by Jonathan Rosenberg
Looking back from the perspective of the 21st century, it is rather hard to imagine that when I first started learning mathematics, relatively few people thought that operator algebras and \( K \)-theory had that much to do with each other. But this was indeed the situation in the early 1970’s, except for major exceptions which I will touch upon shortly. It is also hard for me to imagine not knowing Paul Baum, and in fact, I’m not totally sure when we first met, but I think it was at the AMS Summer Institute in Kingston in 1980.
Even though these days I am probably more of an algebraic topologist and \( K \)-theorist (with an emphasis on applications of those subjects to differential geometry and mathematical physics) than anything else, I ended up in my current line of research via a continuous path, but certainly not via a geodesic. But I did make the lucky decision to go to Berkeley for graduate school and to ask to work with Marc Rieffel on what were then “modern” approaches to the unitary representation theory of Lie groups via \( C^* \)-algebras. Rieffel [e15] had recently introduced the theory of Morita equivalence in the context of \( C^* \)-algebras, and used it to rework Mackey’s theory of induced representations.1 This ultimately grew into a huge theory which is nicely summarized in [e38].
In the days before the internet, learning about new developments in math depended on one of a few things:
being in the right place at the right time, or
being on the right mailing list, or
having an advisor who knew what one should read.
As I mentioned, I had the good fortune to go to Berkeley to work with Marc Rieffel, for reasons (1) and (3). While I was in grad school, Larry Brown and Ed Effros visited for a while in Berkeley, and I had a chance to learn quite a bit from both of them. On top of that, Marc Rieffel was super-organized and had an encyclopedic knowledge of the literature. And finally, there was a wonderful group of fellow students to talk to, including (among many others) Bruce Blackadar, Ken Davidson, and Phil Green.
While most people nowadays think of \( C^* \)-algebras and \( K \)-theory as being inseparable, this connection only emerged slowly. In the late 60’s, work of Jänich [e2], Wood [e3], Karoubi [e6], and Atiyah [e7] showed that the Bott periodicity theorem, which is the foundation of computations in topological \( K \)-theory, has a natural home in the context of Banach algebras. Furthermore, Jänich’s work showed that the space of Fredholm operators on a Hilbert space, which up until this time was mostly only known to operator theorists, can be identified with a classifying space for \( K \)-theory. These developments attracted attention largely because of the Atiyah–Singer index theorem, which came along a bit earlier, but for which the most elegant proofs (starting with [e5]) are based on \( K \)-theory.
It seems that Taylor [e17], [e20] was the first one to use topological \( K \)-theory to study structural properties of Banach algebras, but his focus was mostly on commutative Banach algebras arising in harmonic analysis, and not on \( C^* \)-algebras. So interest in connections between \( C^* \)-algebras and \( K \)-theory came instead from three main sources:
the work of Brown–Douglas–Fillmore (BDF) [e10], [e11], [e26], described in [e49];
the work of Kasparov [e12], [e18] on trying to set up the homology/cohomology theory on Banach algebras (covariant on spaces, contravariant on algebras) that is dual to topological \( K \)-theory; and
Elliott’s discovery [e19] that \( K_0 \) (together with its order structure) provides the invariant needed for the classification of AF-algebras in a canonical way.
Let me briefly try to describe these.
The original work of BDF dealt just with \( C^* \)-algebra extensions of \( C(X) \) by the Calkin algebra, when \( X \) is a compact metric space. However, I believe it was Larry Brown (the B of BDF) who first attempted a more general theory of extensions of separable \( C^* \)-algebras in general [e16], [e24], and who began to study its properties and how they relate to \( K \)-theory. Making this theory work for arbitrary separable \( C^* \)-algebras required some important technical theorems of Voiculescu [e21], Choi–Effros [e22], and Arveson [e25]. I heard lectures on this material when I was a graduate student, and the fact that it seemed so exciting was one of the main reasons why I started to get interested in \( K \)-theory.
The second main development, as listed above, was the work of Kasparov on trying to find a rigorous analytic theory of \( K \)-homology, based on ideas of Atiyah, by abstracting the key properties of elliptic pseudodifferential operators in a way that would generalize to arbitrary spaces. This work was carried out simultaneously with (and independently of) the work of BDF, and at first it attracted little attention outside of the Soviet Union. But Kasparov was paying attention to the BDF theory, and soon [e27], [e29] he was able to show how to unify the BDF work with his own approach, and to show that (for separable nuclear \( C^* \)-algebras) they give isomorphic functors. The result was \( K\mkern-3mu K \)-theory, which is now not only the universally accepted framework for \( C^* \)-algebraic index theory, but also a key tool for studying extensions of \( C^* \)-algebras and for classification problems. But this is getting a bit ahead in the story.
Still back in the early 70’s, Ola Bratteli had generalized the UHF (uniformly hyperfinite) algebras of Glimm [e1] and the “matroid” algebras of Dixmier [e4] to give a much larger class of separable \( C^* \)-algebras, usually very noncommutative, for which the classification problem was accessible by algebraic tools. This was the class of AF (approximately finite-dimensional) algebras [e9]. Bratteli approached the classification problem via what are now called “Bratteli diagrams”, but the map from diagrams to isomorphism classes of AF-algebras, while surjective, is not at all injective, and it was not clear that Bratteli’s isomorphism criterion (for two algebras given by different-looking diagrams) was effectively computable. At this point George Elliott took up the problem, and showed [e19] that the \( K_0 \) group of algebraic \( K \)-theory, along with the natural order and unit structure on it (the latter only in the unital case) provided a simpler way to describe the classification. This work was completed by Effros, Handelman, and Shen, who gave a simple description [e28] of the ordered abelian groups that can arise from AF-algebras. This work was the beginning of what has since become known as the Elliott program, of describing the classification of suitable classes of \( C^* \)-algebras in terms of algebraic invariants coming from \( K \)-theory. Incidentally, Bratteli’s original classification via diagrams and Elliott’s classification via \( K_0 \) look so different that it is hard to see any resemblance between them. The relationship between the two classifications was only clarified a few years ago [e48].
So, with apologies for omissions, of which I’m sure I have made many, this basically describes the state of the relationship between \( C^* \)-algebras and \( K \)-theory as it was understood around 1975. This was just about the time that I started working in the subject, and only a few years before Paul Baum began his important collaboration with Ron Douglas on relating topological and analytic approaches to \( K \)-homology (again, see Douglas’s article [e49] for more details).
My own interest in \( K \)-theory of \( C^* \)-algebras started from the attempt to use \( K \)-theory as a tool for studying the structure of group \( C^* \)-algebras. Only in retrospect can I see that this was really groping toward what eventually became the Baum–Connes conjecture. The first example of this that I know of was the paper [e14], which I remember puzzling over when it first appeared (first in Russian and a few months later in not terribly good English translation). This paper used BDF theory and explicit calculation of Fredholm indices of integral operators to study the group \( C^* \)-algebra of the \( ax+b \) group (the affine motion group of the line). I then tried using the same methods for other groups [e23]. The methods were somewhat ad hoc, but this got me interested in studying the \( C^* \)-algebra extensions that show up in group \( C^* \)-algebras, for example of solvable Lie groups [e33], and ultimately in the \( K \)-theory of these group \( C^* \)-algebras [e34], and in \( K \)-theory of \( C^* \)-algebras in general [e32]. Meanwhile Kasparov had been able to use his \( K \)-theoretic theory of extensions of \( C^* \)-algebras [e27], [e29] to prove nonsplitting of the extension \begin{equation} 0\to C_0(\mathbb{R}\smallsetminus \{0\})\otimes \mathcal{K} \to C^*(H) \to C_0(\mathbb{R}^2) \to 0 \label{eq:Heis} \end{equation} describing the group \( C^* \)-algebra of the 3-dimensional Heisenberg group, a result also proved by Voiculescu [e30] in sharper form using operator-theoretic methods.
Right around the time of these developments I met Paul Baum at the AMS Summer Institute in Kingston, which was a two-week meeting in which all the participants lived together on a college campus, and at some other conferences, too (such as the one leading to the book [e31]). Paul’s jovial and relaxed appearance always put everyone at ease. That was when I started to learn about Paul’s unique approach to mathematics (and to life, for that matter). Ron Douglas describes in his article [e49] what led to the important Baum–Douglas papers on \( K \)-homology [2], [1]. I think it’s fair to say that Paul’s important contribution to this subject, which probably can be traced back through Paul’s papers on Riemann–Roch for singular varieties to Alexander Grothendieck, is the emphasis on having the right (functorial) definitions, so that the proofs of the main theorems will flow effortlessly once the right lemmas are in place.2 This is certainly true of the Baum–Douglas definition of geometric \( K \)-homology, as it is also true of the formulation of the Baum–Connes conjecture [3], [4], [5]. Before Baum–Connes was formulated, I noticed [e34] that there appeared to be a very close connection between the \( K \)-theory of a classifying space \( BG \) and the \( K \)-theory of the group \( C^* \)-algebra \( C^*(G) \) or \( C^*_r(G) \), but it was the lack of a good functorial approach that impeded progress for a while. The Baum–Connes formalism makes attacking relationships like this much easier, and applies to a much wider variety of situations. Just as an example, it is quite easy to prove the Baum–Connes conjecture for simply connected solvable Lie groups, and it immediately follows that for the Heisenberg group \( H \), \( K_j(C^*(H))\cong \mathbb{Z} \) for \( j \) odd, and \( \cong 0 \) for \( j \) even. Then the long exact \( K \)-theory sequence of \eqref{eq:Heis} immediately implies that the extension cannot split.
By the way, I mentioned earlier the importance in the pre-internet era of getting on the right mailing lists. Even if what I did on \( K \)-theory of \( C^* \)-algebras in these early days didn’t have much significance, it did get me on Kasparov’s mailing list, and when [e37] first came out in preprint form in 1981 (the published versions [e35], [e37] came along much later), I was fortunate to find out about it right away. In this crucial paper, Kasparov began to develop the technology for proving many cases of the Baum–Connes conjecture, before the published version of the conjecture was even available.
I will not attempt a formal history of the subject of \( C^* \)-algebras and \( K \)-theory from the mid-1980’s onward, except to say that for more on the Baum–Connes conjecture, one can consult, for example, the book [e42] or surveys like [e43], [e40], [e44], [e45]. For a textbook of \( K \)-theory of \( C^* \)-algebras, see [e39]; on \( K \)-homology, see [e41]; on \( K\mkern-3mu K \)-theory and its variants, see [e36] or [e46]. For applications of \( K \)-theory to the classification of \( C^* \)-algebras, a good reference is [e50].
In my many years of interacting with Paul, one of the most enjoyable times I can remember is the CBMS conference in Boulder, Colorado, in the summer of 1991, at which Paul was the principal speaker. Paul gave a week-long course on “\( K \)-homology and index theory”, and it’s a shame that the book which was supposed to have resulted from this course never got written. However, the weather and the scenery were marvelous, and everyone had a good time and got to know Paul’s special approach to the subject, focused on his work with Douglas and Connes. This conference also served as Paul’s 55th birthday party. But when one looks back on what has happened since then, one can see that the subject matter of \( K \)-theory, \( C^* \)-algebras, and index theory was then, if not still in infancy, at least only in early adulthood. There is every indication that it, and Paul Baum, will continue to flourish.
