Review of Noncommutative Geometry by Alain Connes Vaughan Jones and Henri Moscovici

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This and the following article are expositions C(X) of all continuous complex-valued func- of Noncommutative Geometry by Alain Connes tions on X (which, by the famous Gelfand- (Academic Press, 1994, 661 pages, $59.95 hard- Naimark theorem, completely captures the topol- cover), reviewed in the Bulletin of the Ameri- ogy of a compact Hausdorff space). Then replace can Mathematical Society, v. 33 no. 4, October the quotient space not by a smaller algebra such 1996, pp. 459–466. as C(X/ ) but by a larger one, the crossed product C(X) o consisting of sums γ fγ γ (with ∈ fγ C(XΓ)) which are multiplied according to “The correspondence between geometric spaces ∈ P the rule γfγΓ 1 = f γ 1. This algebraΓ is non- and commutative algebras is a familiar and basic − ◦ − idea of algebraic geometry. The purpose of this commutative and should be completed in some book is to extend the correspondence to the non- fashion, but when this is done, it provides a commutative case in the framework of real analy- powerful tool for the study of X/ . In the con- sis.” text of operator algebras there is nothing patho- Thus begins Connes’ book Noncommutative logical about C(X) o which can beΓ thought of Geometry. The central thesis is that the usual no- as a noncommutative desingularization of X/ . tion of a “space”—a set with some extra struc- In our example theΓ noncommutative algebra ture—is inadequate in many interesting cases was obtained from a group acting on a com-Γ and that coordinates may profitably be replaced mutative one, but once one admits the legitimacy by a noncommutative algebra. Here is a simple of such algebras as “coordinate rings”, one may but central example. Suppose the compact Haus- expect to find new examples devoid of any com- dorff space X is acted on by a discrete group . mutative origin. If the action is sufficiently simple, one may put Connes explores many examples, as diverse the quotient topology on the orbit space, X/Γ, as the space of Penrose tilings and the Brillouin thus capturing the orbit structure in an entirely zone in the quantum Hall effect. In so doing he satisfactory fashion. But as the action becomesΓ develops a myriad of techniques, sometimes, as more complicated, the quotient topology will in the case of his Chern character, as necessary fail to separate orbits, and in extreme cases like generalizations of known tools and sometimes, the action of Z by powers of an irrational rota- as in the case of his quantized calculus, of largely tion of the circle the quotient topology has no noncommutative inspiration. information whatsoever. The noncommutative The technical background of Connes’ pro- approach is first to replace X by the algebra gram is the theory of operator algebras on Hilbert space, closed in some topology. Why should op- Vaughan Jones is professor of mathematics at the Uni- erator algebras be the vessel of noncommutative versity of California, Berkeley. Henri Moscovici is pro- geometry? After all, the Gelfand-Naimark theo- fessor of mathematics at The Ohio State University. rem is not about operators, and, according to

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what we have said so far, it is little more than a mann algebra is X⊕ M(ξ)dµ(ξ) for some mea- curiosity that may faithfully represent C(X) on sure space (X,µ) withR each M(ξ) a factor. Fac- a Hilbert space. There is a simple but telling rea- tors themselves were classified in [22] accord- son for operator algebras. The first noncom- ing to algebraic properties of the projections mutative algebras are the n n matrix algebras (operators p with p2 = p = p ) which they con- × ∗ Mn(C), which are algebras of operators on finite- tain. In a von Neumann algebra M we say p q dimensional Hilbert space. So operator algebra if pq = p and p q if there is a u M with≤ ∼ ∈ is just “large matrix algebra”. Or we could take uu∗ = p and u∗u = q. There is a direct analogy a cue from quantum physics, where the classi- with cardinalities of sets, so a projection is called cal observables—continuous functions on phase “infinite” if it is equivalent to a proper subpro- space—are replaced by algebras of operators, in jection, and “ ” induces an order “.” on equiv- general unbounded, on the Hilbert space of wave alence classes≤ of projections in M. If M were the functions. According to von Neumann’s re- n n matrices, we would have p q if and only × ∼ markable insight, algebras of bounded opera- if rank(p)=rank(q). In a factor, . is a total tors, if sufficiently well understood, would in fact order. The factor’s type is determined by the ex- be enough to understand the unbounded ones. istence of infinite projections type II and III, ∞ This led to “algebraic quantum field theory” ini- and the type of the ordered set . according to tiated by Haag and Kastler in [14]. But we digress. the following table (the Hilbert space is assumed In algebraic geometry access to the power of separable). the theory requires nontrivial results in pure commutative algebra. In noncommutative geometry, then, we expect to need a serious dose of operator algebraic results right at the start. Far more so indeed than algebraic geometry needs commutative algebra, since the intuition af- forded by the set of points in a variety has dis- appeared and there is no reason to expect that all the interesting phenomena of noncommuta- Here is an enormously condensed summary tive geometry arise as mere perturbations of of the work of Murray and von Neumann. commutative ones. Thus a substantial, but by no • Any type I factor is of the form ( 1) id B H ⊗ means dominant, portion of Connes’ book is de- in some tensor “factorization” voted to the abstract structure theory of opera- = 1 2 (where we use ( ) to de- H H ⊗H B · tor algebras. note the algebra of all bounded operators). Operator algebra theory splits almost dis- • Type II 1 factors may be constructed as the jointly into two halves: von Neumann algebras, commutant of the regular representation of initiated by Murray and von Neumann in the a discrete group all of whose conjugacy 1930s [22], and C*-algebras, initiated by Gelfand classes are infinite. and Naimark in the 1940s [12]. Technically the • Type II factors are always of the form II 1 ∞ ⊗ split is according to the topology in which the I (i.e., infinite matrices over a II 1 factor). ∞ algebras are closed-von Neumann algebras in • Type III factors are more difficult but were the topology of pointwise convergence and C*- constructed in [30]. algebras in the norm topology. But morally the • Type II 1 factors afford a trace—a linear split is the noncommutative version of the split functional tr: M C with tr(x∗x) > 0 for → between measure theory and general topology. x =0 and tr(xy)= tr(yx). Type II factors 6 ∞ If functions on [0,1] are represented on have an infinite trace analogous to that on L2([0, 1],dx) as multiplication operators, the ( ), and type III factors have no trace B H continuous functions form a C*-algebra and the whatsoever. L∞ functions form a von Neumann algebra. • There is, up to isomorphism, a single II 1 Every von Neumann algebra is a C*-algebra, but factor with the “hyperfinite” property; i.e., it is seldom wise to think of it as such. any finite set of elements can be arbitrarily Let us first discuss von Neumann algebras, well approximated by elements in a finite- where Connes has been such a dominant figure dimensional subalgebra. And there are II 1 for the last twenty-five years. Murray and von factors without this property. Neumann quickly recognized the importance of The period from the mid 1940s to the late von Neumann algebras whose center contains 1960s was a period of digestion of the results only scalar multiples of the identity, which they of Murray and von Neumann. Highlights include called “factors”, and von Neumann wrote a paper the discovery by Dixmier of a plethora of max- [30] reducing any von Neumann algebra to a fac- imal abelian subalgebras of a II 1 factor [10], the tor by a “direct integral”—a continuous ana- work of Dye on orbit equivalence [11], estab- logue of the direct sum of algebras. Any von Neu- lishing the hyperfiniteness of crossed products

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of abelian algebras by Z, and the characteriza- the kernel of the map from R to Out(M). Now tion by Sakai of von Neumann algebras as C*-al- T (M) can be quite complicated, and Connes also gebras which are duals as Banach spaces [26]. Let defined an invariant S(M) as me cite two results as being the ultimate re- ϕ spectrum( ϕ) 0 , which is a closed multi- \{ } + finement of the Murray-von Neumann ideas—the plicativeT subgroup of R . A type III factor is said construction by McDuff of uncountably many II 1 to be of type∆ III , 0 λ 1 , according to λ ≤ ≤ factors [21] and the construction by Powers of whether S(M) is a continuum of type III factors [25]. + Many of the seeds of future developments a) R type III1 n were sown during this period, but the first idea b) λ n Z type IIIλ, 0 <λ<1 { | ∈ } to truly go beyond Murray and von Neumann was c) 1 type III0 the Tomita-Takesaki theory, first proposed by { } Tomita. The trace property tr(x∗x)= tr(xx∗) One way to obtain a type III λ factor (0 <λ<1) means that is an isometry on the Hilbert space is to take an automorphism of a II factor N, ∗ ∞ completion tr of a factor M with respect to the scaling the trace by λ, and form the crossed H inner product x, y =tr(y∗x). This implies a product N oZ. Connes showed that any type III λ certain symmetryh betweeni a type I or II factor factor is of this form and gave a similar de- and its commutant. The absence of this sym- composition in the III 0 case [4]. Takesaki further metry had caused type II factors to be somewhat showed that a III 1 factor is the crossed product inaccessible. Tomita envisaged as an un- of a II factor by a trace-scaling action of R [27]. ∗ ∞ bounded (anti) linear operator on ϕ, where ϕ The extraordinary conclusion of all this is H is any positive (i.e., ϕ(x∗x) > 0) linear functional that type III factors can be completely under- on M and ϕ is the completion of M with re- stood by knowing type II factors and their au- H spect to x, y = ϕ(y∗x). The -operator is pre- tomorphism groups. So Connes began a pene- closed, soh onei may form the ∗polar decomposi- 1 trating study of automorphisms of type II factors, tion S = J 2 of its closure S. Here is a positive introducing entirely new techniques. The first definite unbounded operator, and J is an isom- step was to classify periodic automorphisms of etry which∆ restores the symmetry between∆ M and the hyperfinite II 1 factor R, then arbitrary au- its commutant: JMJ = the commutant of M on tomorphisms of R ( ). In order to com- ⊗BH ϕ, where M acts by left multiplication. But of plete the classification of hyperfinite III factors, H λ perhaps greater significance is the second part 0 λ<1, a serious snag had to be overcome, ≤ of the Tomita-Takesaki theorem, which asserts: for it was not known that a hyperfinite II fac- ∞ it it tor was automatically isomorphic to R ( ). M − = M (t R), ⊗BH ∈ Connes solved this in [5] by introducing a new approach to hyperfiniteness: injectivity of the which shows∆ the∆ existence of a 1-parameter automorphism group on the von Neumann algebra, von Neumann algebra M, meaning that M has called the modular group or canonical flow, writ- a Banach space complement in ( ). The re- ϕ it it B H ten σt (x)= x − . The modular group is en- viewers consider [5] to be one of the highlights tirely invisible in the abelian case and trivial for of twentieth-century mathematics. The result the types I and∆ II,∆ whereas type III factors are dy- (injective) (hyperfinite) and work of Krieger ⇐⇒ namic objects. allowed Connes to complete the classification of It is at this point that the work of Connes en- hyperfinite III factors, 0 λ<1. Haagerup λ ≤ ters the history. If one had to find a single word showed uniqueness in the III 1 case some ten to sum up his vast contribution to von Neu- years later using a condition discovered by mann algebras, it would be the word “automor- Connes. phisms”. His first step answered the question: The ideas involved in the injective factors ϕ “How does the modular group σt depend on theorem spawned a vast number of new results ϕ?” He showed that, given ϕ and : M C on factors: in particular, an invariant χ(M) — an → with ϕ(x∗x) > 0, (x∗x) > 0 (called “states” be- abelian group which Connes has shown to be cause of the quantum connection), there is a quite arbitrary, thereby constructing in a natural continuous map t ut from R to the unitary way an uncountable family of II1 factors and, for 7→ ϕ 1 group of M such that σ (x)=ut σ (x)u− and example, II factors not isomorphic to their ten- t t t 1 ut σt (us )=ut+s. This is a noncommutative ana- sor products with themselves and II 1 factors dis- logue of the Radon-Nikodym theorem compar- tinct from their opposite algebras (see [6]). Such ing two measures and can be made into a gen- problems were long-standing ones that had been uine extension of it by considering the important considered intractable before Connes. His book Kubo-Martin-Schwinger condition from quan- contains a detailed account of this story. tum statistical mechanics. We see that if Out(M) Having thus dispatched injective von Neu- is the quotient group of all automorphisms by mann algebras, Connes began his investigation ϕ inner ones, σt yields an invariant, called T (M), of noncommutative geometry. The first insight

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was the existence of a canonical von Neumann But Connes’ work, and hence his book, has had algebra associated to a smooth foliation of a little to do with the structure theory of an ab- manifold. The idea is to form the algebra of sec- stract C*-algebra. Rather, he has looked for the tions of the bundle over the leaf space which as- interesting examples and how to understand sociates to each leaf its canonical L2-space of them in a geometric framework. half-densities. These sections must be measur- In the commutative world, the most immedi- able in some sense, and that is where the inter- ate and powerful tools are homology and the fun- est lies: if the leaves wrap around themselves a damental group. Unfortunately, these do not lot, the leaf space is like the orbit space for an have straightforward noncommutative general- irrational rotation, already discussed, and one izations. It is topological K-theory—the study of may not be able to find sections arbitrarily. The vector bundles—that has been the most fruitful, notion of measurability is with respect to the as it passes immediately to the noncommutative transverse measure class of Lebesgue measure; world. A vector bundle on a manifold is, by a re- and Connes saw that if the leaves wind together sult of Swan, the same thing as a projection in in an ergodic way, then his von Neumann alge- a matrix algebra over the smooth functions on bra is a factor. Using noncommutative measure the manifold. So vector bundles are generalized theory, Connes showed how to obtain real-val- by the algebraic K0-group (equivalence classes ued Betti numbers for a foliation with holonomy- of projections in matrix algebras) of a C*-alge- invariant transverse measure and gave a Gauss- bra. Thus, geometry as approached by vector Bonnet formula with purely geometric bundles does admit a noncommutative version. consequences. This set the stage for a full gen- However, this merely represents a starting point eralization of the Atiyah-Singer index theorem in the quest to build a full-fledged extension of to foliations and more general structures. the “commutative” homological apparatus. In At this stage noncommutative measure the- both algebraic geometry and algebraic topology, ory begins to become insufficient, and the next the contravariant K-group based on vector bun- step towards noncommutative geometry is non- dles goes hand in hand with a covariant coun- commutative topology. In the case of foliations terpart, and this duality mutually enhances their this allows us to study individual leaves rather power. A decisive step in the direction of defin- than just the average leaf. ing the K-homological group in operator-theo- As we have said, a C*-algebra—abstractly just retic terms was taken around 1969 by Atiyah [1], a Banach -algebra with x x = x 2—is the ∗ k ∗ k k k who proposed a construction of the Whitehead noncommutative version of a locally compact dual to K-theory based on axiomatizing the con- Hausdorff space. So it is natural that it took cept of elliptic operator. His ideas were taken up longer before the interesting examples in C*-al- and fully developed by Brown, Douglas, and Fill- gebras became clear, since we have to sift among more [3], who turned them into a theory of C - the noncommutative versions of the sometimes ∗ tiresome pathological spaces of point-set topol- algebra extensions, and by Kasparov. The latter ogy. (By contrast, there is essentially only one then succeeded in merging, in the general frame- measure space, so that any von Neumann alge- work of C∗-algebras, the contravariant K-func- bra should be interesting.) The founding work tor with the covariant Ext-functor into a bivari- was done by Gelfand and Naimark, who showed ant KK-functor, endowed with a remarkable that the C*-algebra axioms actually characterize intersection product [15]. Besides Kasparov’s norm-closed sub -algebras of ( ). A hardy theory, two other fundamental results, dealing ∗ B H group of pioneers—notably including Dixmier, with crossed products, provided a catalyst for Kadison, and Kaplansky—kept the theory of C*- the quest to understand the K-groups of non- algebras growing during a period when their fu- commutative C∗ -algebras: the Pimsner- ture importance was not yet appreciated. But the Voiculescu [23] exact sequence for crossed prod- first major structural result was that of Glimm ucts by Z and Connes’ Thom isomorphism [7] [13], who showed that either a C*-algebra is quite for crossed products by R. simple, in that all its representations generate The newly developed K-theoretical apparatus type I von Neumann algebras, or there is a non- led to the formulation of a host of exciting gen- type I representation and all hell breaks loose, eralizations of the Atiyah-Singer index theorem there being representations of all kinds whose [2], extending the scope of the fertile interaction complete classification is impossible. Examples between topology and geometry on one side and of type I C*-algebras are abelian ones and the functional analysis on the other. These devel- algebra of all compact operators. The C*-alge- opments were synthesized by Baum and Connes bra of almost any nonabelian discrete group is into a powerful guiding principle known as the not of type I. Something like a classification of Baum-Connes conjecture. Under the form of an type I C*-algebras could be envisaged, but such analytical analogue for the assembly map of a thought would be folly away from type I. surgery theory, it provides a geometric descrip-

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tion to the K-group of a C∗-algebra represent- [F], [e] =( 1)nSupertrace e[F,e]2n , ing a “noncommutative space”. h i − p +1 This constitutes the main theme of Chapter II, n> . where the principle is formulated in the language ∀ 2 of smooth groupoids and is richly illustrated. Regarding de =[F,e] as a quantized differ- The special cases treated in detail include: the tan- ential, the right-hand side bears a striking re- gent groupoid of a manifold as an embodiment semblance to the formulae expressing the Chern of the Atiyah-Singer index theorem, the orbit classes of a bundle in terms of curvature. The space of a group action, the leaf space of a folia- caveat is that the above expression starts mak- tion and the longitudinal index theorem, the ing sense precisely at the stage where the usual case of Lie groups and the equivariant index one becomes trivial, i.e., when the degree of the theorem on homogeneous spaces of Lie groups. noncommutative “curvature form” surpasses With all the advances made on the K-theo- the dimension of the “space”. The polarized retical flanck in the early 1980s, the lack of a co- form of the right-hand side, homological companion to K-theory remained 0 1 2n a serious impediment. In the absence of effec- τF (a ,a ,... ,a ) tive computational devices, potentially inter- =( 1)nSupertrace a0[F,a1] ...[F,a2n] , − esting generalizations of the index theorem aj , would run the risk of becoming little more than ∈A tautology. Noncommutative topology, reared in encodes the defining features of the cyclic co- the rarefied environment of topological K-the- homology theory and typifies the quintessential ory, was now in need of its own, down-to-earth cyclic cocycle. As an element of the cyclic cohomology theory which would allow making homology group HC ( ) (which is a graded ∗ A concrete calculations. The breakthrough came in module over the polynomial ring HC∗(C)), the 1981 with the discovery by Connes of cyclic co- cohomology class of τF represents the Chern homology and of the spectral sequence relating character of ( ,F) in Connes’ sense. Using this H it to Hochschild cohomology [8]. Far from being character-pairing as a guiding principle, Connes serendipitous, Connes’ discovery came as a log- was able to uncover the fundamental properties ical outcome of his inspired search for a purely and operations of cyclic cohomology, starting algebraic de Rham-type theory for noncommu- with a crucial periodicity operator implicit in the above discussion. For methodological reasons, tative algebras, which should pair with the Ext- these topics are covered in the book in reverse functor and with its “even” mate in a manner sim- chronological order: cyclic cohomology in III.1–3 ilar to the Chern character. and the Chern character of Fredholm modules To elaborate, an abstract elliptic operator, or in IV.1. Fredholm module, over a unital -algebra ∗ A It is worth noting that in the framework of over C consists of a pair ( ,F), where is a H H noncommutative spaces the integrality of the separable Hilbert space on which acts by a - A ∗ index pairing acquires new potency. A sugges- representation π and F = F∗ is a bounded op- tive example is the elegant argument (cf. IV.5) erator on such that F2 I together with the given to the Pimsner-Voiculescu theorem [24] H − commutators [F,π(a)], a , are “small”, i.e., that settled a long-standing conjecture of Kadi- ∀ ∈A compact. The prototypical model consists of an son, asserting that the reduced C∗-algebra of a elliptic (0th order) pseudodifferential operator free group has no nontrivial idempotents. The on a closed smooth manifold M, together with relevant “space” is the “dual” to a free group ; an approximate inverse (i.e., parametrix). In that i.e., the reduced C∗-algebra Cr∗( ), whose canon- case, = C (M), and the commutators [F,π(a)] ical trace τ( g agg)=a1, happens to be pre-Γ A ∞ ∈ are not just compact but of Schatten p-class: cisely the ChernP character ofΓ a geometric Γ Trace [F,π(a)] p < , for any p>dimM. The 1 -summable Fredholm module. Thus, | | ∞ τ(K0(C∗( ))) Z. latter condition, viewed as a “finite dimension- r ⊂ The relationship with the index theory of el- ality” axiom, can be incorporated into the gen- liptic operatorsΓ on manifolds is quite transpar- eral definition of a p-summable Fredholm mod- ent. It does involve, though, a highly nontrivial ule. Furthermore, there is no essential loss of step: the identification of the periodic continu- generality in requiring F2 = I. ous cyclic cohomology of the algebra C∞(M), With these amendments, the pairing [F], [e] h i equipped with its Fréchet space topology, with between the K-homology class represented by the de Rham homology of M (cf. III.2.α). ( ,F) and the K-theory class [e] K0( ) of an As often happens with an idea whose time has H 2 ∈ A idempotent e = e q( ), given by the Fred- come, cyclic cohomology was quickly recognized ∈M A holm index, can be expressed as as relevant to other branches of mathematics.

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Inspired by Connes’ work and motivated by al- signature in K0 is then identified, by means gebraic K-theory, Loday and Quillen [18] devel- R of a “higher index theorem”, as the higher sig- oped the dual theory, cyclic homology, and re- nature correspondingΓ to the given group cocy- lated it to the Lie algebra homology of matrices cle. This could have been the end of the story, over a ring. Starting from a similar standpoint, except that the homotopy invariance of the gen- B. Tsygan [28] arrived on his own at the concept eralized signature is known to take place only of cyclic homology as an additive version of al- in K0(C∗( )). The transition from the first K- gebraic K-theory. group to the second requires a suitable ana- The role of cyclic cohomology as a noncom- logue of theΓ Harish-Chandra-Schwartz algebra. mutative de Rham theory of currents has found The latter is known to exist when belongs to spectacular applications in connection with one the remarkable class of hyperbolic groups in of the central problems in topology, the Novikov Gromov’s sense, which thereforeΓ satisfy the conjecture. In its best-known form, this conjec- Novikov conjecture. ture states that the “higher signatures” of a non- Another remarkable application of cyclic co- simply connected oriented manifold M, formed homology, in its capacity of de Rham homology out of the L-classes of M and the cohomology of closed currents on the space of leaves, is to of π1(M), are homotopy invariants. When the the theory of foliations (cf. III.6–7). To begin manifold is simply connected, this amounts to with, it allows the introduction on a transversely a consequence of Hirzebruch’s signature for- oriented foliation (V, ) of a fundamental cyclic F mula, the homotopy invariance of the L-genus. cohomology class [V/ ]. As in the case of the F For manifolds M whose fundamental group Novikov conjecture, the topological invariance = π1(M) is free abelian, Lusztig [19] devised a is not automatic and raises in fact a major issue. seminal method of proof, based on the Atiyah- Connes’ solution involves a new technique—the SingerΓ index theorem for families of elliptic op- passage to a longitudinal frame bundle of leaf- erators. The family involved is that of signature wise metrics in a manner which preserves the operators on M twisted by flat line bundles, pairing of the transverse fundamental class with ˆ hence parametrized by the Pontryagin dual . the K-theory. A number of interesting and purely The index bundle of this family, which belongs geometric consequences flow out of this. The in- ˆ to K∗( ), is a homotopy invariant, and the Atiyah-Γ herently noncommutative nature of a leaf space Singer theorem allows the identification of its manifests itself with a vengeance in the sur- Chern Γcharacter as the higher signature. prising interaction between the differential-geo- If is not abelian, the Pontryagin dual ceases metric and the measure-theoretic aspects. The to exist as an ordinary space, but the group C∗- most striking example is the equality algebraΓ C∗( ) remains a natural substitute for ˆ C( ). Moreover, work of Mishchenko and Kas- d parov gave aΓ precise meaning to the “noncom- GV = iδ [V/ ] dt F mutative”Γ incarnation of the above index bundle, as an element of K (C∗( )), and established relating the Godbillon-Vey class of a codimen- its homotopy invariance.∗ Various K-homol- sion 1 foliation to the time evolution of the ogy/cohomology devices, inΓ the context of C∗- transverse fundamental class under the modu- algebras, have been successfully used by lar automorphism group of the corresponding Mishchenko [20], Kasparov [16], Kasparov-Skan- von Neumann algebra. dalis [17], and quite recently by Higson-Kas- Classical differential geometry began with a parov to prove the validity of the Novikov con- local calculus and evolved to be able to treat jecture in a wide variety of cases, such as when global issues. In stark contrast, the noncom- is a discrete subgroup of a Lie group (Kas- mutative version has progressed from global to parov) or when is amenable (Higson-Kasparov). local! In fact, the very meaning of the notion of AllΓ these proofs circumvent the last step of “local” in noncommutative geometry was prob- Lusztig’s originalΓ argument, which can only be lematic before the advent of Connes’ quantized restored by means of cyclic cohomology. calculus (cf. Chapter IV). While it is not surpris- The cyclic cohomological approach (due to ing that the formalism of quantum mechanics Connes) is treated in III.4–5. Its salient features should play a foundational role, it took Connes’ are the following. First, one constructs a refined extraordinary insight to realize that it can be ac- variant of the Mishchenko-Kasparov signature, tually turned into an organized collection of as an element of the K-group of the group ring “local” tools, ready to assume the part of the in- , where the ground ring consists of infi- finitesimal calculus. Put in succinct dictionary R R nite matrices of rapid decay. Moreover, any (nor- form, the quantized calculus starts from the fa- malized)Γ group cocycle gives rise canonically to miliar quantum-mechanical interpretation of ob- a cyclic cocycle in HC . The result of the servables, with the replacement of complex vari- ∗ R pairing between this cocycle and the generalized ables (resp. real variables) by operators (resp. Γ

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1 selfadjoint operators) on a (separable) Hilbert [[f,ds− ],g]=0, f,g C∞(M) space and continues with the substitution of ∀ ∈ compactH operators for classical infinitesimals, with an additional relation, of degree n = dim M, the order of a quantum infinitesimal T being arising from the homological nature of the vol- measured by the rate of decay of its character- ume form on M. An example of how classical istic values µn(T ). Motivated by the Fredholm geometric features are recovered from this per- module picture, the quantized differential is de- spective is the “dual” formula giving the geodesic fined by the commutator with a fixed bounded distance operator F = F , F2 = I that splits into two ∗ H dist(x, y) = sup f (x) f (y); [D, f] 1 , isomorphic orthogonal subspaces: − k k≤ n o dT =[F,T]. which retains its meaning for discrete spaces as well. Finally, the integral of a measurable variable As the mathematical underpinning for Ein- T 0 is defined as the logarithmic divergence stein’s general relativity theory, Riemann’s geom- ≥ of the trace of T, etry (in its Lorentzian guise) is the geometry of the space-time at macroscopic scale. By placing N 1 1 − it on equal footing with the geometry of discrete µn(T ) T, log N −→ spaces, in a manner directly compatible with 0 Z X the formalism of quantum mechanics, Connes’ the meaning of measurability being that the spectral approach gains the ability to reach right-hand side is independent of the underly- below the Planck scale and attempt to decipher ing limiting procedure. the fine structure of space-time. This is pre- With the stage thus set, a bewildering array cisely what the author sets out to do in the very of applications of the new calculus follows in last section, devoted to a geometric explanation short order. They range from formulae for the of the best available phenomenological model of Minkowski content of Cantor sets and for the particle physics, the Standard Model. Hausdorff measure on Julia sets—as applica- The book does have its faults. Some parts re- tions of the one-variable quantized calculus—to quire much more mathematical sophistication computing a 4-dimensional analogue of the than others. The index should be much more de- Polyakov action of 2-dimensional conformal field tailed. And the publishers should be censured theory and making sense of the “area” of a 4-di- for binding the book in a way that will not with- mensional manifold passing through a canoni- stand the many readings the book requires. But cal quantization of the calculus on conformal 2n- as a book of genuine vision, Noncommutative manifolds. Geometry renders insignificant such minor quib- The sixth and last chapter could lead to a revolutionary change in geometric thinking, com- bles. parable to the transition from classical to quan- References tum mechanics. The notion of a spectral space [1] M. F. Atiyah, Global theory of elliptic operators, X, in Connes’ sense, is a quantum version of a Proc. Internat. Conf. Funct. Anal. and Related Riemannian space. Thus, the coordinates of X are Topics (Tokyo, 1969), Univ. of Tokyo Press, Tokyo, given by an involutive—although not necessar- 1970, pp. 21–30. ily commutative—algebra of bounded oper- A [2] M. F. Atiyah and I. M. Singer, The index of ellip- ators on a Hilbert space . The quantized length tic operators I and III, Ann. Math. 87 (1968), H element of X is an “infinitesimal” of the form 484–530, 546–604. [3] L. G. Brown, R. G. Douglas, and P. A. Phillmore, 1 ds = D− , Extensions of C∗-algebras and K-homology, Ann. Math. 105 (1977), 335–348. where D = D∗ is an unbounded selfadjoint op- [4] A. Connes, Une classification des facteurs de type erator on such that the commutators [D, a] III, Ann. Sci. École. Norm. Sup. 6 (1973), 133–252. H are bounded a . While ds no longer com- [5] , Classification of injective factors, cases II1 , ∀ ∈A ——— mutes with the coordinates, the algebra they II , IIIλ, λ =1, Ann. Math. 104 (1976), 73–115. ∞ 6 generate does satisfy nontrivial commutation [6] ———, Sur la classification des facteurs de type II, relations. C. R. Acad. Sci. Paris 281 (1975), 13–15. In this picture, a Riemannian (spin) manifold [7] ———, An analogue of the Thom isomorphism for crossed products of a C -algebra by an action of M is described (cf. [9]) by the algebra C (M) rep- ∗ ∞ , Adv. in Math. 39 (1981), 31–55. resented by multiplication operators on the L2- R [8] ———, Noncommutative differential geometry, space of sections of the spin bundle over M and Part I: The Chern character in K-homology with the quantized length element given by the (preprint IHES M/82/53); Part II: De Rham ho- 1 Dirac propagator ds = D− (the ambiguity cre- mology and noncommutative algebra (preprint ated by the possible zero modes is inconse- IHES M/83/19), Inst. Hautes Études Sci. Publ. quential). The commutation relations are Math. 62 (1986), 257–360.

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[9] ———, Gravity coupled with matter and the foun- dation of noncommutative geometry, Comm. Math. Phys. 182 (1996), 155–176. [10] J. Dixmier, Sous-anneaux abéliens maximaux dans les facteurs de type fini, Ann. Math. 59 (1954), 279–286. [11] H. Dye, On groups of measure preserving trans- formations, I, II, Amer. J. Math. 81 (1959), 119–159; 85 (1963), 551–576. [12] I. Gelfand, and M. Neumark, On the imbedding of normed rings into the ring of operators in Hilbert space, Mat. Sb. 12 (1943), 197–213. [13] J. Glimm, Type I C∗-algebras, Ann. Math. 73 (1961), 572–612. [14] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5 (1964), 848. [15] G. G. Kasparov, The operator K-functor and extensions of C∗-algebras, Math. USSR-Izv. 16 (1981), 513–572. [16] ———, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147–201. [17] G. G. Kasparov and G. Skandalis, Groups acting on buildings, operator K-theory and Novikov’s conjecture, K-Theory 4 (1991), 303–337. [18] J. L. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), 569–591. [19] G. Lusztig, Novikov’s higher signature and families of elliptic operators, J. Differential Geom. 7 (1972), 229–256. [20] A. S. Mishchenko, Infinite-dimensional representations of discrete groups and higher signatures, Math. USSR-Izv. 8 (1974), 85–112. [21] D. McDuff, Uncountably many II 1 factors, Ann. Math. 90 (1969), 372–377. [22] F. J. Murray and J. von Neumann, On rings of operators, Ann. Math. 37 (1936), 116–229. [23] M. Pimsner and D. Voiculescu, Exact sequences for K-groups and Ext-groups of certain cross- product C∗-algebras, J. Operator Theory 4 (1980), 93–118. [24] ———, K-groups of reduced crossed products by free groups, J. Operator Theory 8 (1982), 131–156. [25] R. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. Math. 86 (1967), 138–171. [26] S. Sakai, A characterization of W ∗-algebras, Pa- cific J. Math. 6 (1956), 763–773. [27] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310. [28] B. Tsygan, Homology of matrix Lie algebras over rings and the Hochschild homology, Uspekhi Math. Nauk 38 (1983), 217–218. [29] J. von Neumann, On rings of operators: Reduction theory, Ann. Math. 50 (1949), 401–485. [30] ———, On rings of operators, III, Ann. Math. 41 (1940), 94–161.

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