Noncommutative Geometry Alain Connes Contents Preface 6 Introduction 7 1. Measure theory (Chapters I and V) 8 2. Topology and K-theory (Chapter II) 14 3. Cyclic cohomology (Chapter III) 19 4. The quantized calculus (Chapter IV) 25 5. The metric aspect of noncommutative geometry 34 Chapter 1. Noncommutative Spaces and Measure Theory 39 1. Heisenberg and the Noncommutative Algebra of Physical Quantities 40 2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics 45 3. Modular Theory and the Classification of Factors 48 4. Geometric Examples of von Neumann Algebras : Measure Theory of Noncommutative Spaces 51 5. The Index Theorem for Measured Foliations 64 Appendix A : Transverse Measures and Averaging Sequences 77 Appendix B : Abstract Transverse Measure Theory 78 Appendix C : Noncommutative Spaces and Set Theory 79 Chapter 2. Topology and K-Theory 84 1. C∗-algebras and their K-theory 86 2. Elementary Examples of Quotient Spaces 90 3. The Space X of Penrose Tilings 94 4. Duals of Discrete Groups and the Novikov Conjecture 99 5. The Tangent Groupoid of a Manifold 104 6. Wrong-way Functoriality in K-theory as a Deformation 111 7. The Orbit Space of a Group Action 116 8. The Leaf Space of a Foliation 123 9. The Longitudinal Index Theorem for Foliations 134 10. The Analytic Assembly Map and Lie Groups 141 Appendix A : C∗-modules and Strong Morita Equivalence 156 Appendix B : E-theory and Deformations of Algebras 163 Appendix C : Crossed Products of C∗-algebras and the Thom Isomorphism 175 3 CONTENTS 4 Appendix D : Penrose Tilings 179 Chapter 3. Cyclic Cohomology and Differential Geometry 183 1. Cyclic Cohomology 187 2. Examples 212 3. Pairing of Cyclic Cohomology with K-Theory 229 4. The Higher Index Theorem for Covering Spaces 238 5. The Novikov Conjecture for Hyperbolic Groups 244 6. Factors of Type III, Cyclic Cohomology and the Godbillon-Vey Invariant 250 7. The Transverse Fundamental Class for Foliations and Geometric Corollaries269 Appendix A. The Cyclic Category Λ 280 Appendix B. Locally Convex Algebras 290 Appendix C. Stability under Holomorphic Functional Calculus 291 Chapter 4. Quantized Calculus 293 1. Quantized Differential Calculus and Cyclic Cohomology 298 2. The Dixmier Trace and the Hochschild Class of the Character 305 3. Quantized Calculus in One Variable and Fractal Sets 320 4. Conformal Manifolds 339 5. Fredholm Modules and Rank-One Discrete Groups 348 6. Elliptic Theory on the Noncommutative Torus T2 and the Quantum Hall θ Effect 356 7. Entire Cyclic Cohomology 375 8. The Chern Character of θ-summable Fredholm Modules 399 9. θ-summable K-cycles, Discrete Groups and Quantum Field Theory 417 Appendix A. Kasparov’s Bivariant Theory 439 Appendix B. Real and Complex Interpolation of Banach Spaces 447 Appendix C. Normed Ideals of Compact Operators 450 Appendix D. The Chern Character of Deformations of Algebras 454 Chapter 5. Operator algebras 458 1. The Papers of Murray and von Neumann 459 2. Representations of C∗-algebras 469 3. The Algebraic Framework for Noncommutative Integration and the Theory of Weights 472 4. The Factors of Powers, Araki and Woods, and of Krieger 475 5. The Radon-Nikody´m Theorem and Factors of Type III 480 λ 6. Noncommutative Ergodic Theory 487 7. Amenable von Neumann Algebras 500 8. The Flow of Weights: Mod(M) 505 9. The Classification of Amenable Factors 512 10. Subfactors of Type II Factors 517 1 11. Hecke Algebras, Type III Factors and Statistical Theory of Prime Numbers523 CONTENTS 5 Appendix A. Crossed Products of von Neumann Algebras 537 Appendix B. Correspondences 539 Chapter 6. The metric aspect of noncommutative geometry 552 1. Riemannian Manifolds and the Dirac Operator 555 2. Positivity in Hochschild Cohomology and Inequalities for the Yang–Mills Action 569 3. Product of the Continuum by the Discrete and the Symmetry Breaking Mechanism 574 4. The Notion of Manifold in Noncommutative Geometry 598 5. The Standard U(1)×SU(2)×SU(3) Model 609 Bibliography 627 Preface This book is the English version of the French “G´eom´etrie non commutative” pub- lished by InterEditions Paris (1990). After the initial translation by S.K. Berberian, a considerable amount of rewriting was done and many additions made, multiplying by 3.8 the size of the original manuscript. In particular the present text contains several unpublished results. My thanks go first of all to C´ecile whose patience and care for the manuscript have been essential to its completion. This second version of the book greatly benefited from the important modifications suggested by M. Rieffel, D.Sullivan, J.-L.Loday, J.Lott, J.Bellissard, P.B.Cohen, R.Coquereaux, J.Dixmier, M. Karoubi, P. Kr´ee, H. Bacry, P. de la Harpe, A. Hof, G. Kasparov, J. Cuntz, D. Tes- tard, D. Kastler, T. Loring, J. Pradines, V. Nistor, R. Plymen, R. Brown, C. Kassel, and M. Gerstenhaber. Patrick Ion and Arthur Greenspoon played a decisive rˆole in the finalisation of the book, clearing up many mathematical imprecisions and consid- erably smoothing the initial manuscript. I wish to express my deep gratitude for their generous help and their insight. Finally, my thanks go to Marie Claude for her help in creating the picture on the cover of the book, to Gilles who took the photograph, and to Bonnie Ion and Fran¸coise for their help with the bibliography. Many thanks go also to Peter Renz who orchestrated the whole thing. Alain Connes 30 June 1994 Paris 6 Introduction The correspondence between geometric spaces and commutative algebras is a familiar and basic idea of algebraic geometry. The purpose of this book is to extend this correspondence to the noncommutative case in the framework of real analysis. The theory, called noncommutative geometry, rests on two essential points: 1. The existence of many natural spaces for which the classical set-theoretic tools of analysis, such as measure theory, topology, calculus, and metric ideas lose their pertinence, but which correspond very naturally to a noncommutative algebra. Such spaces arise both in mathematics and in quantum physics and we shall discuss them in more detail below; examples include: a) The space of Penrose tilings b) The space of leaves of a foliation c) The space of irreducible unitary representations of a discrete group d) The phase space in quantum mechanics e) The Brillouin zone in the quantum Hall effect f) Space-time. Moreover, even for classical spaces, which correspond to commutative algebras, the new point of view will give new tools and results, for instance for the Julia sets of iteration theory. 2. The extension of the classical tools, such as measure theory, topology, differential calculus and Riemannian geometry, to the noncommutative situation. This extension involves, of course, an algebraic reformulation of the above tools, but passing from the commutative to the noncommutative case is never straightforward. On the one hand, completelynewphenomenaariseinthenoncommutativecase, suchastheexistenceofa canonical time evolution for a noncommutative measure space. On the other hand, the constraint of developing the theory in the noncommutative framework leads to a new point of view and new tools even in the commutative case, such as cyclic cohomology and the quantized differential calculus which, unlike the theory of distributions, is (cid:82) perfectly adapted to products and gives meaning and uses expressions like f(Z)|dZ|p where Z is not differentiable (and p not necessarily an integer). 7 1. MEASURE THEORY (CHAPTERS I AND V) 8 Let us now discuss in more detail the extension of the classical tools of analysis to the noncommutative case. 1. Measure theory (Chapters I and V) IthaslongbeenknowntooperatoralgebraiststhatthetheoryofvonNeumannalgebras and weights constitutes a far reaching generalization of classical measure theory. Given a countably generated measure space X, the linear space of square-integrable (classes of) measurable functions on X forms a Hilbert space. It is one of the great virtues of the Lebesgue theory that every element of the latter Hilbert space is represented by a measurable function, a fact which easily implies the Radon-Nikody´m theorem, for instance. There is, up to isomorphism, only one Hilbert space with a countable basis, and in the above construction the original measure space is encoded by the represen- tation (by multiplication operators) of its algebra of bounded measurable functions. This algebra turns out to be the prototype of a commutative von Neumann algebra, which is dual to an (essentially unique) measure space X. In general a construction of a Hilbert space with a countable basis provides one with specific automorphisms (unitary operators) of that space. The algebra of operators in the Hilbert space which commute with these particular automorphisms is a von Neumann algebra, and all von Neumann algebras are obtained in that manner. The theory of not necessarily commutative von Neumann algebras was initiated by Murray and von Neumann and is considerably more difficult than the commutative case. The center of a von Neumann algebra is a commutative von Neumann algebra, and, as such, dual to an essentially unique measure space. The general case thus decomposes over the center as a direct integral of so-called factors, i.e. von Neumann algebras with trivial center. In increasing degree of complexity the factors were initially classified by Murray and von Neumann into three types, I, II, and III. The type I factors and more generally the type I von Neumann algebras, (i.e. direct integrals of type I factors) are isomorphic to commutants of commutative von Neumann algebras. Thus, up to the notion of multiplicity they correspond to classical measure theory. The type II factors exhibit a completely new phenomenon, that of continuous dimen- sion. Thus, whereas a type I factor corresponds to the geometry of lines, planes, ..., k-dimensional complex subspaces of a given Hilbert space, the subspaces that belong to a type II factor are no longer classified by a dimension which is an integer but by a dimension which is a positive real number and will span a continuum of values (an interval). Moreover, crucial properties such as the equality dim(E∧F)+dim(E ∨F) = dim(E)+dim(F) 1. MEASURE THEORY (CHAPTERS I AND V) 9 remain true in this continuous geometry (E∧F is the intersection of the subspaces and E ∨F the closure of the linear span of E and F). The type III factors are those which remain after the type I and type II cases have been considered. They appear at first sight to be singular and intractable. Relying on Tomita’s theory of modular Hilbert algebras and on the earlier work of Powers, Araki, Woods and Krieger, I showed in my thesis that type III is subdivided into types III , λ λ ∈ [0,1] and that a factor of type III , λ (cid:54)= 1, can be reconstructed uniquely as a λ crossed product of a type II von Neumann algebra by an automorphism contracting the trace. This result was then extended by M. Takesaki to cover the III case as well, 1 using a one-parameter group of automorphisms instead of a single automorphism. These results thus reduce the understanding of type III factors to that of type II factors and their automorphisms, a task which was completed in the hyperfinite case and culminates in the complete classification of hyperfinite von Neumann algebras presented briefly in Chapter I Section 3 and in great detail in Chapter V. The reduction from type III to type II has some resemblance to the reduction of arbi- trary locally compact groups to unimodular ones by a semidirect product construction. There is one essential difference, however, which is that the range of the module, which is a closed subgroup of R∗ in the locally compact group case, has to be replaced for + type III factors by an ergodic action of R∗: the flow of weights of the type III fac- 0 + tor. This flow is an invariant of the factor and can, by Krieger’s theorem (Chapter V) be any ergodic flow, thus exhibiting an intrinsic relation between type III factors and ergodic theory and lending support to the ideas of G. Mackey on virtual subgroups. Indeed, in Mackey’s terminology, a virtual subgroup of R∗ corresponds exactly to an + ergodic action of R∗. + Since general von Neumann algebras have such an unexpected and powerful structure theory it is natural to look for them in more common parts of mathematics and to start using them as tools. After some earlier work by Singer, Coburn, Douglas, and Schaeffer, and by Shubin (whose work is the first application of type II techniques to the spectral theory of operators), a decisive step in this direction was taken up by M.F. Atiyahand I. M. Singer. They showedthat the type II vonNeumann algebra generated by the regular representation of a discrete group (already considered by Murray and von Neumann) provides, thanks to the continuous dimension, the necessary tool to measure the multiplicity of the kernel of an invariant elliptic differential operator on a Galois covering space. Moreover, they showed that the type II index on the covering equals the ordinary (type I) index on the quotient manifold. Atiyah then went on, with W. Schmid, to show the power of this result in the geometric construction of discrete series representations of semisimple Lie groups. Motivated by this result and by the second construction of factors by Murray and von Neumann, namely the group-measure-space construction, I then showed that a foliated manifold gives rise in a canonical manner to a von Neumann algebra (Chapter I Section 1. MEASURE THEORY (CHAPTERS I AND V) 10 4). A general element of this algebra is just a random operator in the L2 space of the generic leaf of the foliation and can thus be seen as an operator-valued function on the badly behaved leaf space X of the foliation. As in the case of covering spaces the generic leaf is in general not compact even if the ambient manifold is compact. A notable first difference from the case of discrete groups and covering spaces is that in general the von Neumann algebra of a foliation is not of type II. In fact every type can occur and, for instance, very standard foliations such as the Anosov foliation of the unit sphere bundle of a compact Riemann surface of genus > 1 give a factor of type III . This allows one to illustrate by concrete geometric examples the meaning 1 of type I, type II, type III ... and we shall do that as early as Section 4 of Chapter λ I. Geometrically the reduction from type III to type II amounts to the replacement of the initial noncommutative space by the total space of an R∗ principal bundle over it. + We shall see much later (Chapter III Section 6) the deep relation between the flow of weights and the Godbillon-Vey class for codimension-one foliations. The second notable difference is in the formulation of the index theorem, which, as in Atiyah’s case, uses the type II continuous dimensions as the key tool. For foliations one needs first to realize that the type II Radon measures, i.e. the traces on the C∗-algebra of the foliation (cf. below) correspond exactly to the holonomy invariant measures. Such measures are characterized (cf. Chapter I Section 5) by a de Rham current, the Ruelle-Sullivan current, and the index formula for the type II index of a longitudinal elliptic differential operator now involves the homology class of the Ruelle -Sullivan current. In contrast to the case of covering spaces this homology class is in general not even rational; the continuous dimensions involved can now assume arbitrary real values, and the index is not related to an integer-valued index. In the case of measured foliations the continuous dimensions acquire a very clear geo- metric meaning. First, a general projection belonging to the von Neumann algebra of the foliation yields a random Hilbert space, i.e. a measurable bundle of Hilbert spaces over the badly behaved space X of leaves of the foliation. Next, any such random Hilbert space is isomorphic to one associated to a transversal as follows: the transver- sal intersected with a generic leaf yields a countable set; the fiber over the leaf is then the Hilbert space with basis this countable set. Finally, in the above isomorphism, the transverse measure of the transversal is independent of any choices and gives the continuous dimension of the original projection. One can then formulate the index the- orem independently of von Neumann algebras, which we do in Section 5 of Chapter I. In simple cases where the ergodic theorem applies, one recovers the transverse measure of a transversal as the density of the corresponding discrete subset of the generic leaf, i.e. as the limit of the number of points of this subset over increasingly large volumes. Thus the Murray and von Neumann continuous dimensions bear the same relation to ordinary dimensions as (continuous) densities bear to the counting of finite sets.