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Question: What are the applications of operator algebras to other areas?

More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by using operator algebras' techniques, or which are corollaries of operator algebras' theorems.

I ask this question for seeing how operator algebras are connected to the other mathematical areas and for better understanding what concrete role it is currently playing in the mathematical world.

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    $\begingroup$ Nice to hear this from a specialist in operator algebras. :) When at the Last Judgment we will be accused of snobbery, we will point at Sébastien Palcoux and say: "Here's a conscientious man among us!" $\endgroup$ – Sergei Akbarov Mar 22 '15 at 9:31
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    $\begingroup$ Ergodic theory, both topological and measure-theoretic (there is an enormous literature on C*- and W*-algebraic applications in ergodic theory). $\endgroup$ – David Handelman Mar 22 '15 at 16:02
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    $\begingroup$ We have two Fields medals: Jones (connections between von Neumann algebras, physics, and knot theory) and Connes (a generalization of the Atiyah-Singer index theorem to foliated manifolds). Other obvious answers that spring to mind are applications to group representations and quantum statistical physics. If the question is reopened I will add this as an answer. $\endgroup$ – Nik Weaver Mar 22 '15 at 17:03
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    $\begingroup$ @Vincent: Probably the part about "I've heard some things about the reputation of this area" and the placement of "operator algebras" in quotes gave this question a very different tone. I will edit the question to give it a more neutral tone. $\endgroup$ – Nik Weaver Mar 22 '15 at 18:48
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    $\begingroup$ Re the OP' comment above: I think it is a vast stretch to claim that Atiyah is "close to operator algebras", unless "close to" means "has done good work related to", in which case Atiyah is a dense point in mathematics $\endgroup$ – Yemon Choi Mar 23 '15 at 14:07
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I'm a little puzzled by the tone of the original question. My personal view is that operator algebras are intrinsically interesting, and if there are good applications to other fields, so much the better ... I think this is a pretty common attitude, probably among people in most areas of pure math.

Anyway, I am not the most qualified to describe some of these applications, but here are a few of the main ones.

(1) Connes' index theorem for foliated manifolds. For instance see here and here. Connes 1982 Fields medal was awarded in part for his work on foliations.

(2) Jones' work connecting von Neumann algebras and geometric topology, which gave rise to a new knot invariant. See here for a nice overview. Jones was awarded the Fields medal in 1990 in part for this work.

(3) Mathematical physics. Many connections, some more established and some more conjectural. The KMS theory is surely one of the standouts.

(4) An early motivation was the theory of group representations, e.g., J. Dixmier, Anneaux d'operateurs et representations des groups, Seminaire Bourbaki, Vol. 1 (1995), 331-336.

(5) As mkreisel mentions, there are applications to the Novikov conjecture. See G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1977), 147-201.

(6) The Kadison-Singer problem originally arose as a problem in operator algebras. It now has connections to many other areas (harmonic analysis, Banach space theory, signal analysis, ...).

I'm sure I am forgetting some important ones.

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    $\begingroup$ While I certainly agree that operator algebras are intrinsically interesting, I have encountered some negativity towards the subject from mathematicians who work in other fields. In the original question, I understood the author as asking for an explanation of this negativity. But perhaps you have never encountered it and it is just my personal misfortune (or perhaps this kind of negativity towards other fields is more common than I expected, though I hope that isn't the case). $\endgroup$ – mkreisel Mar 23 '15 at 3:22
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    $\begingroup$ Huh. No, I haven't encountered this. But no doubt negativity towards other fields is fairly common --- I can be guilty of this myself --- though I suppose I feel it's something one shouldn't take too seriously. That's just human nature. $\endgroup$ – Nik Weaver Mar 23 '15 at 3:56
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    $\begingroup$ Being myself convinced in the interest of operator algebras (intrinsic and extrinsic), my first reaction about these rumors was to think that my interest for the operator algebras blinded me to the reality of the feelings of other mathematicians for it, as if it was becoming an ivory tower. This is what pushed me to post this question (which originally asked in preliminary, some explanations about that). Now I am satisfied by your explanation on the human nature, stating that such phenomena are fairly common. I think this view will be confirmed by mathematicians outside of operator algebras. $\endgroup$ – Sebastien Palcoux Mar 23 '15 at 5:59
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    $\begingroup$ Sébastien, the problem exists. Not only in this concrete field, but in many fields in mathematics. You were right when asking this question, this is a normal reaction of a human (contrary to robot). The discussions like this must be common in science, otherwise this activity turns into a religion (where references to "tone" replace doubts :). $\endgroup$ – Sergei Akbarov Mar 23 '15 at 7:48
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My guess as to why operator algebras gets a bad rap: In my mind there are certain areas which mathematicians are inherently drawn to, chief among them number theory, topology/geometry, and any area which has a physical interpretation/application (pure analysis, PDE, etc). Mathematicians also tend to like problems which have a low barrier of entry, and so appreciate them even if they don't immediately contribute to understanding the big areas.

Operator algebras sits at the other polar extreme. There is a high barrier of entry (functional analysis, algebra, algebraic topology, geometry, physics) which would be forgivable if it had led to the solution of deep problems in one of the big areas. Unfortunately, the big problems it has contributed to haven't been finished off in an entirely satisfactory way (I'll explain some examples below). Part of this is attributable to the attitudes of those working in the field. For example, much of the work in noncommutative geometry is about finding the right notion of a "quantum geometry" which can involve quite a bit of wheedling with definitions and exploring examples. Many papers are about giving good definitions, the kind of bedrock stuff that's been established in e.g. topology/geometry for decades now. And even simple examples can be difficult; it's still not understood how to appropriately extend Riemannian geometry to the noncommutative torus.

So the field is still young, requires a high entry barrier, and the impetus to break that barrier isn't high enough for those outside the field because there aren't big name fundamental problems that operator algebras have solved. I know this is somewhat reductive, but it's my best attempt for now.

Finally some examples of the successes of operator algebras (please correct me if you see mistakes here or have additions):

All recent results on the Novikov Conjecture that I know of use operator algebras in an essential way, and often prove the stronger Baum-Connes Conjecture (see e.g. work of Gennadi Kasparov and Vincent Lafforgue). At this point, we know that the Baum-Connes Conjecture holds for many classes of groups (e.g. hyperbolic, amenable, and even some property-T groups).

Elliott's program for classifying C^*-algebras using K-theoretic invariants has found wide applications in classifying dynamical systems. I get the sense that dynamicists often rephrase these K-theoretic invariants in more familiar dynamical language, which can obscure the influence of operator algebras, but I believe there are some cases where you really do need to use the language of K-theory to fully describe classification results. Here are two recent papers: http://arxiv.org/abs/1406.2382, http://arxiv.org/abs/1502.06658. Generally one can look for work of George Elliott, N. Christopher Phillips and papers by Giordano, Putnam, and Skau.

I should also add that there is a large body of work on quasicrystals and the gap labeling conjecture of Jean Bellissard where operator algebras have played a major role. References are easily found on Bellissard's website.

The work of Packer/Rieffel/Luef shows many connections between frame theory and operator algebras. On a related note, some of the widest reaching results on the HRT Conjecture were proven by Linnell using operator algebraic methods (see http://arxiv.org/abs/math/9807057). This result is in a similar spirit to the zero-divisor conjecture for group rings, which can also be approached using operator algebraic methods (see Luck's work on L^2-torsion and this paper http://arxiv.org/abs/1202.1213).

There's also Connes' work on foliations, but unfortunately I don't know enough to describe that in any detail.

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  • $\begingroup$ Can you give some sample results or references for classification results in dynamical systems theory? $\endgroup$ – ThiKu Mar 22 '15 at 17:54
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    $\begingroup$ It is sweeping, presumptuous, and opinionated, and I did not mean to put words in anyone's mouth. I don't really work on any of these areas myself, this is just my ambient sense of the mathematical community at large. It's hard to understand and describe why mathematicians find a certain subject or result (un)interesting. $\endgroup$ – mkreisel Mar 23 '15 at 15:42
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    $\begingroup$ Yet I have consistently encountered people outside operator algebras (and NC geometry in particular) who ask "Well what's it all for? Do I really have to learn it?" to the point where I begin to feel guilty if a result I can prove is "internal" to NC geometry. I don't know whether people in number theory or analysis feel the same way. I was attempting to explain why NC geometry may inspire this more than other fields, but perhaps I'm wrong and it doesn't. $\endgroup$ – mkreisel Mar 23 '15 at 15:49
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    $\begingroup$ @mkreisel (and whoever else might be reading), here are some words of Rota: “What can you prove with exterior algebra that you cannot prove without it”? Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz’ theory of distributions, idèles and Grothendieck’s schemes, to mention only a few. A proper retort might be: “You are right. There is nothing in yesterday’s mathematics that could not also be proved without it. (cont) $\endgroup$ – Todd Trimble Mar 23 '15 at 16:56
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    $\begingroup$ (cont.) "Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures.” Now: instead of 'exterior algebra', substitute 'operator algebras and noncommutative geometry', and I think Rota's point still applies. $\endgroup$ – Todd Trimble Mar 23 '15 at 16:58

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