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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 14th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated with this content: ## Definition A __Hilbert W\*-module__ is a [[Hilbert C*-module]] $M$ over a [[von Neumann algebra]] $A$ such that $M$ admits a [[predual]] as a [[Banach space]]. ## Properties For any [[von Neumann algebra]] $A$, the category of [[Hilbert W*-modules]] over $A$ is equivalent to the category of [[W*-representations]] of $A$. The equivalence is implemented by the following functors. Given a [[Hilbert W*-module]] $M$, we send it to the [[completion]] of $M\otimes_A L^2(A)$, where $L^2(A)$ is the Haagerup standard form of $A$. Given a [[W*-representation]] $R$, we send it to the [[internal hom]] $Hom_A(L^2(A), R)$, which is a [[Hilbert W*-module]] over $A$. ## Related concepts * [[W*-representation]] * [[Hilbert C*-module]] * [[von Neumann algebra]] * [[duality between geometry and algebra]] <a href="https://ncatlab.org/nlab/revision/Hilbert+W%2A-module/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+W%2A-module">current</a>

   Created with this content:

   Definition

   A Hilbert W*-module is a Hilbert C*-module $M$ over a von Neumann algebra $A$ such that $M$ admits a predual as a Banach space.

   Properties

   For any von Neumann algebra $A$, the category of Hilbert W*-modules over $A$ is equivalent to the category of W*-representations of $A$.

   The equivalence is implemented by the following functors.

   Given a Hilbert W*-module $M$, we send it to the completion of $M\otimes_A L^2(A)$, where $L^2(A)$ is the Haagerup standard form of $A$.

   Given a W*-representation $R$, we send it to the internal hom $Hom_A(L^2(A), R)$, which is a Hilbert W*-module over $A$.

   Related concepts

   • W*-representation

   • Hilbert C*-module

   • von Neumann algebra

   • duality between geometry and algebra

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorTom Mainiero
   • CommentTimeMay 9th 2022
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   Author: Tom Mainiero
   Format: MarkdownItexAdded a few references, including the first reference I know of to W*-modules (Paschke). <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+W%2A-module/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+W%2A-module/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+W%2A-module">current</a>

   Added a few references, including the first reference I know of to W*-modules (Paschke).

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorTom Mainiero
   • CommentTimeMay 9th 2022
   • PermaLink
   Author: Tom Mainiero
   Format: MarkdownItexAdded extra technical condition needed in definition, along with appropriate reference. <a href="https://ncatlab.org/nlab/revision/diff/Hilbert+W%2A-module/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Hilbert+W%2A-module/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Hilbert+W%2A-module">current</a>

   Added extra technical condition needed in definition, along with appropriate reference.

   diff, v4, current