#### 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; andElliott’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.