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The question here inspires my present question.

Reyes proves here that the contravariant functor Spec from the category of commutative rings to the category of sets cannot be extended to the category of noncommutative rings in such a way that every noncommutative ring is assigned to a nonempty set. Reyes also proves that it is impossible to suitably extend the Gelfand spectrum functor to the category of noncommutative C* algebras.

If one loosens the demand a Set-valued functor, then there are nice analogues of Gelfand duality. Please allow me to philosophize for a moment (I do so in order for someone to correct my perhaps inaccurate viewpoint). Even in the commutative case the need for "more open sets" in the Zariski topology led to the development of topos theory by Grothendieck in order to support étale cohomology. Toposes extend the notion of locale, which has the noncommutative relative quantale. It is possible to associate quantales to étale groupoids, to which there have been associated homology theories. Simon Henry's work on Boolean toposes (focusing on the von Neumann algebra/measure theoretic setting...his work goes beyond this) uncovers deeper important connections between von Neumann algebras and toposes.

I have begun to wonder if to find a good homology/cohomology theory for von Neumann algebras will require extracting a topos-like geometric object from the projection lattice of the von Neumann algebra and computing some sort of homology/cohomology of that object. I've read in Henry's papers that the kinds of object coming from projection lattices of von Neumann algebras are substantially different from Grothendieck toposes (in some way that I don't know enough about to ask for). The following question is a bit pie-in-the sky, and most likely completely hopeless, but I wonder if there is an "orienting answer":

Question: Is there hope of associating a "nice" topos to a von Neumann algebra?

This question is laughable, but I ask it nevertheless. What I mean by "nice" here is something like "has a computable cohomology of some kind". The philosophy being that toposes may be the right "noncommutative spaces" that may stand in counterpoint to von Neumann algebras.

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(I'm going to be a bit informal to be able to go to the point relatively directly, but if you want more details on some specific aspect. I can try to add them)

Toposes are closely related to topological groupoids, in fact, they can be seen as a special type of localic groupoids or localics stack, the "étale-complete localic groupoids". (see the other answer)

So because we know very well how to attach a C* algebra or Von Neuman algebra to a groupoid it is very natural to expect that one can attach C* or Von Neuman algebra to a topos. Maybe not in full generality as topos corresponds to very general topological spaces and C*-algebras are attached to locally compact topological groupoids, but at least for 'nice topos' it should be possible. And also topos corresponds to Groupoid up to morita equivalence only, so the algebra we produce in general is only well defined up to Morita equivalence.

In some sense my work on this topic at the time was an attempt to give a direct description of the C* algebra or Von Neuman algebra one can attach to a topos (without going through groupoids) or to describe some properties of the Von Neuman algebra directly in terme of the topos (for example its modular time evolution).

And in fact it is possible:

To get a von Neuman algebra you should start with a Boolean topos that satisfies some 'measurability' condition, consider an 'internal Hilbert space object' in the topos and look at its algebra of endomorphisms. The construction works better if one assume that the topos $T$ is in addition 'locally separated' and take an Hilbert space of the form $L^2(X)$ for $X$ such that $T/X$ is separated. In this case you get a close connection between what I call measure theory over $T$ and the modular time evolution of the corresponding Von Neuman Algebra. This is essentially what I study in the paper you linked. For C* algebra things are a bit more complicated, the best construction I could get to is described here.

Now, to go back to your question: can we go the other way and attach a topos to a von Neuman algebra or C algebra ?*

Essentially, no. At least not in a very interesting way if we do not have some additional structures. Of course it is not possible to give a definitive negative answer to this kind of question, so I'll say "probably not".

The problem is better understood in terms of groupoids than in terms of topos: the convolution algebra of a groupoids contains a lot of information on the groupoids, but if you consider it as a mere C*-algebras clearly a lot of information is lost.

For example, let's consider a groupoid $BG$ with only one object $*$ and $Hom(*,*)=G$ a group (Corresponding to the topos $BG$ of sets with a $G$-action). The kind of Von Neuman algebra or C* algebra you will attach to this topos is a Groupe algebra of $G$. Now if $G$ is abelian you will obtain an abelian Von Neuman algebra. But Abelian Von Neuman algebras corresponds to ordinary measurable spaces, so in this case you get two very different types of toposes that corresponds to exactly the same von Neuman algebra (a BG, and a topos of sheaves over a Boolean locale). The isomorphisms between the two Von Neuman algebra you get is induced by a kind of "Fourier transform" whose origin is purely analytic and non-geometric (at least in this picture).

What I read on this type of example is that if you want to construct a topos (or groupoid) out of an algebra you need something more. What this "something more" is can vary a lot, to give two example:

  • For C*-algebra the notion of Cartan subalgebra sometime allow to reconstruct a groupoids, I don't know the literature on this topic but these slides will give you an idea. I assume a similar theory for Von Neuman algebra might be possible.

  • One expects there will some connection between module for the algebra that one obtains a some kind of bundle of vector space/hilbert space on the topos. These bundle of vector spaces on the topos generally have a "pointwise tensor product". So one expect the Algebra we obtain to have an additional structure that corresponds to this tensor product, i.e. some sort of "generalized bi-algebra structure". One also expect that this tensor product is enough to recover the geometric object (this is very similar to Tanaka theory). I have a draft that I never finished on this topics if want to see a precise statement.

  • There are probably other similar story that can be told.

So in some sense I see that C*-algebra/Von Neuman algebra attached to a topos as some kind of invariant, like a homology theory. That reveals a lot of important and sometime hidden informations, but definitely not all the informations.

To finish, I would like to comment on the Bohr topos mentioned in the other answer, as it is is the only such construction present in the literature. I want to emphasize that it does not really answer the question in a satisfying way because the "Bohr topos" is not reall a topos, it is only an ordinary topological space. Indeed, because it is a topos of sheaves on a poset, it actually is a topos of sheaves on a locale, and in fact on a topological space due to a compactness argument, so it will never exhibit any "non-commutative" phenomenon. The construction has been formulated in the language of toposes because many people hope it might be possible to modify the construction to actually produce a topos, and maybe it is, but at the present time what is constructed is really just an ordinary topological space.

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  • $\begingroup$ Thank you for the answer, Simon, and for the clarification about the Bohr topos construct. $\endgroup$ – Jon Bannon Jul 9 at 16:43
  • $\begingroup$ You may be able to accelerate my understanding of why the Bohr topos/topological space is not a good candidate for an invariant of von Neumann algebras. The hope is not so much that the object exhibit noncommutative phenomena, but that it is a "classical" object sensitive enough to distinguish noncommutative objects. Crudely, one might hope that the poset structure (the way abelian subalgebras sit inside the von Neumann/C* algebra) would remember enough about the von Neumann algebra structure to distinguish certain von Neumann algebras from one another. $\endgroup$ – Jon Bannon Jul 9 at 17:29
  • $\begingroup$ Said more briefly, are you saying that this "Bohr topos" topological space cannot even detect noncommutative phenomena? According the the above link, it remembers the Jordan algebra structure, and JBW algebras have some connection to cohomology math.uci.edu/~brusso/ChuRus111114final.pdf $\endgroup$ – Jon Bannon Jul 9 at 17:31
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    $\begingroup$ What I mean is just that by Gelfand duality, ordinary topological space essentially corresponds to commutative algebras. So when you construct the Bohr topos, maybe it will encode some information about your non-commutative algebra (obviously it does) but it encodes it in a way that is "purely commutative" (in the sense that it is an ordinary space). The Bohr topos itself is not a "non-commutative object". The sort of construction that I think your question ask for should somehow encode the "generalized spaces" aspect of VN algebra (i.e. these related to non-commutativity) in the... $\endgroup$ – Simon Henry Jul 9 at 18:45
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    $\begingroup$ "Generalized space" aspect of toposes (which are encoded in the categorical aspect beyond the poset of subterminal objects). $\endgroup$ – Simon Henry Jul 9 at 18:50
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You may want to read about the so called BOHR TOPOS, ie a topos built on a C*-algebra. Here is a reference on nLab

https://ncatlab.org/nlab/show/Bohr+topos

and here is a great discussion on the n-Category Cafe:

https://golem.ph.utexas.edu/category/2011/07/bohr_toposes.html

I may be wrong, but once you are into the Bohr topos, the original algebra appears as * algebra object of the ambient category. At that point you have all the topos-related artillery, and you can do your cohomology there.

PS Another related thing you may want to look into are QUANTALES. Basically, a quantale is the "quantum version" of a locale, and the prototype is built out of subspaces of an algebra. The guy who invented them was Mulvey: see Mulvey and Pellettier. Perhaps they are useful for your endeavor

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    $\begingroup$ Thanks for this, Mirco! I'll have a look. $\endgroup$ – Jon Bannon Jul 8 at 23:35
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    $\begingroup$ I really like the Bohr topos direction. Thanks again! I am not yet sure to accept this as the definitive answer (since I am naive), but feel like doing so as it is certainly an "orienting answer" and points to something quite nice. In a while I will accept this if nothing even more striking springs up. Thanks again! $\endgroup$ – Jon Bannon Jul 9 at 0:26
  • $\begingroup$ In fact, some of this stuff appears in the dissertation linked to in the question... $\endgroup$ – Jon Bannon Jul 9 at 3:09
  • $\begingroup$ @JonBannon I have added the link to Mulvey's article. As for my "answer", it was more of an enlarged comment, but glad to help $\endgroup$ – Mirco A. Mannucci Jul 9 at 9:27

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