Timeline for Can we characterize the spatial tensor product of von Neumann algebras categorically?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jan 10, 2011 at 9:32 | answer | added | Thomas Timmermann | timeline score: 4 | |
| Jan 9, 2011 at 5:40 | comment | added | Dmitri Pavlov | @Theo: It appears that you construction again gives the categorical tensor product. Therefore your approach gives rise to yet another reformulation of the problem: How can we characterize the full subcategory of A-Mod⊠B-Mod that corresponds to representations of the spatial tensor product (and not just categorical tensor product)? | |
| Jan 9, 2011 at 5:19 | comment | added | Theo Johnson-Freyd | (continuation) Namely, let $A,B$ be rings-up-to-Morita, and $\cal A=A\text{-Mod}$ and $\cal B=B\text{-Mod}$ their categories of modules. Then there is a category $\cal A\boxtimes\cal B$, which is the universal cocomplete category receiving a functor from $\cal A \times \cal B$ (the cartesian product in CAT) that is separately cocontinuous in each variable. By a lemma/exercise, $\cal A\boxtimes\cal B\simeq(A\otimes B)\text{-Mod}$, where $A\otimes B$ is the usual tensor product of rings. So this determines $A\otimes B$ up to Morita equivalence. Maybe something similar works for vN algebras? | |
| Jan 9, 2011 at 5:13 | comment | added | Theo Johnson-Freyd | Just an aside, and it comes with the caveat that I know some noncommutative algebra, but absolutely nothing about von Neumann algebra: the description you gave of the tensor product of noncommutative algebras is correct, but isn't entirely "categorical" in the following sense. Namely, a priori your "external tensor product" requires some more data --- at the least, it requires having a "forgetful" functor so that you can talk about underlying vector spaces of the algebras. But, you can give a "purely" categorical description in the Morita framework, which might be all you want. (continued) | |
| Jan 8, 2011 at 23:36 | history | edited | Dmitri Pavlov | CC BY-SA 2.5 |
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| Jan 8, 2011 at 23:34 | comment | added | Dmitri Pavlov | @Chris: Well, Guichardet's article describes the categorical tensor product and actually it is the place where I learned it. I will add a reference to his paper right now. | |
| Jan 8, 2011 at 21:52 | comment | added | Chris Heunen | Proposition 8.2 and further of "Sur la categorie des algebres de Von Neumann" by Alain Guichardet (Bull. Sci. Math. 90:41-64, 1966) defines a tensor product of von Neumann algebras that (i) reverts to the coproduct for commutative algebras that (ii) is not quite a coproduct but does satisfy a (categorical) universal property. He also develops some basic results about it. It is not quite the spatial tensor product, but might resemble it enough to cover what you're after? | |
| Jan 8, 2011 at 19:26 | history | asked | Dmitri Pavlov | CC BY-SA 2.5 |