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1. • CommentRowNumber1.
   • CommentAuthorspitters
   • CommentTimeApr 9th 2018
   • PermaLink
   Author: spitters
   Format: MarkdownItexPhD-thesis on categorical aspects of vN-algebras. We'll probably want to include more results from it. <a href="https://ncatlab.org/nlab/revision/diff/von+Neumann+algebra/50">diff</a>, <a href="https://ncatlab.org/nlab/revision/von+Neumann+algebra/50">v50</a>, <a href="https://ncatlab.org/nlab/show/von+Neumann+algebra">current</a>

   PhD-thesis on categorical aspects of vN-algebras. We’ll probably want to include more results from it.

   diff, v50, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 14th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded this: ## The category of von Neumann algebras The category of von Neumann algebras is a [[locally presentable category]]. The [[forgetful functor]] from von Neumann algebras to [[sets]] that sends a von Neumann algebra to its [[unit ball]] is a [[right adjoint functor]]. In fact, it is a [[monadic functor]] and preserves all [[sifted colimits]]. Thus, [[limits]] and [[sifted colimits]] of von Neumann algebras can be computed on the level of underlying unit balls. Small [[coproducts]] of von Neumann algebras exist. There is also a “reduced” version of small [[coproducts]], known as [[free products]], which can be defined in a manner analogous to the [[spatial tensor product]]. ## Monoidal structures There are two different [[tensor products]] one can define on von Neumann algebras. First, one can use the usual universal property of tensor products and postulate that morphisms $A\otimes B\to M$ are in a natural bijection with pairs of morphisms $A\to M$ and $B\to M$ whose images commute in $M$. This yields a [[symmetric monoidal structure]] on von Neumann algebras. This monoidal structure is not [[closed]]. Secondly, one can also define a “reduced” version, known as the _spatial tensor product_. Given two von Neumann algebras $A$ and $B$, their spatial tensor product is the von Neumann algebra generated by $A\otimes 1$ and $1\otimes B$ in the von Neumann algebra $B(L^2 A\otimes L^2 B)$, where $L^2 A$ and $L^2 B$ are the Haagerup standard form of $A$ and $B$ respectively. This also results in a [[symmetric monoidal structure]]. Furthermore, passing to the [[opposite category]] yields a [[closed monoidal structure]]. <a href="https://ncatlab.org/nlab/revision/diff/von+Neumann+algebra/51">diff</a>, <a href="https://ncatlab.org/nlab/revision/von+Neumann+algebra/51">v51</a>, <a href="https://ncatlab.org/nlab/show/von+Neumann+algebra">current</a>

   Added this:

   The category of von Neumann algebras

   The category of von Neumann algebras is a locally presentable category.

   The forgetful functor from von Neumann algebras to sets that sends a von Neumann algebra to its unit ball is a right adjoint functor. In fact, it is a monadic functor and preserves all sifted colimits.

   Thus, limits and sifted colimits of von Neumann algebras can be computed on the level of underlying unit balls.

   Small coproducts of von Neumann algebras exist. There is also a “reduced” version of small coproducts, known as free products, which can be defined in a manner analogous to the spatial tensor product.

   Monoidal structures

   There are two different tensor products one can define on von Neumann algebras.

   First, one can use the usual universal property of tensor products and postulate that morphisms $A\otimes B\to M$ are in a natural bijection with pairs of morphisms $A\to M$ and $B\to M$ whose images commute in $M$. This yields a symmetric monoidal structure on von Neumann algebras. This monoidal structure is not closed.

   Secondly, one can also define a “reduced” version, known as the spatial tensor product. Given two von Neumann algebras $A$ and $B$, their spatial tensor product is the von Neumann algebra generated by $A\otimes 1$ and $1\otimes B$ in the von Neumann algebra $B(L^2 A\otimes L^2 B)$, where $L^2 A$ and $L^2 B$ are the Haagerup standard form of $A$ and $B$ respectively. This also results in a symmetric monoidal structure. Furthermore, passing to the opposite category yields a closed monoidal structure.

   diff, v51, current

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 30th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded references. <a href="https://ncatlab.org/nlab/revision/diff/von+Neumann+algebra/54">diff</a>, <a href="https://ncatlab.org/nlab/revision/von+Neumann+algebra/54">v54</a>, <a href="https://ncatlab.org/nlab/show/von+Neumann+algebra">current</a>

   Added references.

   diff, v54, current

4. • CommentRowNumber4.
   • CommentAuthornLab edit announcer
   • CommentTimeAug 21st 2024
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexchanged [[higher algebra - contents]] to [[algebra - contents]] in context sidebar Anonymouse <a href="https://ncatlab.org/nlab/revision/diff/von+Neumann+algebra/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/von+Neumann+algebra/61">v61</a>, <a href="https://ncatlab.org/nlab/show/von+Neumann+algebra">current</a>

   changed higher algebra - contents to algebra - contents in context sidebar

   Anonymouse

   diff, v61, current