George Elliott on the Classification of C*-algebras

Emerging Research Fronts Commentary, August 2011

George Elliott
Left to right: Guihua Gong, George Elliott, Liangqing Li.

Article: On the classification of simple inductive limit C*-algebras, II: The isomorphism theorem

Authors: Elliott, GA;Gong, GH;Li, LQ
Journal: INVENT MATH, 168 (2): 249-320, MAY 2007
Addresses: Univ Toronto, Dept Math, Toronto, ON M5S 2E4, Canada.
Hebei Normal Univ, Dept Math, Shijiazhuang, Peoples R China.
Univ Puerto Rico, Dept Math, Rio Piedras, PR 00931 USA.

George Elliott talks with and answers a few questions about this month's Emerging Research Front paper in the field of Mathematics.

SW: Why do you think your paper is highly cited?

The paper is a landmark in a line of inquiry begun twenty years ago—indeed, at a simpler level, forty years ago.

My two groundbreaking papers on this subject, in 1976 and almost twenty years later in 1993, were considered surprising, and led to work by many others in the last twenty years, of which the present paper by Gong, Li, and me may be considered in some sense the culmination. (A parallel stream of work led to another highly cited result, due to Kirchberg and Phillips. Both of these streams of work have since been pursued further and also come closer together.)

SW: Does it describe a new discovery, methodology, or synthesis of knowledge?

My 1976 paper (in the Journal of Algebra) reported the discovery that it was possible to classify a broad class of C*-algebras (Bratteli's approximately finite-dimensional, or AF, C*-algebras) using a very old and familiar invariant, or label, introduced by Murray and von Neumann at the very beginning of the subject of operator algebras in Hilbert space, forty years earlier. (This in turn was a natural outgrowth of quantum mechanics, based on the non-commuting nature of the observables appearing in the Heisenberg Uncertainty Principle.)

Bratteli had given a classification of AF algebras in terms of the data coming into their construction, which was useful but conceptually much less elegant, and more special, as my invariant was defined for an arbitrary algebra (not just an AF algebra).

In my 1993 paper (in Crelle's Journal), I showed how important my abstract approach to the AF algebra classification was by extending this classification to a considerably larger class of algebras (real rank zero AT algebras)—by introducing an additional abstract invariant, or label, again a familiar object known to be closely related to the invariant I used for AF algebras.

In terms of standard terminology the invariants I used were the even and odd parts of what was called K-theory. My work on AF algebras had in fact begun the transfer of K-theory from geometry, topology, algebra, and number theory back to operator algebras—where it had originated (with Murray and von Neumann) before being, with very fruitful consequences, absorbed by these other branches of mathematics.

"There is perhaps a modest implication in the context of society as a whole, given the deep-lying tendency of people to wish to classify things, events, and phenomena—and even other people!"

The 2007 Inventiones paper with Gong and Li, roughly speaking, extended this classification to the largest class for which the invariants I introduced could be effective, and which was still disjoint from the Kirchberg-Phillips class—the classification of which also used the same invariants. (The two classes, that considered by Gong, Li, and me, and that considered by Kirchberg and Phillips, were as different as finite and infinite numbers, although it is possible to state the two results in a common framework.)

SW: Would you summarize the significance of your paper in layman's terms?

The answer to the preceding question is pertinent to this, as the present paper concerns, in effect, all three categories mentioned there: the discovery of a new phenomenon (that an extremely large class of C*-algebras—i.e., of collections of Hilbert space operators closed under the natural operations—is extremely well behaved—which was a complete surprise); the development of a new methodology (the use of functorial invariants, or labels, to effect the classification—never before had a classification of any family of objects depended on such a sophisticated labeling process—in which two objects which were essentially the same might have labels which were not the same, but only essentially the same, in the same rather subtle sense as for the original objects, but in a considerably simpler setting; the theory of AF algebras was a revolution!—carried further within the theory of C*-algebras in the later papers in question, but also carried over to completely different classes of objects); and, finally, the 2007 paper with Gong and Li represented a far-reaching synthesis of dozens of papers—involving the three of us but also many others—developing the original ideas of the earlier two papers of mine referred to into a complex literature. (It should also be said that this final paper necessarily at the same time solved a number of new technical problems.)

From a general point of view, the C*-algebra classification could conceivably turn out to be as important as well-known earlier classifications—such as the Linnaeus classification of biological species, the Mendeleev periodic table of the chemical elements, the standard model organization of elementary particles in physics, or, in mathematics, the classification of the finite simple groups.

A mathematical classification which is more analogous to that (still incomplete) of separable amenable C*-algebras is the famous (now complete) classification of separable amenable von Neumann algebras, due to Connes and others (which was obtained after the AF C*-algebra result, and used a somewhat similar functorial approach, but otherwise involved quite different techniques, as the setting of von Neumann algebras—which are assumed to be closed in a stronger sense than C*-algebras—while analogous to that of C*-algebras is quite different).

SW: How did you become involved in this research, and how would you describe the particular challenges, setbacks, and successes that you've encountered along the way?

I became familiar with the early work of Glimm and Dixmier on certain special AF algebras as a graduate student, and so realized the importance of the work of Bratteli a little later, classifying arbitrary AF algebras by their construction data, expressed in a form which is now referred to as a Bratteli diagram. It was not at all obvious, however, that this data was an invariant in the sense of being a functor, until I realized that it was essentially equivalent to the K-theory functor.

Whereas the AF algebra theorem was mainly a conceptual advance, and at least in retrospect was only mildly complex from a technical point of view, there were formidable technical challenges to be overcome at various stages of the broader classification program.

"...the basic framework at the level of the very notion of classification which had to be developed for the present project is novel—not to say iconoclastic!"

For instance, in the main case that has been considered so far (simple algebras, meaning no non-trivial two-sided ideals), whereas the main classification statement could be stated rather simply, typically using the most elementary K-theoretical invariants, the proof in different cases often required the introduction of considerably more complicated invariants—and it was not even obvious at the beginning that this was necessary, not to mention the question of what the appropriate additional invariants might be!

Another, even more difficult, challenge was related to combinatorial subtleties arising in connection with dimension phenomena. These difficulties were dealt with by a combination of early work of Li in her Ph.D. thesis and a 260-page paper of Gong, completed only shortly before our final joint work!

In short, the work has consisted necessarily of a long sequence of successes, with perhaps just the single—quite serious, but I hope only temporary—setback that we can so far only deal with what might be referred to as the well-behaved case. (Fortunately, this is not a tautology!)

SW: Where do you see your research leading in the future?

So far, there has been a distinct division of objects in the class under study into well-behaved and non-well-behaved ones, with the classification of the well-behaved ones, one feels, almost complete, but with the non-well-behaved ones still a complete mystery. Perhaps these ones will turn out after all to be well behaved, if one stares at them fiercely enough! One does have a potential candidate after all for the label (incorporating what is known as the Cuntz semigroup), and the problem may just be that no one has been able to show that this describes things completely.

For the record, the question is whether the class of objects known as separable amenable C*-algebras can be classified in simple terms (perhaps by means of invariants which are K-theoretical in nature). By the well-behaved ones, one means those which have the somewhat technical property that they absorb tensorially the Jiang-Su algebra (but these are also known to be well behaved in other ways).

As mentioned earlier, there is also the question of applying this new approach to possibly classifying other objects, such as countable groups. Already a class of groups has been identified that has the same classification pattern as AF algebras. (A new complication is that the classifying functor can no longer be viewed as K-theory, but must be constructed more indirectly.)

SW: Do you foresee any social or political implications for your research?

There is perhaps a modest implication in the context of society as a whole, given the deep-lying tendency of people to wish to classify things, events, and phenomena—and even other people! As mentioned, the basic framework at the level of the very notion of classification which had to be developed for the present project is novel—not to say iconoclastic! (One expects labels to be easy to compare, and not to be subtle in character or hard to distinguish, even if the objects themselves under study are like fine wines.)End

Professor George Elliott
University of Toronto
Toronto, Ontario, Canada

Professor Guihua Gong
University of Puerto Rico
Rio Piedras, Puerto Rico

Professor Liangqing Li
University of Puerto Rico
Rio Piedras, Puerto Rico



   |   BACK TO TOP