Celebratio Mathematica

For me, the year 1978 was a good year! In the winter I de­livered the Her­mann Weyl Lec­tures [e3] at the In­sti­tute for Ad­vanced Study in Prin­ceton. In the sum­mer I gave an in­vited talk at the In­ter­na­tion­al Con­gress of Math­em­aticians held in Hel­sinki. And in Decem­ber, I met Paul Baum. I had to nav­ig­ate the snowy roads between Stony Brook and Prin­ceton, and I lost my lug­gage on the trip to Hel­sinki, but the year ended on a good note!

On the plane trip re­turn­ing from Hel­sinki, my then-col­league Jeff Chee­ger told me I should look up Paul Baum in con­nec­tion with my work on $K$-ho­mo­logy. Earli­er that sum­mer, for the same reas­on, Mi­chael Atiyah had sug­ges­ted to Paul that he look me up. Both Paul and I had come up with con­crete real­iz­a­tions of $K$-ho­mo­logy, and Mi­chael Atiyah be­lieved re­lat­ing the real­iz­a­tions could be very im­port­ant.

My work had star­ted with a prob­lem in ab­stract op­er­at­or the­ory seem­ing to have little con­tact with $K$-the­ory. It was the out­come of this joint ef­fort with Larry Brown and Peter Fill­more, now called BDF the­ory, about which I spoke in Prin­ceton and Hel­sinki. An ex­tant write-up can be found in [e2], [e3].

In the 60’s a top­ic of con­sid­er­able in­terest among op­er­at­or the­or­ists con­cerned com­pact per­turb­a­tions and prop­er­ties of op­er­at­ors which held mod­ulo them. In par­tic­u­lar, a res­ult of Her­man Weyl and John von Neu­mann showed that self-ad­joint op­er­at­ors with the same es­sen­tial spec­trum were unit­ar­ily equi­val­ent mod­ulo the com­pact op­er­at­ors. Here, es­sen­tial spec­trum is defined as lim­it points of the spec­trum or ei­gen­val­ues of in­fin­ite mul­ti­pli­city. People were in­ter­ested in the ana­log of this res­ult for nor­mal op­er­at­ors. And, in par­tic­u­lar, Paul Hal­mos had asked if one could char­ac­ter­ize such op­er­at­ors. Dav­id Berg had shown that nor­mal op­er­at­ors with the same es­sen­tial spec­trum were unit­ar­ily equi­val­ent mod­ulo com­pacts. But where­as an op­er­at­or that is self-ad­joint mod­ulo the com­pacts must be a self-ad­joint plus a com­pact, the ana­log­ous state­ment is no longer true for nor­mal op­er­at­ors. The uni­lat­er­al shift provides a counter­example. In par­tic­u­lar, Paul Hal­mos raised two spe­cif­ic ques­tions. First, when is an op­er­at­or that is nor­mal mod­ulo the com­pacts the sum of a nor­mal and a com­pact? Second, is the col­lec­tion of sums of a nor­mal op­er­at­or and a com­pact op­er­at­or closed in the norm to­po­logy? These are some of the prob­lems that Larry Brown, Peter Fill­more, and I were in­vest­ig­at­ing.

In tack­ling these ques­tions, fol­low­ing earli­er work of Lewis Coburn, we even­tu­ally fo­cused on the as­so­ci­ated short ex­act se­quence of $C^{\ast }$-al­geb­ras $$0\to \mathcal{K}(\mathcal{H})\to \mathcal{E}(T)\to C(X_T)\to 0.$$ Here $\mathcal{K}(\mathcal{H})$ is the two-sided ideal of com­pact op­er­at­ors on $\mathcal{H}$, $T$ is an op­er­at­or on the Hil­bert space $\mathcal{H}$ sat­is­fy­ing $$[T,T^{\ast }]=T{T}^{\ast }-T^{\ast }T\in \mathcal{K}(\mathcal{H}),$$ $\mathcal{E}(T)$ is the $C^{\ast }$-al­gebra gen­er­ated by $T$ and $I$, and $X_T$is the es­sen­tial spec­trum of $T$ or the spec­trum of $\pi (T)$ in the Calkin al­gebra $\mathcal{L}(\mathcal{H})/\mathcal{K}(\mathcal{H})$. We real­ized that the col­lec­tion of such ex­ten­sions formed a com­mut­at­ive semig­roup with iden­tity in a nat­ur­al fash­ion and that the ques­tion of when $T=N+K$ is equi­val­ent to the ques­tion of when the cor­res­pond­ing ex­ten­sion is trivi­al or splits. But of course, this only re­placed one prob­lem with an­oth­er prob­lem; that is, when is such an ex­ten­sion trivi­al?

One ob­struc­tion to tri­vi­al­ity is the ex­ist­ence of a $\lambda$ not in the es­sen­tial spec­trum of $T,{\sigma }_e(T)$, for which the Fred­holm in­dex of $T-\lambda$ is not zero. (Re­call that an op­er­at­or is said to be Fred­holm if it has closed range and both its ker­nel and coker­nel are fi­nite-di­men­sion­al. Moreover, its in­dex is the dif­fer­ence of the di­men­sions of the ker­nel and the coker­nel.) My res­ults with Larry Brown and Peter Fill­more ul­ti­mately showed that the con­verse held; that is, the ex­ten­sion is trivi­al if and only if there is no such $\lambda$. But prov­ing that took some do­ing.

In con­sid­er­ing the ana­log­ous ques­tion of tri­vi­al­ity for spaces $X\subseteq {\mathbb{C}}^m$ or for $m$-tuples of es­sen­tially nor­mal op­er­at­ors $(T_1,\dots ,T_m)$ that com­mute mod­ulo the com­pacts, we real­ized that one needed to con­sider tensor­ing the cor­res­pond­ing ex­ten­sion by $M_m(\mathbb{C})$. Even­tu­ally, this led us to the fact that the semig­roup was a group that had a nat­ur­al pair­ing with $K$-the­ory. Ul­ti­mately, this showed that such ex­ten­sions yiel­ded a con­crete real­iz­a­tion for odd $K$-ho­mo­logy and that, for $X$ a sub­set of the plane, the Fred­holm in­dex men­tioned above was enough to de­cide wheth­er or not an ex­ten­sion was trivi­al.

Fol­low­ing Al­ex­an­der Grothen­dieck’s in­tro­duc­tion of al­geb­ra­ic $K$-the­ory, Mi­chael Atiyah and Fritz Hirzebruch defined to­po­lo­gic­al $K$-the­ory, and Mi­chael Atiyah later showed that an ab­strac­tion of el­lipt­ic op­er­at­ors yiel­ded a con­crete ana­lyt­ic real­iz­a­tion of even $K$-ho­mo­logy. However, he was un­able to de­scribe the equi­val­ence re­la­tion. The BDF work provided a real­iz­a­tion of the odd $K$-ho­mo­logy group and a work­able set of equi­val­ence re­la­tions. Us­ing the re­la­tion between the odd and even $K$-ho­mo­logy groups, one could com­plete Mi­chael Atiyah’s real­iz­a­tion of the even $K$-ho­mo­logy. About the same time, Gen­nadi Kas­parov was at­tempt­ing to fol­low up on Mi­chael Atiyah’s work to ob­tain an equi­val­ence re­la­tion and use it to study the Novikov con­jec­ture, which was Atiyah’s ori­gin­al mo­tiv­a­tion.

Paul Baum had star­ted out try­ing to provide a con­crete geo­met­ric real­iz­a­tion of $K$-ho­mo­logy for a space $X$. After sev­er­al at­tempts, he had come up with a pic­ture that Mi­chael Atiyah liked. It in­volved the no­tion of a ${\mathrm{spin}}^c$-man­i­fold, a no­tion known to be re­lated to $K$-the­ory. Let $X$ be a fi­nite com­plex. For each group $K_i(X),i=0,1$, one con­siders every com­pact ${\mathrm{spin}}^c$ man­i­fold $M$ with di­men­sion of the same par­ity as $i$, every com­plex vec­tor bundle $E$ over $M$, and every con­tinu­ous map $f$ from $M$ to $X$. The triples $(M,E,f)$ are the cycles for $K_i(X)$. Paul Baum had an equi­val­ence re­la­tion built from three steps. The crit­ic­al step was what Paul called “vec­tor bundle modi­fic­a­tion”. It cap­tures Bott peri­od­icity and the fact that $K_i(X)$ is peri­od­ic in $i$ with peri­od 2. One might con­sider Paul’s cycles to be ana­log­ous to those in sin­gu­lar ho­mo­logy the­ory, with $(M,f)$ play­ing the role of sin­gu­lar sim­plex and $E$ the role of mul­ti­pli­city.

Mi­chael Atiyah had shown the ex­ist­ence of $K$-ho­mo­logy us­ing Span­i­er–White­head du­al­ity and had provided a con­crete real­iz­a­tion for it. But to be use­ful one needed to be able to re­cog­nize nat­ur­ally oc­cur­ring cycles for this the­ory and know when they are equi­val­ent. In BDF we had ana­lyt­ic cycles, while in Paul Baum’s de­scrip­tion we had geo­met­ric ones. Mi­chael Atiyah’s be­lief that an ex­pli­cit iso­morph­ism between these two pic­tures could be very im­port­ant led to our meet­ing.

At the end of the 70’s, an air­line flew from Provid­ence, Rhode Is­land, to Islip, New York, and back, stop­ping at New Haven and Bridge­port in Con­necti­c­ut, be­fore re­turn­ing to Provid­ence. Paul and I flew those routes reg­u­larly, once every month or two for the next year and a half. Ini­tially most of our time was de­voted to ex­plain­ing to each oth­er our real­iz­a­tions of $K$-ho­mo­logy. Since each of us had lim­ited ex­pert­ise in the oth­er’s field, this ex­er­cise re­quired con­sid­er­able ef­fort, and the con­sump­tion of much wine and good food, es­pe­cially at a now-closed res­taur­ant called Na­po­leon’s in Port Jef­fer­son, New York.

In Mi­chael Atiyah’s real­iz­a­tion of even $K$-ho­mo­logy, $K_0(X)$, for $X$ a fi­nite com­plex, he defined cycles in terms of an ab­stract no­tion of an el­lipt­ic op­er­at­or: $(T,{\sigma }_1,{\sigma }_2)$, where ${\sigma }_1$ and ${\sigma }_2$ are $\ast$-rep­res­ent­a­tions of $C(X)$ on Hil­bert spaces ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ and $T$ is a Fred­holm op­er­at­or from $H_1$ to $H_2$ which in­ter­twines ${\sigma }_1$ and ${\sigma }_2$ up to the com­pacts. The co­v­ari­ance re­quired of a ho­mo­logy the­ory arises from the fol­low­ing con­struc­tion. Let $f:X\to Y$ be a con­tinu­ous map and let $(T,{\sigma }_1,{\sigma }_2)$ be an ab­stract el­lipt­ic op­er­at­or on $X$. Define $f^{\ast }:C(Y)\to C(X)$ by $f^{\ast }(\xi )=\xi \circ f$. Then $(T,{\sigma }_1\circ f^{\ast },{\sigma }_2\circ f^{\ast })$ is an ab­stract el­lipt­ic op­er­at­or on $Y$. If $X$ is a com­pact man­i­fold, if $T$ is giv­en by an el­lipt­ic pseudodif­fer­en­tial op­er­at­or of or­der 0 between sec­tions of vec­tor bundles $E_1$ and $E_2$ over $X$, and if con­tinu­ous func­tions on $X$ act on sec­tions of these vec­tor bundles by point­wise mul­ti­plic­a­tion, then one ob­tains an “Atiyah cycle” from this data, and Mi­chael Atiyah showed there were enough of those cycles to gen­er­ate $K_0(X)$. However, he was un­able to de­scribe ex­pli­citly the ne­ces­sary equi­val­ence re­la­tion.

Paul and I real­ized that the same setup with ${\mathcal{H}}_1={\mathcal{H}}_2$, ${\sigma }_1={\sigma }_2$, and $T$ a self-ad­joint Fred­holm op­er­at­or en­ables one to ob­tain an odd cycle fol­low­ing BDF. In par­tic­u­lar, if $P$ is the or­tho­gon­al pro­jec­tion onto the pos­it­ive spec­tral space for $T$, then com­press­ing a mul­ti­pli­er $\phi$ in $C(X)$ by $P$ to ob­tain a “Toep­litz-like op­er­at­or”, $P\sigma (\phi )P$, one ob­tains a unit­al $\ast$-ho­mo­morph­ism from $C(X)$ to the Calkin al­gebra of the pos­it­ive spec­tral sub­space, which is an­oth­er ver­sion of a BDF cycle. In par­tic­u­lar, this showed that the odd ana­log of the Atiyah–Sing­er in­dex the­or­em could be viewed as the clas­sic­al in­dex the­or­em for Toep­litz op­er­at­ors such as those defined on the unit circle by Is­rael Go­hberg and Mark Krein. This ob­ser­va­tion also en­abled us to define the sought-after iso­morph­ism between the ana­lyt­ic and geo­met­ric real­iz­a­tions of $K$-ho­mo­logy.

Re­call that for a fi­nite-di­men­sion­al Hil­bert space $G$, one can define the as­so­ci­ated Clif­ford al­gebra $\mathrm{Cl}(G)$ on which $G$ acts by mul­ti­plic­a­tion. This ac­tion is not ir­re­du­cible since $\mathrm{Cl}(G)$ splits in­to an or­tho­gon­al dir­ect sum of sub­mod­ules. Let $M$ be a smooth closed man­i­fold with a Rieman­ni­an met­ric. The fibers of the co­tan­gent bundle, $T^{\ast }M$, to $M$ can be used to define a bundle of com­plex Clif­ford al­geb­ras $\mathrm{Cl}(T^{\ast }M)$ over $M$. The bundle $T^{\ast }M$ acts on $\mathrm{Cl}(T^{\ast }M)$ but not ir­re­du­cibly. The state­ment that $M$ is a ${\mathrm{spin}}^c$ man­i­fold means that not only does one have this point­wise ac­tion on sec­tions of the co­tan­gent bundle but there is a sub-bundle of the Clif­ford bundle $\mathrm{Cl}(T^{\ast }M)$, called the spinor bundle, on which $T^{\ast }M$ acts ir­re­du­cibly. This en­ables one to define the Dir­ac op­er­at­or, $D_M$, on smooth sec­tions of this spinor bundle to ob­tain a first-or­der el­lipt­ic dif­fer­en­tial op­er­at­or. When $M$ is odd-di­men­sion­al, the Dir­ac op­er­at­or is self-ad­joint and the con­struc­tion de­scribed in the pre­ced­ing para­graph, ap­plied to $D(1+D^2{)}^{-1/2}$, yields an odd BDF cycle. Moreover, if we tensor $D$ with the iden­tity op­er­at­or $I_E$ on a vec­tor bundle $E$, we can pro­ceed as above to make from $D\otimes I_E$ a BDF cycle con­struc­ted from the pair $(M,E)$. A choice of con­nec­tion on $E$ is re­quired, but the res­ult­ing odd cycle defines a class in $K_1(M)$ that is in­de­pend­ent of the choice of con­nec­tion. Call this class $[D\otimes I_E]$. For a giv­en triple $(M,E,f)$ with $\dim M$ odd, us­ing the co­v­ari­ance defined by com­pos­ing the $C(M)$-rep­res­ent­a­tion with $f^{\ast }$, we can map $[D\otimes I_E]\in K_1(M)$ to a class $$f_{\ast }([D\otimes I_E])\in K_1(X)$$ that is defined by a BDF cycle as­so­ci­ated with an ab­stract el­lipt­ic op­er­at­or on $X$. The map $$(M,E,f)\mapsto f_{\ast }([D\otimes I_E])$$ yields the de­sired iso­morph­ism between geo­met­ric and ana­lyt­ic odd $K$-ho­mo­logy.

In the set­ting of the pre­ced­ing para­graph, but with $M$ even-di­men­sion­al, the bounded op­er­at­or con­struc­ted from $D\otimes I_E$ de­com­poses to $$\left(\begin{array}{cc}0& T^{\ast }\\ T& 0\end{array}\right)$$ re­l­at­ive to the de­com­pos­i­tion of the spinor bundle in­to even and odd spinors. Us­ing $T$, we ob­tain an Atiyah cycle on $M$ and we can push its class in­to $K_0(X)$, as be­fore, to ob­tain the map $$(M,E,f)\mapsto (T,{\sigma }_0\circ f^{\ast },{\sigma }_1\circ f^{\ast }),$$ where the ${\sigma }_i$’s are the rep­res­ent­a­tions of $C(M)$ on the sec­tions of $M$’s even and odd spinor bundles. This map defines the de­sired iso­morph­ism in the even case.

By the time Paul and I at­ten­ded the AMS Sum­mer In­sti­tute in Op­er­at­or Al­geb­ras in King­ston, Ontario, in the sum­mer of 1980, we had un­der­stood the iso­morph­ism and how to es­tab­lish it. It is closely re­lated to the Atiyah–Sing­er in­dex the­or­em. These res­ults were de­scribed in [1].

The King­ston meet­ing was Paul’s de­but as an “op­er­at­or al­geb­ra­ist”. After that he was no longer able to be an­onym­ous in that com­munity. He drove from Provid­ence, took the ferry to Long Is­land, and picked me up at Stony Brook on the way to King­ston. He was one of the “geo­met­er-to­po­lo­gists” who got in­ter­ested in what is now known as non­com­mut­at­ive geo­metry. Alain Connes was at the King­ston meet­ing also.

I had planned to vis­it Alain Connes in Par­is for the fall semester. As it turned out, I could make it for only a week and Paul joined me. Dur­ing that time, Paul and Alain star­ted work on what has be­come known as the Baum–Connes con­jec­ture. Paul and I, on the oth­er hand, star­ted work­ing on un­der­stand­ing an ana­lyt­ic real­iz­a­tion for the even re­l­at­ive $K$-ho­mo­logy. Some mo­tiv­a­tion for this was provided by the proof of the Atiyah–Sing­er in­dex the­or­em giv­en in the Pal­ais notes [e1]. We got to­geth­er fre­quently over the next two or three years, work­ing to un­der­stand this real­iz­a­tion. We also worked to un­der­stand the re­la­tion of our work to that of Gen­nadi Kas­parov, who was at­tempt­ing to com­plete Mi­chael Atiyah’s ori­gin­al ef­fort to re­solve the Novikov con­jec­ture. In his work, a two vari­able “$KK$-the­ory” ap­peared and he had in­cor­por­ated the $C^{\ast }$-ex­ten­sion frame­work.

In de­fin­ing $K$-ho­mo­logy the­ory there are es­sen­tially four groups: the even group $K_0(X)$, the odd group $K_1(X)$, the even re­l­at­ive group $K_0(X,A)$, and the re­l­at­ive odd group $K_1(X,A)$, where $X$ is a com­pact met­riz­able space and $A$ is a closed sub­set of $X$. While it is pos­sible to define the re­l­at­ive groups ab­stractly in terms of the ab­so­lute groups, we be­lieved that identi­fy­ing a con­crete real­iz­a­tion of re­l­at­ive cycles would lead to ap­plic­a­tions and new con­nec­tions. Our first an­nounce­ment of what we thought to be true was giv­en at the U.S.-Ja­pan op­er­at­or al­geb­ras meet­ing held in Kyoto in the sum­mer of 1983. For Paul and for me this was a first vis­it to Ja­pan. On the way to Kyoto I stopped at Sendai at the in­vit­a­tion of Masami­chi Take­saki be­fore con­tinu­ing to Kyoto. In Kyoto we stayed at a Hol­i­day Inn and we re­coun­ted our ad­ven­tures, both math­em­at­ic­al and oth­er­wise, dur­ing break­fast each morn­ing. Paul was par­tic­u­larly in­trigued by the Ja­pan­ese res­taur­ants and the “geisha-like wait­resses”. There was also a cafe on the roof at which “post-World War II Hawaii­an songs” were the or­der of the day. The con­fer­ence en­abled us to dis­cuss our re­l­at­ive $K$-ho­mo­logy pro­gram with oth­er par­ti­cipants at the work­shop.

There are two parts to the real­iz­a­tion of even re­l­at­ive $K$-ho­mo­logy: one for the cycles in $K_0(X,A)$ and a second for the bound­ary map $\partial :K_0(X,A)\to K_1(A)$. To be use­ful the lat­ter map needs to be very ex­pli­cit. The even re­l­at­ive cycle is a vari­ation of the ab­so­lute Atiyah cycle $(T,{\sigma }_1,{\sigma }_2)$ in which $T$ is now as­sumed only to be semi-Fred­holm, which means the ker­nel of $T$ can be in­fin­ite-di­men­sion­al. Ad­di­tion­ally, one as­sumes that for $\phi \in C(X)$ the com­pres­sion of a mul­ti­pli­er ${\sigma }_1(\phi )$ to the ker­nel of $T$ is com­pact if $\phi$ van­ishes on $A$. This al­lows one to define the $K_1$ cycle that is the im­age of the bound­ary map as fol­lows. Take $\xi$ in $C(A)$, ex­tend it ar­bit­rar­ily to $X$ to ob­tain $\phi$ in $C(X)$, and com­press mul­ti­plic­a­tion by ${\sigma }_1(\phi )$ to the ker­nel of $T$. The res­ult­ing op­er­at­or is well-defined up to com­pacts and, with $Q$ de­not­ing pro­jec­tion onto the ker­nel of $T$, the map $\xi \mapsto Q{\sigma }_1(\phi )Q$ defines a unit­al $\ast$-ho­mo­morph­ism from $C(A)$ to the Calkin al­gebra of the ker­nel of $T$; i.e., it defines a BDF cycle rep­res­ent­ing a class in $K_1(A)$. This cycle rep­res­ents the im­age of $(T,{\sigma }_1,{\sigma }_2)$’s class un­der the de­sired bound­ary map $\partial :K_0(X,A)\to K_1(A)$.

Show­ing that this defin­i­tion yields the re­l­at­ive even group along with the bound­ary map was tackled by Paul and my­self dur­ing the spe­cial year 1984–85 at MSRI in Berke­ley. Part of the proof res­ted on match­ing up these groups with cor­res­pond­ing $KK$-groups of Gen­nadi Kas­parov. The pro­gram at MSRI was or­gan­ized by Alain Connes, Masami­chi Take­saki, and my­self. Be­cause Paul had re­cently moved from Brown Uni­versity to Penn State, he was un­able to join the group for the whole year. However, he was a fre­quent vis­it­or for sev­er­al weeks at a time.

Al­though our work provides a rather ex­pli­cit de­scrip­tion of re­l­at­ive even cycles and the bound­ary map, the de­scrip­tion is not ter­ribly use­ful in the form giv­en above. The prob­lem is to de­cide how one re­cog­nizes a re­l­at­ive cycle and the im­age of the bound­ary map when the cycle arises from a dif­fer­en­tial op­er­at­or on a smooth man­i­fold. To ac­com­plish this, we were joined by Mi­chael Taylor, who at that time was my col­league at Stony Brook and who is an ex­pert on PDEs. We fo­cused on dif­fer­en­tial op­er­at­ors, and a gen­er­al­iz­a­tion of them, called pseudodif­fer­en­tial op­er­at­ors, defined from smooth sec­tions of a vec­tor bundle $E_0$ to smooth sec­tions of a vec­tor bundle $E_1$ on a smooth man­i­fold $M$ with bound­ary $\partial M$.

Since the dif­fer­en­tial op­er­at­or defines an un­boun­ded op­er­at­or on the space of smooth sec­tions, one has to pro­ceed care­fully. In par­tic­u­lar, one must con­sider ex­ten­sions of this op­er­at­or that are closed. We showed that some of them define even re­l­at­ive cycles. Be­cause such op­er­at­ors have more than one pos­sible closed ex­ten­sion, the is­sue of unique­ness arises, but un­der broad hy­po­theses it turns out that the class rep­res­en­ted by the re­l­at­ive cycle is in­de­pend­ent of the choice of closed ex­ten­sion. One can also identi­fy the odd bound­ary cycle ex­pli­citly, now as an el­lipt­ic pseudodif­fer­en­tial op­er­at­or $D_{\partial }$ on $\partial M$. The cor­res­pond­ence $$[D]\to [D_{\partial }]$$ defines the bound­ary map from $K_0(M,\partial M)$ to $K_1(\partial M)$.

The fact that dif­fer­ent closed ex­ten­sions of $D$ define the same ele­ment of $K_0(M,\partial M)$ im­plies that their cor­res­pond­ing im­ages, un­der the bound­ary map, in $K_1(\partial M)$ are equal, which has far-reach­ing con­sequences. For ex­ample, if $D$ is the Dir­ac op­er­at­or $D_M$ on $M$ (where we as­sume $M$ has a ${\mathrm{spin}}^c$ struc­ture) then the im­age of $[D_M]$ is the class defined by the Dir­ac op­er­at­or on $\partial M$ defined by the ${\mathrm{spin}}^c$ struc­ture on $\partial M$ in­duced by the ${\mathrm{spin}}^c$ struc­ture on $M$. If one takes the max­im­al closed ex­ten­sion of $D_M$, this im­age un­der the bound­ary map is the odd cycle defined by the Calderón pro­jec­tion. This iden­ti­fic­a­tion yields the Toep­litz in­dex the­or­em of Boutet de Mon­vel and, in fact, al­lows one to gen­er­al­ize it. Many more res­ults in­volving in­dex the­ory fol­low from this setup.

This real­iz­a­tion is par­tic­u­larly in­ter­est­ing in the case $X$ is a strictly pseudo­con­vex com­plex man­i­fold and $D$ is the $\bar{\partial }$-op­er­at­or. In this case, one can identi­fy the ker­nel $\bar{\partial }$ as the fi­nite dir­ect sum of sub­spaces of mixed forms with the first sum­mand be­ing the Berg­man space on $X$ and with oth­er sum­mands, which in­volve high­er-de­gree forms, be­ing fi­nite-di­men­sion­al. Thus, the Berg­man space on $X$ is iden­ti­fied up to a fi­nite-di­men­sion­al sub­space with the ker­nel of the re­l­at­ive even $K_0(X,\partial X)$ cycle. If one uses the above re­cipe to cal­cu­late the bound­ary map, one finds that $[{\bar{\partial }}_{\partial X}]=[D_{\partial X}]$. This en­ables one to con­clude that the cycles defined by the Berg­man space on $X$ and the Dir­ac op­er­at­or on $\partial X$ rep­res­ent the same class in $K_1(\partial X)$. Since the Atiyah–Sing­er in­dex the­or­em or the iso­morph­ism that we ob­tained earli­er de­scribes the odd cycle defined by the Dir­ac op­er­at­or, one is able to cal­cu­late the odd cycle defined by the Berg­man space. This is es­sen­tially the res­ult of Boutet de Mon­vel at this level of gen­er­al­ity, with a proof slightly dif­fer­ent from the one men­tioned above.

This work, which was pub­lished in [2], was es­sen­tially com­pleted for the 1988 AMS Sum­mer In­sti­tute held in New Hamp­shire and or­gan­ized by Wil­li­am Arveson and my­self. Many of the themes of the sub­ject now called non­com­mut­at­ive geo­metry were dis­cussed there as well as at the meet­ing in Bowdoin, Maine, that fol­lowed. The loc­a­tion of the meet­ings was chosen to take ad­vant­age of the sum­mer weath­er, but moth­er nature thwarted us. It was hot and fans were in short sup­ply. However, one con­sequence of these meet­ings be­ing held in New Eng­land was the op­por­tun­ity to meet Paul’s fath­er, an artist. I had already met his moth­er, also an artist, on sev­er­al oc­ca­sions be­cause she lived in New York City. This op­por­tun­ity provided some in­sight in­to Paul’s per­son­al­ity.

It is also of in­terest to con­sider the odd re­l­at­ive group. Here the re­l­at­ive cycles would be pairs $(T,\sigma )$, where $T$ is a sym­met­ric semi-Fred­holm op­er­at­or on a Hil­bert space $\mathcal{H}$ and $\sigma$ is a $\ast$‑rep­res­ent­a­tion of $C(X)$ on $\mathcal{H}$. Again, one as­sumes that for a mul­ti­pli­er $\sigma (\phi )$ arising from a $\phi \in C(X)$ that van­ishes on $A$ the com­pres­sion of $\sigma (\phi )$ to the ker­nel of $T$ is com­pact. One needs also to define the bound­ary map ef­fect­ively and the ques­tion is some­what del­ic­ate, in­volving the clas­sic­al the­ory of ex­ten­sions of sym­met­ric op­er­at­ors. Sub­sequent work by Mi­chael Taylor fin­essed these dif­fi­culties by tak­ing a product $X\times S^1$ but at a cost. In par­tic­u­lar, the bound­ary map is less ex­pli­cit, as is match­ing up the bound­ary map in the case of dif­fer­en­tial op­er­at­ors. Al­though we dis­cussed how to com­plete this de­vel­op­ment dir­ectly, we nev­er car­ried it out. Again, there are pos­sible ap­plic­a­tions that I re­lated to Sturm–Li­ouville the­ory in high­er di­men­sions.

It had been ten years at this point since I met Paul Baum. Al­though our re­search in­terests di­verged sub­sequently, we re­main good friends and of­ten re­min­isce about our roles in the de­vel­op­ment of $K$-ho­mo­logy.