This is a continuation of a previous post where, to start the discussions, I quoted a portion of a recent interview with Alain (incidentally the full interview is now available and I highly recommend it to the readers of this blog). After a couple of busy weeks now I have some time to follow up on this important question. My aim is to gradually explore some of the themes discussed in the introduction to Alain's 1985 paper: "Noncommutative Differential Geometry". Without any doubts this will take many posts.

When the question ``what is a noncommutative space?" is put to a mathematician you may hear one of the following answers, among others:

1. A noncommutative space is just an associative algebra which may or may not be commutative,

2. A noncommutative space is just a C*-algebra which may or may not be commutative,

3. A noncommutative space is a von Neumann algebra which may or may not be commutative,

4. A noncommutative space is a `spectral triple',

5. A noncommutative space is an abelian category, possibly with some extra structure,

6. No one knows what a noncommutative space is.

I don't agree with item 6 and I would like to add that items 1-5 all have some elements of truth in them. The problem is that of course the very classic question " what is a space ?" is not so well defined and its answer depends on the context. So the best answer would be something like this: a space is a set endowed with some extra structure like a topology, a measure, a smooth structure, a metric, a sheaf, etc. etc...

A similar working definition can be adopted to noncommutative geometry and in fact this turns out to be essential for the whole subject. To start let us agree to call a not necessarily commutative associative algebra a noncommutative algebra. Then we can adopt the thesis:

``A noncommutative space is a noncommutative algebra possibly endowed with some extra structure"

This is a very powerful idea and can be fully justified as we shall see a bit later. For the moment

we should note that examples 1-4 in the above list all fall within the scope of this definition. For example a C* algebra is an algebra over the field of complex numbers equipped with an involution and a norm satisfying some conditions.

Proposal 5 however looks very different from the rest. This becomes relevant when we need to discuss morphisms between noncommutative spaces. Definitions 1-4 suggest that a morphism should be something like an algebra homomorphism. This is however too rigid and there may not be enough morphisms between noncommutative spaces in this rigid sense. This is in fact a purely noncommutative phenomenon to which we shall return later. For the moment it suffices to say that given an algebra A we can pass to the abelian category A-mod of representations of A. There is less information in A-mod than in A and in fact A-mod characterizes A only up to Morita equivalence but the topological information in A is not lost in this process. In many cases, like Hochschild homology, cyclic homology, and K-theory, it can be fully defined in terms of A-mod. More on this later.

Talking about a good definition of a noncommutative space inevitably brings up the issue of the role of definitions in the development of mathematics. The Orthodox view of mathematics as an exact science (generally held by non-mathematicians!) assumes that first come precise definitions of objects of study and then their main properties are explored by proving theorems

about them. In reality, however, it often happens that the main results of a subject, up to a certain point, are proved before a general agreement is reached on foundational issues. One can give so many examples of this but the development of differential geometry along with the notion of a manifold from Riemann to Weyl and Whitney is a case in point. Up to the early 1960's one of the main textbooks on differential geometry was Élie Cartan's classic, Geometry of Riemannian Spaces. Even in its second enlarged 1946 French edition Cartan says

'' The general notion of manifold is quite difficult to define with precision"

(see also this) and simply refrains from giving a general definition. He had much more important and interesting things to say!

Back to the main topic of this post, what we should do next is to justify our notion of a noncommutative space: why this is a reasonable idea and in what sense it extends the classical idea of a geometric space. This I will do in the next post.

## 10 comments:

The fact that you mention #5 is interesting to me, because that is the direction the purely algebraic side of noncommutative geometry is headed. Is there an analogous approach on the functional analytic side?

very classic question "what is a space?" is not so well definedThis is why option 6 should be considered more carefully. To a topos theorist it is clear that 1-5 are unsatisfactory because a 'space' is a topos, but this only covers classical logic and we clearly need to generalise the foundational notion of topos using quantum logic.

Many noncommutative spaces are commutative spaces quotiented out by a groupoid action (or even the action of a category), although this statement has different meanings in the different contexts 1-5 above. My question is Does every noncommutative space arise in such a way? Or perhaps more accurately vague, Are there important noncommutative spaces that cannot be viewed as quotients of commutative spaces in any reasonable way?

Dear Walt,

Thanks for your comments. ``Is there an analogous approach on the functional analytic side?"

Yes there is and it is a major issue. I shall be brief but I hope to be able to return to this soon. One reason is that in general there aren't that many morphisms between, say, C* algebras, e.g. when the source is a simple algebra and the target is commutative. One approach to remedy this is to consider the larger class of completely positive maps but let us not discuss it now. Along the lines of item 5, the right category of representation to choose in this case are Hilbert space reps (of a given, say, C*-algebra as your NC space). To be able to induce a B-(Hilbert) module from an A-module via tensoring puts some strong conditions on the type of A-B bimodules you need to consider (Hilbert bimodules). We can consider these Hilbert A-B bimodules as defining correspondences between NC topological spaces represented by C* algebras A and B. There is also a notion of correspondence, due to Connes, between V N algebras with composition defined by, now famous, Connes fusion. Both issues are very well discussed in Alain's 1994 book (pages 156, 539, in the on line version). There is also a homotopy category of NC spaces, Kasparov's KK categoty whose objects are C* algebras and its morphisms are KK groups (Hilbert bimodules up to homotopy equivalence), but this can be the subject of a new post!

Anonymous: Yes there are noncommutative spaces that do not arise as quotients of commutative ones. The latter all give rise to algebras that are antiisomorphic to themselves and there are von-Neumann algebras which are not antiisomorphic to themselves. Thus even at the measure theory level the "quotients" do not exhaust all possibilities. To see where they come from you can use the twisting by a two cocycle in the crossed product construction, exactly as quaternions emerge as a twisted cross product of complex numbers by the complex conjugation and a non-trivial two cocycle....

Dear Anonymous,

There is of course an advantage of quotienting with respect to groupoid actions (as opposed to general categories) in that the resulting groupoid algebras over C will carry a * structure and one can then complete them into a C* algebra and eventually you can use powerful methods of functional analysis. Also, groupoids are closer to the idea of ``equivalence relation" since the symmetry property of an equivalence relation has its analogue in the groupoid world which is the invertibility of every morphism ( I am not sure about this, but I guess sometime even "pseudo equivalence relation" used to to refer to a groupoid...) Now as Alain said in his comment, quotients of a commutative (and commutativity is important here) algebra by a groupoid action have the property that the resulting algebra is isomorphic to its opposite algebra (or anti-isomorphic to itself) by simply sending g to g^-1. But if you allow quotienting by categories, as you seem to, e.g. in quiver algebras, then the quotient does not have this property. I suspect one can still show that not all (even finite dimensional) algebras are quotients of commutative ones by category actions. I guess for people in NCAG working with quiver algebras this must be quite well known and hopefully they can help here....In fact I think the finite dimensional example of Alain should work here too.

Does the Nature paper 26 April 2007 influence #3 von Neumann Algeba relative to NGC?

Robert D. MacPherson & David J. Srolovitz

'The von Neumann relation generalized to coarsening of three-dimensional microstructures' [p1053]

doi:10.1038/nature05745

Editor's Summary

http://www.nature.com/nature/journal/v446/n7139/edsumm/e070426-09.html

Since the MiniMax Theorem of von Nemann is the foundation equation of mathematical game theory, perhaps game theory may have a role in NCG or other geometries?

The Max-Plus or Tropical Algebras use node in lieu of vertex.

Are these algebras Node Operator Algebras [NOA] related to Vertex Operator Algebras [VOA]?

Dear AC and Masoud,

Thanks for your replies. I have a follow up question then. What are the most important examples of noncommutative spaces that don't arise as groupoid quotients of commutative spaces?

I am asking because I haven't been convinced that the formal set up of noncommutative geometry is the best one. (I admit that this probably says more about my own ignorance than about the subject.) Probably the best way for me to become convinced would be to see the key examples. I've asked some people this question and they mention the increased flexibility of quotienting. This is fine, but in my world (algebraic geometry) this can be handled perfectly well by using sheaves which are equivariant with respect to the groupoid action and so on. Therefore the extra flexibility of quotienting is not a very strong attraction for me.

So it is good to hear that these aren't the only examples. It's also not surprising since equivariant geometry seems much closer to usual geometry than arbitrary algebras are to commutative ones. On the other hand, I myself can't imagine any notion of space which is more general than equivariant space (i.e. more or less a topos) and which would be amenable to geometric intuition. So I'm hoping that after seeing a key example or two, I'll have some idea of the point.

I hasten to add that I do not doubt that there is a point -- it's just that I feel like I'm missing it. (I also think many subjects are worth investigating purely on the grounds of formal attractiveness, but I doubt this is an example where that's happening.) And thanks for indulging such basic questions.

Anonymous,

a few remarks from a very NCAG-person (with the stress on A for algebraic).

All Alain is saying is that there are other C^*-algebras than skew-group constructions over commutative ones and for Massoud : of course there are plenty of non-anti-isomorphic algebras (central simple algebras whose class is not of order two in the Brauer group or as he asked for quiver examples : any path algebra of a quiver whose opposite quiver, (the one obtained by reversing all arrows) is not isomorphic to the quiver you started off with).

But thats not the real issue, for lets take the quiver case, then in NCAG we would like to view the isoclasses of finite dimensional representations as our 'noncommutative manifold' to which Anonymous would reply 'but hey, isoclasses of n-dimensional representations are just orbits under GL(n) on a commutative variety (the representation space) and we can handle this quite well with a Thomasson-like topos of GL(n)-equivariant and isotrivial etale covers' (or something similar, I'm not quite up to point with current topos-fashion) and of course (s)he is right.

It all depends on what your math-habitat is whether one approach or another gives you a better insight in the real problem. I'm born,raised and educated in Antwerp making noncommutative algebras and their representations my natural habitat, but if I would have gotten stacky-mother milk (if I would have been born on the US-east coast for instance) then the topos-point of view might be more natural to me. So, anonymous I'm afraid NCAG cannot offer you something entirely different (though we have a few interesting tricks and btw. its quite a difficult problem to distinguish representation-schemes among all GL(n)-schemes so NCAG is not 'just' GL(n) equivariant geometry...).

As for NCG : their point is that they do not have to consider a space at all! To them a 'noncommutative space' is a spectral triple made out of a C^*-algebra and their approach to things is something like 'hey, there are interesting orbit-problems that have an extremely ill behaved orbit-space (such as all real-numbers modulo SL(2,Z)-action by Moebius-transfos) but these problems still

have a natural C^*-algebra associated to them (a skew-group construction or a Cuntz-Krieger algebra) and we are pretty good at computing K-theoretic (and other) invariants of such algebras and as such invariants in the commutative case give us topological (and other) info about the space, these calculations give us some way into these horrible orbit-spaces' (or at least thats my outsider interpretation of what they do).

hope this helps a bit.

Has anyone of the NCG experts here had a look at Döring-Isham's considerations?

They try to find a useful description of elements of a noncommutative algebra as "functions internal to a non-classical topos", namely as morphisms from a "space object" to a "value object" in a suitable topos (of presheaves over commutative subalgebras of a fixed algebra, in their case).

Their motivation is quantum mechanics and the "noncommutative phase space" appearing there, but it looks like their constructions are not really specific to the context of quantum mechanics but rather address the general issue of noncommutative algebras.

I am not entirely convinced yet that what they achieve is what one would be looking for (in part certainly because I don't fully oversee what exactly it is they achieve), and in particular it seems that their topos does not know enough about the sum and product operations on the noncommutative algebra.

But apart from that, it seems to me that they are headed in a direction which, when followed to the end, would maybe allow noncommutative geometry to be considered as ordinary geometry internalized away from the topos of sets into a more non-classical topos (say of presheaves over commutative algebras).

If that would indeed work out (which I don't dare to estimate the chances for), it would seem to give a very nice answer to the question "What is a noncommutative space?" (or of any of the more concrete versions of this question).

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