### c*-algebras

In the past couple of weeks, Masoud Khalkhali and I have been reading and discussing this paper by Marcolli and Al-Yasry. Along the way, I’ve been explaining some things I know about bicategories, spans, cospans and cobordisms, and so on, while Masoud has been explaining to me some of the basic ideas of noncommutative geometry, and (today) K-theory and cyclic cohomology. I find the paper pretty interesting, especially with a bit of that background help to identify and understand the main points. Noncommutative geometry is fairly new to me, but a lot of the material that goes into it turns out to be familiar stuff bearing unfamiliar names, or looked at in 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 $\mathbb{C}$-linear combinations of “elements” in a groupoid. But these “elements” are actually morphisms, and this picture is commonly drawn without objects at all. I’ve mentioned before some ideas for how to deal with this (roughly: $\mathbb{C}$ is easy to confuse with the algebra of $1 \times 1$ matrices over $\mathbb{C}$), but anything special I have to say about that is something I’ll hide under my hat for the moment.

I must say that, though some aspects of how people talk about it, like the one I just mentioned, seem a bit off, to my mind, I like NCG in many respects. One is the way it ties in to ideas I know a bit about from the physics end of things, such as algebras of operators on Hilbert spaces. People talk about Hamiltonians, concepts of time-evolution, creation and annihilation operators, and so on in the algebras that are supposed to represent spaces. I don’t yet understand how this all fits together, but it’s definitely appealing.

Another good thing about NCG is the clever elegance of Connes’ original idea of yet another way to generalize the concept “space”. Namely, there was already a duality between spaces (in the usual sense) and commutative algebras (of functions on spaces), so generalizing to noncommutative algebras should give corresponding concepts of “spaces” which are different from all the usual ones in fairly profound ways. I’m assured, though I don’t really know how it all works, that one can do all sorts of things with these “spaces”, such as finding their volumes, defining derivatives of functions on them, and so on. They do lack some qualities traditionally associated with space – for instance, many of them don’t have many, or in some cases any, points. But then, “point” is a dubious concept to begin with, if you want a framework for physics – nobody’s ever seen one, physically, and it’s not clear to me what seeing one would consist of…

(As an aside – this is different from other versions of “pointless” topology, such as the passage from ordinary topologies to, sites in the sense of Grothendieck. The notion of “space” went through some fairly serious mutations during the 20th century: from Einstein’s two theories of relativity, to these and other mathematicians’ generalizations, the concept of “space” has turned out to be either very problematic, or wonderfully flexible. A neat book is Max Jammer’s “Concepts of Space“: though it focuses on physics and stops in the 1930’s, you get to appreciate how this concept gradually came together out of folk concepts, went through several very different stages, and in the 20th century started to be warped out of all recognition. It’s as if – to adapt Dan Dennett – “their word for milk became our word for health”.I would like to see a comparable history of mathematicians’ more various concepts, covering more of the 20th century. Plus, one could probably write a less Eurocentric genealogy nowadays than Jammer did in 1954.)

Anyway, what I’d like to say about the Marcolli and Al-Yasry paper at the moment has to do with the setup, rather than the later parts, which are also interesting. This has to do with the idea of a correspondence between noncommutative spaces. Masoud explained to me that, related to the matter of not having many points, such “spaces” also tend to be short on honest-to-goodness maps between them. Instead, it seems that people often use correspondences. Using that duality to replace spaces with algebras, a recurring idea is to think of a category where morphism from algebra $A$ to algebra $B$ is not a map, but a left-right $(A,B)$-bimodule, $_AM_B$. This is similar to the business of making categories of spans.

Let me describe briefly what Marcolli and Al-Yasry describe in the paper. They actually have a 2-category. It has:

Objects: An object is a copy of the 3-sphere $S^3$ with an embedded graph $G$.

Morphisms: A morphism is a span of branched covers of 3-manifolds over $S^3$:

$G_1 \subset S^3 \stackrel{\pi_1}{\longleftarrow} M \stackrel{\pi_2}{\longrightarrow} S^3 \supset G_2$

such that each of the maps $\pi_i$ is branched over a graph containing $G_i$ (perhaps strictly). In fact, as they point out, there’s a theorem (due to Alexander) proving that ANY 3-manifold $M$ can be realized as a branched cover over the 3-sphere, branched at some graph (though perhaps not including a given $G$, and certainly not uniquely).

2-Morphisms: A 2-morphism between morphisms $M_1$ and $M_2$ (together with their $\pi$ maps) is a cobordism $M_1 \rightarrow W \leftarrow M_2$, in a way that’s compatible with the structure of the $lateux M_i$ as branched covers of the 3-sphere. The $M_i$ are being included as components of the boundary $\partial W$ – I’m writing it this way to emphasize that a cobordism is a kind of cospan. Here, it’s a cospan between spans.

This is somewhat familiar to me, though I’d been thinking mostly about examples of cospans between cospans – in fact, thinking of both as cobordisms. From a categorical point of view, this is very similar, except that with spans you compose not by gluing along a shared boundary, but taking a fibred product over one of the objects (in this case, one of the spheres). Abstractly, these are dual – one is a pushout, and the other is a pullback – but in practice, they look quite different.

However, this higher-categorical stuff can be put aside temporarily – they get back to it later, but to start with, they just collapse all the $hom$-categories into $hom$-sets by taking morphisms to be connected components of the categories. That is, they think about taking morphisms to be cobordism classes of manifolds (in a setting where both manifolds and cobordisms have some branched-covering information hanging around that needs to be respected – they’re supposed to be morphisms, after all).

So the result is a category. Because they’re writing for noncommutative geometry people, who are happy with the word “groupoid” but not “category”, they actually call it a “semigroupoid” – but as they point out, “semigroupoid” is essentially a synonym for (small) “category”.

Apparently it’s quite common in NCG to do certain things with groupoids $\mathcal{G}$ – like taking the groupoid algebra $\mathbb{C}[\mathcal{G}]$ of $\mathbb{C}$-linear combinations of morphisms, with a product that comes from multiplying coefficients and composing morphisms whenever possible. The corresponding general thing is a categorical algebra. There are several quantum-mechanical-flavoured things that can be done with it. One is to let it act as an algebra of operators on a Hilbert space.

This is, again, a fairly standard business. The way it works is to define a Hilbert space $\mathcal{H}(G)$ at each object $G$ of the category, which has a basis consisting of all morphisms whose source is $G$. Then the algebra acts on this, since any morphism $M'$ which can be post-composed with one $M$ starting at $G$ acts (by composition) to give a new morphism $M' \circ M$ starting at $G$ – that is, it acts on basis elements of $\mathcal{H}(G)$ to give new ones. Extending linearly, algebra elements (combinations of morphisms) also act on $\mathcal{H}(G)$.

So this gives, at each object $G$, an algebra of operators acting on a Hilbert space $\mathcal{H}(G)$ – the main components of a noncommutative space (actually, these need to be defined by a spectral triple: the missing ingredient in this description is a special Dirac operator). Furthermore, the morphisms (which in this case are, remember, given by those spans of branched covers) give correspondences between these.

Anyway, I don’t really grasp the big picture this fits into, but reading this paper with Masoud is interesting. It ties into a number of things I’ve already thought about, but also suggests all sorts of connections with other topics and opportunities to learn some new ideas. That’s nice, because although I still have plenty of work to do getting papers written up on work already done, I was starting to feel a little bit narrowly focused.

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 $l^2$, the shift operators can be taken as operators on a Hilbert space. So in particular, they generate an algebra of operators – a $C^*$-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 $C^*$-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 $C^*$-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 $C^*$-algebras coming from transformation groups: situations where a (locally compact topological) group $G$ acts on a (locally compact Hausdorff) space $X$. These can automatically be thought of as groupoids, taking objects to be points in the space, and morphisms from $x \in X$ to $y \in X$ to be group elements whose action takes $x$ to $y$. Now as for $C^*$-algebras, you can build them by taking algebras $C_c(X \times G)$ of compactly supported complex functions on $X$. This becomes an algebra with the convolution product, given by integrating over the group (so we’re assuming $G$ has a nice invariant measure like Haar measure on a Lie group):

$f \star g (x,t) = \int_G f(x,s)g(xs,s^{-1}t) ds$

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 $G$ of all morphisms, with $G^{(2)} \subset G \times G$ being the set of composable pairs. Given a topology on $G$, this $G^{(2)}$ 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 $x$ (with $x = x^{-1} = x^2$), which again gets the subspace topology automatically (which makes source and target maps continuous).

Then we’d like to again define a $C^*$-algebra on $G$ 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 $G$ (the set of all morphisms), one for each object: $\{ \lambda^{u} \}$, which we’ll use to do convolution at the $x \in X$ which correspond to the object $u$. The measure $\lambda^{u}$ is supported on the component of the object $u$. The whole system needs to have some nice properties. One is that for any function $f$, the function taking $u$ to the integral of $f$ with respect to $\lambda^{u}$ should be in $C_c(G)$9. The other is that $\lambda$ is equivariant, in the sense that if $x : u \rightarrow v$,

$\int f(xy) d \lambda^{u}(y) = \int f(y) d \lambda^{v}(y)$

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

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 $G$ is locally compact, Hausdorff, and so on, we get $C_c(G)$, the set of smooth, compactly supported complex functions with a convolution product:

$f \star g (y) = \int f(yx)g(x^{-1})d \lambda^{s(y)}(x)$

(where $s(y)$ is the source object of the morphism $y$). 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 $C*$-algebras in their own right, or using groupoids to think of already existing algebras as coming from this setup for some groupoid $G$. 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 $C^*$-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.

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