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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.

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.

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

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