Noncommutative topology in geometry & physics, lecture 1

0:00 Thank you. Thank you. Thank you. Okay. I'm Sean at Rosenberg, and this is the title of my course. Um, um, um, uh, peut-être que je vais dire, avant de commencer, pour les francophones,

2:30 si c'est plus convenable de poser des questions français, euh, euh, euh, pour le travail. So that's the title of the course, and I've tried to write out notes. I'm not sure whether I'm going to be able to write out everything, but the notes are going to be on the web at that address, so you don't need to furiously copy everything down and you can go look at them at your leisure and in particular I've tried to put the references down so some of the references that I'll mention are some of the same ones that don't have me mentioned yesterday in this class. There'll be a little bit of overlap with what he discussed yesterday. Today's lecture is going to be pretty general, nothing very technical at all. I just want to sort of explain some of the general philosophy of what non-computer topology is and how it gets used for various things. So that's my goal today. And then you'll see the other lectures will deal with more specific topics. There are actually plenty of seats up here. I'm just standing back here because the projector's here. I'm going to talk about the general subject of non-commutative topology, and to a lesser extent, non-commutative geometry, and not so much for their own sake as for the sake of what they can be used for. So I'm going to talk about some applications. In particular, I want to talk about how the philosophy of non-community topology, non-community geometry, can be used for studying problems in ordinary topology

5:00 in the sense that people who call themselves topologists work on. And actually, I should say that I myself have a kind of split personality. So to some extent, I'm a non-community of geography person, but I also to some extent think of myself as a topologist in the usual sense and will try to go back and forth between these two points of view. So this is sort of the basic idea of what I wanted to cover. of this class, the first thing is to sort of explain what noncommutative topology is, what noncommutative geometry is, and what some of the basic techniques are for studying them. Now, this is a program on a trimester on grupoids, so I'm supposed to mention grupoids somewhere, there, so here they are. So I'm going to talk about non-commuter topology, especially group-oids, and how they can be used to study various problems in traditional geometry and topology. So here are some things that ordinary topologists and geometers study. Group actions on manifolds, geometry of collation, topology of stratified spaces, and topology of singular spaces. And I'll explain what all of these things are. Actually, there is some significant overlap between all of these various categories, but I'll explain to you later what I mean by that. At some point I'll talk about specificity theory, why it appears in physics and what it has to do with non-creative topology. I think my point of view will be somewhat different from Jean-Louis, so I don't think that either of our courses will completely subsume the other, but I'll talk about that a little bit here. I put in a reference if you want to see a current reference on what Twisted K-Theory is. There's actually a new preprint by Karubi that just came out last week, or at least he sent it to me last week. It's actually publicly available yet, but you can ask him for a sort of quick description

7:30 This is sort of a convenient reference. Then I want to talk about some applications of non-commutative topology, non-commutative geometry that have shown up in the recent physics literature. Actually, if you go to the PET-TH Physics Archive, and you just do a search on the word non-commutative, you'll find, I believe, thousands of current papers. So there's all sorts of stuff going on, and I can't pretend to give you a survey of anything more than a sort of tiny, tiny fraction of it, but I wanted to talk about some things that I'm aware of that I find interesting, so that'll be probably most in the last lecture. Okay, so today's lecture is on non-commutative topology, and the basic philosophy of non-commutative topology, since we're in Paris, it's appropriate to think about the French classics, so I think really the philosophy of non-commutative topology in some sense goes back to Descartes in 1638, because he was the first person to point out that really you should use techniques of algebra for studying geometry. I mean, at that time, that was quite a new idea, and that's sort of really what underlines everything we're going to talk about today. So, basically, Descartes' idea, sort of translated into modern language, is that you study the geometry of a space by looking at the algebraic properties of functions on that space. And, so in other words, spaces are equivalent but dual to algebras, in some sense, and so non-computed and plot evolved applying this principle in the algebra function, so to speak, is non-commutative.

10:00 Now, there are lots of different kinds of non-commutative geometry. And, for example, one subject of great current interest, which I know practically nothing about, is non-commutative algebraic geometry, where you study non-commutative algebras that are something like algebraic. I mean, non-commutative analogs are the algebras of functions on algebras of regular functions on algebraic varieties. So there, the algebras are typically an Ethereum and they have certain other kinds of nice homological properties and stuff, and there's a lot of interesting stuff going on in that area. As I said, I know nothing about it. I'm not going to talk about it. So when I say non-commutative geometry, I don't, I mean, maybe some of the ideas apply to that too, I just don't know anything about it. But here, I'm going to be talking about non-commutative geometry in the same sense that Joel and Louise mentioned yesterday. In other words, I'm going to talk about C-star algebras and their sub-algebras as the kinds of non-commutative algebras that I'm interested in. And now, there's at least one physicist here, so what would be the reason for restricting attention to C-star algebras, well, C-star algebras are algebras that can be realized as algebras of operators on a helper class, and it seems to be a natural principle of quantum mechanics that observables of quantum mechanical systems have to be operators on overspaces so that, in some sense, forces you to look at this category. Okay, now, this is also the so let's mention something about Borrell's philosophy. I don't know if you can read this back there, but this is from the introduction to Borrell's book that he wrote about a hundred years ago. And a lot of what he said in this little preface here is actually remarkably still not out of date. So let me mention a couple things that he mentions here. So one is the very first sentence,

12:30 which I think is really quite important, just translating for those people probably in the French because you maybe just have trouble reading the French because the French is so small it says that mathematical science is based on observation and experience and in other words, basically his point is that mathematical abstraction shouldn't be abstraction just for the sake of abstraction action for the sense, for the purpose of understanding, organizing principles in the physical world. And he has also, another thing that I think is great that you see, he has a little complaint about new trends in teaching mathematics, so people were complaining even a hundred years ago about that. All right, so these are some things that Jean-Louis mentioned yesterday. So let me just tell you what a C-star algebra is for those people who aren't already familiar with them. C-star algebra is a Bonhoeff algebra, that means it's both a Bonhoeff space and an algebra, and the norm of a product is less than or equal to the product of the norms. You have to have a conjugate, I'm just going to work over C in this course for simplicity. I mean, c-star authors of the reals, as you mentioned, are actually a legitimate subject of interest, and actually they're quite interesting from the point of view of non-community geometry, but the problems involved in dealing with them are more complicated because of the fact that the reals are not out there. So I'm just going to ignore that for now and just talk about complex authors. So you have a conjugate linear involution star. So star is a kind of an abstraction of the adjoined operator for operators on the Hilbert space. So it's conjugate linear and reverses the order of multiplication. And then you have the basic identity that the norm of A star A is equal to the norm of A squared. We're all on this in the algebra. And as Jean-Louis mentioned yesterday, that has lots of very strong implications. for example, it means that if A commutes with A star, if A is normal, in particular if A is self-adjoint,

15:00 then the norm and the spectral radius coincide. So for that reason, you can compute norms totally from the algebraic properties of the algebra. But the algebraic properties of the algebra, I should emphasize and include the involution. It's not just the A as a bonnet algebra, but the star operation as well. Okay, so now there are two basic classical theorems which go back to the 40s that are sort of the foundation of the whole subject. So one is the Galvan-Neumark theorem, which I already mentioned, that if Bonner's star algebra is that it's C-star algebra, if and only if it's isometrically star isomorphic, in other words, isomorphic preserving the star operation, to a star-closed, norm-closed algebra of bounded operators on public space. And the other theorem, which is also due to Galpond, is the fact that every commutative C-star algebra is isomorphic to C0 of x for x of locally compact housework space. So C sub 0 means continuous complex value functions that vanish in infinity. In other words, if you take the one-point compactification of x, these are functions that extend to the one-point compactification and take the value of zero. And in fact, you can even phrase that in categorical language, there's a contravariant equivalence of categories between the category of locally compact house-core spaces and proper maps in the category of commutative C-star algebras and the star on the left axis. So the topology of locally compact house door spaces is the same thing, basically, as the study of commutative C-star algebras. Okay, that should be pretty clear, I think, so far. That's exactly what Jean-Louis mentioned yesterday. Is it possible? No, it doesn't matter. I mean, if the, what is extra, the extra statement you could add is that if the algebra is separable, then x is second countable and vice versa. Okay, now, um, because of that last statement,

17:30 So, you see, once again, locally compact spaces correspond to commutative C-star algebra, so for that reason, non-commutative C-star algebra should, in some sense, correspond to non-commutative spaces. And here are a few basic examples, and again, Jean-Louis We mentioned these yesterday, but let me go over them again. So one of these is the C-star algebra of a locally compact group. So here, what you do is you take the convolution algebra of L1 functions on the group with respect to harm measure and take the largest C-star algebra completion of that. And you should view that as the algebra of functions on a certain non-commutative space, G-hat, which is the unitary dual of G. In other words, it's a set of all unitary representations of G up to equivalence with some extra structure. And if G is locally compactabelian, then this g-hat is actually also a locally compact group. This is the Pontryag and Zuhl group. And as Jean-Louis mentioned yesterday, the Fourier transform sets up an isomorphism between c star g and the functions vanishing infinity on g-hat. So once again, you see that in the Abelian case, this algebra is the algebra of functions on this space. but in a non-community case we sort of take that as a kind of organizing principle even though jihad is not a space in the usual sense it's sort of a non-community so that's example number one another example which is very important in theory part of the reason why it's important is it's just one of the one of the few very non-community through which it's easy to do calculations. And that's a non-commutative torus. So the way this is defined, you take a skew-symmetric N-by-N matrix, capital theta,

20:00 and you take the largest C-SERR algebra that has N unitary generators, U sub J, and you require them to satisfy this sort of Heisenberg commutation relation uj and uk almost commute, but when you commute them across each other, you pick up this factor of eta to pi i theta jk. And you can see that obviously the state of jk has to be skeosimetric because of that relation. And for different matrices, you get different algebras. So there are lots and lots of these things. But if theta is identically zero, then this whole factor goes out, and you just have N unitary generators that commute with each other, and you just get the algebra of continuous functions on the N-torus. So for that reason, this thing is called a non-commutative N-torus, you know, parameterized by the matrix capital theta. So this A theta, you can also see, it's a completion of a certain algebra of non-commutative Laurent polynomials and N variables, because the general element of the algebra looks something like a series expansion like that, and because of the commutation relation, you can always reorder the factors to put them, let's say, in standard order like that. Okay, one case which is especially important is a case when n is equal to 2, but you can write theta in this sort of standard form. And then, this is usually called a little theta instead, and that's called a rotation algebra. It's called an irrational rotation algebra if theta is irrational, and in that case, the algebra is actually simple. Alright, now these non-commutative tori actually are closely related to group C-stair algebras because of the following construction. If you take the group z to the n, the free amelian group on n generators, you can take theta and e to the 2 pi i theta gives you a co-cycle on that group, which defines a certain no-potent central extension, g, of z to the n.

22:30 and A theta is a certain canonical quotient of the group C star algebra of G. Basically, it's the same thing as the part of C star... You see, C star of G, if you have any irreducible representation of G, when you restrict it to T, since T is central, it has to restrict to a character of T, So the dual space of G splits up according to the characters of T. So C star algebra just splits as a direct sum of various pieces. And this thing over here, the rotation of the non-commuted torus is just the piece corresponding to the identity character. Z goes to Z on T. What? What are the other characters? correspond to different choices of theta, actually, it turns out. I mean, all of those are isomorphic to things of the same type also, but the difference of theta is similar to the theta multiplied by whatever the parameter of the character itself is. Okay. Any questions so far? So these are just a few very basic examples. All right. Now, again, since this is a program on grouploids, we're supposed to talk about grouploids. And now, having mentioned groups, we can move on to grouploids. And as Jean-Louis mentioned yesterday, you can take the C-star algebra of a locally compact grouploid with a R-system. So these were defined by Renaud, and this number 10 here is a reference to Renaud's book, which I already mentioned yesterday. And again, there are several ways to think about group voids. One way is to think about G as consisting of the morphisms in a small category in which

25:00 are invertible, and then you have two maps from G to the set of objects of G, the range and the source. What I call range is what John Lee called the target yesterday, so I'm calling R and S instead of T and S, but it's exactly the same thing. All right, so now the simplest case of the groupoid is just an equivalence relation. So in that case, there's one and only one morphism between any two objects. If you have two objects in a set, either they're equivalent or they're not equivalent. R cross S, this isn't a case where all the elements are equivalent to each other, sorry. I mean, in general, R cross S sets up a bijection between G. this isn't I didn't write something in there it sets a bidection between G and the subset of G0 across G0 declined by the FOMOS the simplest example of this which actually does show up in physics is the case of a spinning particle in quantum mechanics so if you have a particle of spin n minus 1 over 2 like the most famous example one-half, so then n is equal to two. Then there are basically two states spin up and spin down in a kind of sort of semi-classical model. There are two states, spin up and spin down. But in quantum mechanics, you're allowed to have transitions between these two states. So then the quantum mechanical algebra of observables is the C-star algebra of the groupoid consisting of two points with the equivalence relation in which the two points are equivalent to each other because you can go back and forth between these two states. So then the algebra of observables is the matrix algebra. Here's a little picture. So here, for particles spin three halves, you actually have four sort of classical states,

27:30 and you're allowed to have transitions between any two of them, and so the groupoid C-star algebra is the matrix algebra. Alright, now another example of the groupoid, which I'm going to use quite a bit, which I think is shown that we didn't mention yesterday we'll probably get to this later in this course is the example of a transformation group. So suppose you have a space X locally compact and you have a locally compact group G and G acts on X. So then you can turn G cross S into a groupoid in which the set of objects and you get a morphism from one point of x to another by means of any group element that sends this point to that point. So therefore, if you think of g, x as the arrow that starts at x, it's given by the group element g, then the source of that group weight element is x, and the target is g dot x. So that's a very important example. And multiplication is fairly obvious. It's this. And the associated groupoid algebra is... so G is the groupoid script G is the groupoid and Roman G is the group that's acting on X and then and then the groupoid C star algebra is just the cross product of the functions on X that C star alpha attacks to the commutative ordinary space X by the action of G. Okay, and in concrete terms, that's the completion of the convolution algebra of how one functions on G cross X. And if X is just a point, this goes out, you have the completion of the convolution algebra of G, and you just get C star G back.

30:00 So in general, this cross product is some kind of mixture and c star of g. In a sense, we'll make more precise later on. Okay, so this is another very basic example of the non-community space and non-community of algebra. And then finally, here's another example. Jean-Louis mentioned these yesterday also. So, this example gives you non-commutative spaces that are not too non-commutative. They're not, in some sense, they're not as non-commutative as the non-commutative torii, at least if they're irrational, because they don't have any simple quotients other than the compact operators. But these are the continuous trace algorithms. So those things were studied by Bell and Bissomier-Douardy. So the paper Bissomier-Douardy was back in the 60s, this famous paper, which is really a great classic. And, alternatively, if you want the contents of this paper, it's almost completely copied over in Dissamier's Seastar Alphabet book, so the paper and the book more or less contain the same material. Um, and these, uh, these manual D algebras, uh, correspond in, in a sort of ring theory literature, or the non-commutative algebra-geometry literature to things called Azamayic algebras. If there are any rank theorists, you might be familiar with the other name. So an algebra of continuous trace is characterized by something called Fell's condition. This goes back to that paper of Fell, which I didn't mention here, but the references you've just seen as well.

32:30 So Fell had started studying these things actually for the purpose of understanding the context in which Fell originally came across them He was trying to understand the structure of the group C-star algebra, SL2C. And he found that the algebra is not, it's a little bit non-commutative, but a big chunk of it satisfied this thing which he called, which is now known as Spell's condition, that the dual space is half-score. The dual space is the set of equivalence classes of aerodustral representations on Hilbert space. And for each point in the dual space, there's an element X in the algebra, which is a local rank 1 projection in the neighborhood of decimal. So that's what characterizes continuous trace algebra. And now if you assume for simplicity that X is second countable, each continuous separable continuous trace algebra can be stabilized and when you stabilize it you get one of these things called a stable continuous trace algebra which is locally isomorphic to the continuous functions with values in the compact operators on an infinite dimensional separable Hilbert space but the algebra is sort of globally twisted up so globally it sort of looks like this but globally there can be some twisting that twisting can be explained as follows you have a locally trivial bundle of algebras over x and the fibers of these algebras are isomorphic p and the structure group of the bundle is the automorphism group in other words, a group of invertible star isomorphisms from K to itself, and that's just the same thing, since all those biomorphisms are induced by unitary operators in the Hilbert space, as the projected unitary group. That's the unitary group divided out by the scalars of norm 1. Why the projected unitary group? Well, that's because if you conjugate by a unitary,

35:00 and you multiply that unitary by a scalar, then the operation of conjugation So it's not the full unitary group, but the quotient of the unitaries by the scalars that matter. So that's the biomorphism group that compacts. And the unitary group of an infinite dimensional Hilward space in the natural topology is actually contractible. So this pathological group is actually contractible. It didn't have a P there. But you divide it out by a circle. So what you have is the topology of a contractible space divided out by a free proper action of a circle. So that's a KZ2 space, in other words, a space with one non-zero homotopy group in dimension 2. So therefore, bundles of this form are classified by maps from x into the classifying space of this group. When you take a classifying space of a group that shifts the homotopy groups up in dimension by 1, EPUH is actually KZ3 space, and maps from X into KZ3 are just H3. So these bundles are classified by an invariant H3, and that's called the DCM-U-IB invariant. And that turns out to be a complete classification. Incidentally, since this came up yesterday, people asked, well, why this word's stable? The problem is, if you have a continuous trace output, even if the fibers are infinite dimensional, it's not necessarily stable, and the continuous field of algebers that's associated to it is also not necessarily locally trivial. That was pointed out in Easton E.I.P.U.D.'s paper, and that's part of the reason why the paper is so long, because they dealt with all sorts of complications like that. But after you stabilize that difficulty, it goes away. Okay, so let's see, any questions up to this point? So just to summarize where we are, I've introduced the notion of a non-commutative space. So non-commutative topology will be, by definition, the study of such spaces. And now the question is, well, what does it mean to study these non-commutative spaces? And you want to study them up to certain equivalence relations,

37:30 the natural equivalence relations. So, here are some natural equivalence relations which you can consider. They're sort of in increasing order of complexity, and depending on the problem you're interested in, you need different equivalence relations. So, this is now going to get a little bit more technical, but it's really essential to go over this because otherwise doesn't really have any content. So one equivalence relation would be star isomorphism of C-star algebras. So this sort of corresponds to studying topological spaces up to homeomorphism, which is usually a very, very difficult problem. Now, for separable C-star algebras, something which is slightly easier, we already saw this come up in the case that the continuous trace algebra is a stable isomorphism. In other words, you tensor your algebras to the compacts first. This washes away a certain amount of structure, which for many purposes doesn't matter anyway. So, for example, in the case of continuous trace algebras, after you tensor the K, the algebra becomes locally trivial, which is a big technical simplification. so that for that reason you can study this equivalence relation of stable isomorphism and isomorphism after tensoring the K and it turns out by theorem of brown, green, and recall that that coincides with the next equivalence relation which is this one right underneath these a priori these two equivalence relations are completely different so this is actually a kind of surprising theorem that they turn out to be the same so that reference there is the paper of brown, green, and recall The paper's actually rather short because actually the proof of the theorem is not entirely contained in this paper. What this paper does is it reduces it down to another paper of Brown, which is right adjacent to this in the same issue. And then you need another ten paper pages to finish it off. So that's the Brown-Green-Riefel theorem. So those two equivalence relations are really the same. So what's Mariae equivalence?

40:00 This is, I put the word strong in parentheses because when Riefel first introduced this he added that word. Nowadays it's mostly, we mostly leave the word strong off because when you're dealing with C-star algebras is that the other version without the strong turns out to be totally useless, and nobody gives it anymore. But also, you might want to put the word strong in to distinguish it from purely algebraic marine equivalents, which is something that ring theorists study, which was introduced by Maria quite a long time ago in ring theory. something about the history of this. Morita was interested in the problem, this is a purely algebraic problem, suppose you have two rings, when are their categories of representations equivalent to each other? And what will implement such an equivalence? So he actually gave a complete answer to that problem, and that answer was in terms of something we now know as Morita equivalence, and what Rieffel did was that he took Maria's paper and modified it for the context of C-star algebras. Since you have to keep track of topologies and so on, it's not a completely straightforward thing to do because you have to put the right analytical conditions on also. But when you do that, you get this notion of Maria equivalence for C-star algebras, and basically what it says is that two algebras are very equivalent if there's something called an equivalence by-module. So that's an A-B by-module, A-X on one side, B-X on the other, and by tensoring with this thing in one direction or the other way, you get equivalences of categories between the categories of representations of A and the representations of B, and you go back and forth. This tense of product, of course, is completed in a certain topology, which I'm ignoring here, and there are certain technical complications that I haven't talked about. And this notion was due to refill. And if you do this correctly, then this condition for separable options turns out to be . And as I said, this was originally due to refill, and refill wrote a nice summary of

42:30 but in the Proceedings of the Kingston Summer Institute. So that's in Proceedings of Symposium, Pure Math, Volume 38, Part 1, published by ANS. And there's a nice little summary in the notion of Marie Equivalence, if you want to read more about it. That turns out to be one of the really key equivalence relations for non-community spaces. We'll see what Maria equivalence means in the context of interpolation. So I'll bring it back to that. Questions on that so far? Sorry, this is probably a very badly posed question, but this definition of Maria equivalence for the map spaces, does this imply that the maps have to be vertical? What maps are you talking about? Well, it's a general definition. I mean, these are algebras of functions on the space. Right. Well, A and B even think of as algebras of functions on two different non-community spaces. On two different non-community spaces, yeah. So in order to satisfy this condition of the marine equivalents, what does that actually say about the general notion of mapping between the generalized spaces? Oh, no, it's much weaker than, for example, the homeomorphism of the spaceships. Yeah, and when you say it's much weaker, is it in the sense that it's much more restrictive? No, no, no, there are a lot more marine equivalences than there are isomorphism. Oh, okay, okay, okay. Any isomorphism is a marine equivalence, but not conversely. But not conversely, okay. I'm sorry, it's a bad question. For example, we talked before about this very simple groupoid algebra. if you just have a finite discrete group weight in which all points are equivalent to each other, and it's doing a full equivalence relation on n points, then any two of these are already equivalent, even though the spaces are obviously quite different. I see. Okay, right. So here the algebra is m2, here the algebra is m4, but these algebras are going to be equivalent to each other. Okay, good. Thanks. Alternatively, they're stably isomorphic. tensor k is just isomorphic to k again, and then 4 tensor k is isomorphic to k again, so that sort of explicit, sort of shows you explicitly what the stable isomorphism is.

45:00 But the isomorphism between this k and this k doesn't give you an isomorphism between this and this, obviously not. No, no, no, no, okay. Okay, that's it. What's the head version of Moritian? Well, that had to do with... That had to do with something... I mean, originally, Riefel studied this notion of Moritian Pobles in the context of anointing algebras, And then, out of that, you constructed something through C-Stero-algebraism. But it turns out, if you set up the theory the wrong way, you get something totally useless. So, there's no point in discussing that. You can read it in Riegel's original paper. It's a paper in the Journal of Pure and Applied Algebra, but nobody's ever used it again. Whereas, strong reading equivalents now, you know, shows up in almost every paper in C-Stero-algebraism Now, since those two equivalence relations are the same, you might wonder why you even bother mentioning both of them. But that's because, and this is something that's important, just knowing that there's a stable isomorphism between two algebras is not really as useful as having an explicit equivalence by module between them. So for many purposes, even though these two notions are equivalent, and this one's easier to state, this one's actually more useful. So for people who actually work in the subject, this is for people who actually use more than that. And that's because having the explicit bimodule round is a very useful thing, because that enables you to carry structure back and forth in one algebra or the other. I mean, you can read about that in the brown-green region paper. It's not terribly important, I would say. I mean, part of the problem is you have to... The k is, in some sense, too small. This k is the algebra of compact operators on a separable inter-dimensional Hilbert space, so we can't expect it to seem non-separable phenomena, so to speak.

47:30 Okay, so now let's think back to where we started. We were trying to figure out how to carry topology over into the non-commutative world. So another equivalence relation for topological spaces, which topologists study, probably much more than homeomorphism, is homotopy-quotalism. So he studied the notion of homotopy, of maps between topological spaces, and homotopy-quotalism between spaces. For example, homotopy theorists deal almost entirely with spaces modulo-homotopy. They don't really care about spaces modulo-homeomorphism. So the notion of homotopy goes over very straightforwardly. Two star homomorphisms from one algebra to another are homotopic if basically there's a one-parameter family of star homomorphisms and they interpolate between them. And that's the same as saying that there's a homomorphism from A to the continuous functions on the unit interval into B, which restricts to f sub j for j equal 0 or j equal 1. So, you know, here you can obviously evaluate any of the points of the interval. And by evaluating different points, you get different homomorphisms. So that's the notion of homotopy, and algebers are homotopy equivalent if they're star homomorphisms from A to B and from B back to A so that the equivalences in both directions are homotopic together. So that's a straightforward translation of the definition for topological spaces. Now, you can combine this with what we had before and consider stable Hematop equivalence. Actually, the word stable has more than one meaning, and this is what should really be called matrix-stable Hematop equivalence in the matrix sense, in other words, tensing with k, which is sort of the dimensional matrix algebra, rather than suspension stability, which is stability after you suspend many times. That operation is also interesting, and the two kinds of stability are not exactly the same,

50:00 but they're sort of related in a complicated way to term it. Anyway, so you could say that a and b are stably homotopy, covalent, if after you stabilize, they become homotoply equivalents. So that's another useful equivalence relation. It sort of blends those two. And finally, now you have a kind of strange equivalence relation called KK equivalence. So what is that? You can describe it the following way. a category called KK that you then construct out of the separable C-star algebras. So take a category of separable C-star algebras for the star homomorphisms between them build in the operations of homotopy invariance, stability, and split exactness. And it turns out so this is sort of a very complicated theorem of Pixin together with this complicated theory of Cospera. So this is not at all easy to set up. It turns out when you divide half by those equivalence relations, on the category of separable C-star algorithms, you get a triangulated abelian category. It's not abelian category. It's a triangulated category added a category called KK. and two alphabets A and B are called KK equivalent if they become isomorphic in this category it's a kind of very weak notion of equivalence but it implies for example that the groups have the same K theory not only they have the same K theory but a specific KK equivalent so that's an element of the KK category that sends A to B that has an inverse going back. So if you have such a thing, then that specific element gives you an explicit isomorphism between the k-theory of A and k-theory of B with arbitrary coefficients, and not just with integral coefficients with, you know, c-like k coefficients for all k, et cetera. So this is, if you're just interested in k-theory,

52:30 then for k-theory purposes, k-k equivalence is really sort of the right equivalence relation. Is there a question now for smooth selection? Um, sort of. I mean, for algebras that aren't C-star algebras, that are more general kinds of topological noncommutative algebras, like smooth subalgebras of C-star algebras, there's a different triangulated category, which is called not capital KK, but little kk. It was introduced by Kutz and Meyer and various other people. And some of the same things work there, but the relationship between little KK and capital KK is very, very complicated. That's still actually a major subject of the current research. Because little KK has all the same formal properties as capital KK, that just has one major drawback, and that is nothing whatsoever is computable. Whereas for capital KK, there's very strong structure theories. So, for example, there's a theorem of Mime with Schochit that completely characterizes KK equivalents on a very large category. See, sort of, I'll talk about that a little later. Okay, so that's about it for these equivalence relations. Any more questions about equivalence relations? You know, this is not an exhaustive list, but these are a few things to get you started. So KK equivalence is given by a bind module, but that bind module is not necessarily a meridian equivalence bind module. A meridian equivalence by-module is a KK by-module that gives a KK equivalence. So this is weaker than meridian equivalence, but not every KK equivalence comes from meridian equivalence. Very far from it, in fact. Okay, so now let's go back to the commutative case and figure out what are all these different equivalence relations in the commutative case. So, if you have a commutative C-star algebra, by Galvin's theorem, it's a form C-zero of X, where X is locally compact.

55:00 So, what are all these equivalence relations? If you have isomorphism, stable isomorphism, read equivalence, it turns out all of these coincide for a billion C-star algebras. And they just give you homeomorphism of the X. So now the C0 of X and C0 of Y are star isomorphic or stably isomorphic or marine equivalent if and only if X is homeomorphic or Y. So, so far I haven't introduced anything that's any different from just homeomorphism for spaces. Homeomorphism for spaces. These currents are with homeomorphism for spaces. Excuse me. Yeah. Does this first statement also contain a few known second capital? The first statement, yes. But KK equivalents, for KK equivalents, you need a second category. So if you want to study KK equivalents, it turns out the two abelian C-stera algebras are k-k equivalents exactly when they have the same complex k-theory groups. So that's a theorem of mine with Schottet in the new journal. Here's the reference. This paper over here in 1987. So, for that reason, KK equivalence on ordinary space is a very, very weak equivalence relation. So, for example, if you have two finite CW complexes with torsion-free homology, then the k-theory and the homology basically coincide so k-k x and y are k-k equivalent if and only if the sum of the many numbers for x is equal to the sum of the many numbers for y and the Euler characteristic of x is equal to the Euler characteristic of y in that case

57:30 see the sum of the many numbers is the sum of the rank of K0 and K1, and the other characteristic is the difference of the rank of K0 and K1. So if you know both of those numbers, you know the rank of K0, you know the rank of K1. If everything's portion free, that determines K0 and K1, so then they're a KK equivalent. So that means the KK equivalents for finite-seedell big complex have just been determined by two managers if everything's portion free, just these So that's a very, very weak kind of equivalence relation. For example, two spheres of different dimensions are KK equivalent to the parodies of the dimensions of the same. So, that's kind of the usual topology. That's a rather strange equivalence relation. No legitimate topologist would ever So let's see. That's been an hour. Maybe we should take a five-minute break so we can escape if we want to. We'll go to the toilet and some coffee and we'll continue in about five minutes. I said five because I know it's going to be serious. Thank you.