Celebratio Mathematica

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 $C^{\ast }$-al­geb­ras. Rief­fel [e15] had re­cently in­tro­duced the the­ory of Mor­ita equi­val­ence in the con­text of $C^{\ast }$-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:

1. be­ing in the right place at the right time, or

2. be­ing on the right mail­ing list, or

3. 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 $C^{\ast }$-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 $C^{\ast }$-al­geb­ras. So in­terest in con­nec­tions between $C^{\ast }$-al­geb­ras and $K$-the­ory came in­stead from three main sources:

1. the work of Brown–Douglas–Fill­more (BDF) [e10], [e11], [e26], de­scribed in [e49];

2. 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

3. 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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-al­geb­ras) they give iso­morph­ic func­tors. The res­ult was $KK$-the­ory, which is now not only the uni­ver­sally ac­cep­ted frame­work for $C^{\ast }$-al­geb­ra­ic in­dex the­ory, but also a key tool for study­ing ex­ten­sions of $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-al­geb­ras star­ted from the at­tempt to use $K$-the­ory as a tool for study­ing the struc­ture of group $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-al­gebra ex­ten­sions that show up in group $C^{\ast }$-al­geb­ras, for ex­ample of solv­able Lie groups [e33], and ul­ti­mately in the $K$-the­ory of these group $C^{\ast }$-al­geb­ras [e34], and in $K$-the­ory of $C^{\ast }$-al­geb­ras in gen­er­al [e32]. Mean­while Kas­parov had been able to use his $K$-the­or­et­ic the­ory of ex­ten­sions of $C^{\ast }$-al­geb­ras [e27], [e29] to prove non­split­ting of the ex­ten­sion $$\begin{array}{}\text{(1)}& 0\to C_0(\mathbb{R}\setminus \{0\})\otimes \mathcal{K}\to C^{\ast }(H)\to C_0({\mathbb{R}}^2)\to 0\end{array}$$ de­scrib­ing the group $C^{\ast }$-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 $C^{\ast }$-al­gebra $C^{\ast }(G)$ or $C_r^{\ast }(G)$, 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 $H$, $K_j(C^{\ast }(H))\cong \mathbb{Z}$ for $j$ odd, and $\cong 0$ for $j$ even. Then the long ex­act $K$-the­ory se­quence of $\text{(1)}$ 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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-al­geb­ras, see [e39]; on $K$-ho­mo­logy, see [e41]; on $KK$-the­ory and its vari­ants, see [e36] or [e46]. For ap­plic­a­tions of $K$-the­ory to the clas­si­fic­a­tion of $C^{\ast }$-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, $C^{\ast }$-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.