.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11695/17118
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