So I’ve posted some slides from my talk at Groupoidfest. I also gave this talk here at Western in the Algebra seminar on Wednesday. It seemed to go over fairly well, although it was a bit of an outlier for the conference. However, I’m getting used to that consequence of trying to talk to both physicists and mathematicians. Anyway, after I got back from GFest (as they call it), it took me a few days to get caught up on lecturing and grading and so forth, but here are some slightly belated comments on what went on there. A lot of the content of the talks went over my head, as happens. However, at lunch of the first day, Arlan Ramsay gave me and a couple other beginning researchers some good advice about learning at conferences where you only grasp about 10% of what’s going on: be like a baby learning to walk. Don’t be afraid of looking stupid – just grab the 10% you understand, and then do it again. (Since I spent part the weekend watching a baby learn to walk, this was quite apropos).

So this I’ll comment a bit on some of the general themes I did manage to pick up, and in a subsequent post I may say more about some of the talks that seemed particularly relevant and/or comprehensible to me.GFest was held at the University of Iowa, in Iowa City – by happenstance, a friend from UCR, Erin Pearse, recently started there as a VIGRE postdoc, so I managed to stay with him and his family while I was in town, which was good. I was a little surprised at first that he was interested in sitting in on the talks at the conference, since his research is mostly in fractal geometry, and I didn’t initally see the relevance. However, I guess it shouldn’t have been too surprising, since part of the great thing about groupoids is their ability to represent symmetry. The kinds of fractals in question are the self-similar kind, which have various interesting types of symmetry.

In particular, Erin explained to me that the connection has something to do with shift operators. These operators, which shift a sequence of numbers and insert a new value in it, can be used iteratively to build up, for instance, the Cantor set. (Which is a set of sequences of 0’s and 2’s in ternary notation – the shift operators take you from a point in the whole set, to a point in one of its pieces, which resemble the whole.)

This was one reflection of a more general theme: since there’s a Hilbert space of sequences, namely , the shift operators can be taken as operators on a Hilbert space. So in particular, they generate an algebra of operators – a -algebra (see also some notes). The general theme is that most of the people at the GFest were interested in groupoids as a way of saying something about -algebras. I probably heard this term bandied about more than the actual term “groupoid” while I was there.

One reason my point of view was an outlier is that I was talking about finite, topologically discrete groupoids. However, this is kind of beside the point, since I’m really more interested in ones that come from Lie groups, and have some interesting topology. But I avoid getting into that so far because I’ve been postponing extending this stuff to smooth groupoids, since that leads to infinite-dimensional 2-Hilbert spaces, and gets more complicated than what I’ve been talking about so far. The theory of these does exist – Crane and Yetter develop a lot of the theory needed under the aegis of “measurable categories” – but it involves a lot more analysis.

In fact, while I’m used to thinking of groupoids as a special kind of category, a lot of the talk about them at GFest emphasized exactly this analysis a lot more. It seems to be bread-and-butter for people who work with groupoids arising in -algebras. Paul Muhly, who organized the conference, kindly gave me the current working draft of a book he’s writing on this stuff, where a lot of the important ideas people were using are collected together and explained. (Note that I’ve only started reading it, so I may be mistaking things here).

One point seems to be that these algebras coming from groupoids are related to the -algebras coming from transformation groups: situations where a (locally compact topological) group acts on a (locally compact Hausdorff) space . These can automatically be thought of as groupoids, taking objects to be points in the space, and morphisms from to to be group elements whose action takes to . Now as for -algebras, you can build them by taking algebras of compactly supported complex functions on . This becomes an algebra with the convolution product, given by integrating over the group (so we’re assuming has a nice invariant measure like Haar measure on a Lie group):

and the “star” operation is just complex conjugation.

You can do something similar for groupoids generally, since groupoids decompose into isomorphism classes, each of which looks just like a set with the action of some particular group on it. For this to really make sense, you must be talking about *topological groupoids*. Here, they think of groupoids as a set of all morphisms, with being the set of composable pairs. Given a topology on , this gets the subspace topology on the product. This is making use of the fact that objects of the groupoid needn’t be defined separately – they correspond to the “identity” morphisms (with ), which again gets the subspace topology automatically (which makes source and target maps continuous).

Then we’d like to again define a -algebra on using something like the above definition. But then we need to define a convolution product, and for that, we needed a Haar measure on the group. Fortunately, for topologically reasonable groups, you’re guaranteed to have one, and it’s unique (maybe up to a scalar multiple); unfortunately, you don’t have either existence or uniqueness guaranteed for groupoids. So instead you need to have a *Haar system*.

This is a family of measures on (the set of *all *morphisms), one for each object: , which we’ll use to do convolution at the which correspond to the object . The measure is supported on the component of the object . The whole system needs to have some nice properties. One is that for any function , the function taking to the integral of with respect to should be in 9. The other is that is equivariant, in the sense that if ,

(shifting which measure we use by is the same as shifting the function by ).

This is a bit obscure to me at the moment, but it’s clear enough that you need some family of measures to define a convolution. The first property just ensures that the algebra is closed under this product. The second is just the kind of property you should expect from groupoids: if you’ve defined something that’s not equivariant, you’re just asking for aggravation. So then finally, making a bunch of assumptions, such as that is locally compact, Hausdorff, and so on, we get , the set of smooth, compactly supported complex functions with a convolution product:

(where is the *source* object of the morphism ). The star operation is still complex conjugation.

So, while I’m running a bit long here, this is the basic setup behind most of what people were talking about at Groupoidfest. Either studying these -algebras in their own right, or using groupoids to think of already existing algebras as coming from this setup for some groupoid . The point, I suppose, is that representations of these algebras, and of the groupoids they come from, are closely related, just as representations of groups and their group algebras are.

This subject – representations of groupoids, is exactly what my talk was about, except that I ignored all the topology to simplify certain things. Right after my talk, Marius Ionescu gave one about irreducible representations of groupoid -algebras, which I’m trying to get up to speed on to see how these things are done in the case with more interesting topology. (For my purposes, it’ll also be necessary to understand infinite-dimensional 2-Hilbert spaces better, but that’s another story…) Maybe when I see what that’s about, I’ll say something further on that subject.

There were a number of other good talks – perhaps soon I’ll see if I can summarize what I gathered from some of them.

November 11, 2007 at 11:44 pm

Jeffrey, I like your blog and I have a question. My favorite kind of topological groupoids are profinite groupoids because, like the late MIT mathematician Gian-Carlo Rota, I am also fascinated by profinite structures in mathematics.

A profinite groupoid is the inverse limit of automorphism finite groupoids. The best and shortest (5 pages) paper on the subject is “Profinite Groupoids and their Cohomology” by Andrew Baker:

http://www.maths.gla.ac.uk/~ajb/dvi-ps/cohgpds.pdf

Do you know if anyone at GFest or elsewhere has thought about potential applications of profinite groupoids? Thanks.

November 12, 2007 at 3:53 am

Hi, Charlie – thanks for the comment!

I don’t remember anyone talking about profinite groupoids explicitly. The first place it occurs to me to look would be at these shift operators, related to construction of self-similar fractals such as the Cantor set, which has some kind of profinite topological structure. (E.g. the open sets are those consisting of sequences which agree for the first terms). I’m not sure if that’s right, though, because although I see how these are operators in a -algebra – left and right shifts on sequences – I don’t see quite where/what the related groupoid is. I’m just assured it’s there somehow.

November 13, 2007 at 6:31 am

Hi! Since you’re the academic grandson of Irving Segal, it’s good that you’re getting into C*-algebras.

One deep thing C*-algebras (and noncommutative geometry) have in common with groupoids is that they give a way to tackle “bad quotients”. If we have a group G acting on a space X and the orbits aren’t closed, the quotient X/G can be non-Hausdorff. This makes life tough.

In this situation, the noncommutative geometry people replace the quotient X/G by a “noncommutative space”, or more precisely a noncommutative C*-algebra. This is formed by taking the algebra of functions on X and throwing in a bunch of new operators that describe how elements of G act on functions on X (by “translation”). The result is a noncommutative C*-algebra. I think you’re describing this C*-algebra in your post when you talk about about C*-algebras coming from transformation groups.

On the other hand, the category theorists deal with nasty quotients by replacing X/G by the “weak quotient” X//G, a topological category with X as objects and morphisms coming from elements of G.

There is some sense, not completely clear to me at the moment, in which the “algebra of functions” on X//G is the same as the noncommutative C*-algebra described above.

I guess the idea of getting noncommutative C*-algebras from groupoids is supposed to make the connection between the two approaches clear.

It would be very nice to straighten out the details. Maybe you can do it.

November 13, 2007 at 8:53 am

Hi, John:

I went to the noncommutative geometry seminar here at UWO today and got some more appreciation for some of this stuff. My next post will probably talk a bit about it.

I don’t know about straightening out those details at the moment, but one thing that had been bothering me about the difference between the category-theorist’s view of groupoids and this more algebraic view was that the objects disappear in the second. So naturally people look at algebras of functions on the (morphisms of) the groupoid – which for a category-theorist seems wrong. Why assign complex numbers to morphisms? At best, that suggests the objects are all assigned the same thing: , as a vector space.

After watching the talk today, I have a feeling about how to make this seem less funny. I also have a much better grip on what “Morita equivalence” means – one of the terms people used a lot at GFest. These are related, since is Morita equivalent to any of its matrix algebras…

I don’t know that this helps deal with the question you mentioned, but I have some hope it may clarify some other things. In any case, I’m even more persuaded that noncommutative geometry can help answer some questions I was wondering about.

May 28, 2008 at 7:10 pm

[…] a somewhat different way than the one I’m accustomed to. For example, as I mentioned when I went to the Groupoidfest conference, there’s a theme in NCG involving groupoids, and algebras of -linear combinations of […]

September 15, 2008 at 7:41 pm

I came across your blog while doing some research on open problems in Groupoids. I noticed some of the things you have posted I did work on in My PhD thesis. I have yet to make it to Groupoid fest. I am working on Constructing the C-star algebra from the topological groupoid. Did you hear any talks on the subject. More particularly the topological groupoid from the Cuntz Graph $E^{\infty}$. If this is your field I would love to discuss some things on the subject.