1、EMS Series of Lectures in MathematicsEdited by Andrew Rai (University of Edinburgh, U.K.)EMS Series of Lectures in Mathematics is a book series aimed ats, professionalmathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied

2、mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature.Previously pu

3、blished in this series:Katrin Wehrheim, Uhlenbeck CompactnessTorsten Ekedahl, Omester of Elliptic CurvesSergey V. Matveev, Lectures on Algebraic TopologyJoseph C. Várilly, An Introduction to Noncommutative GeometryReto Müller, Differential Harnack Inequalities and the Ricci FlowEustasiBarr

4、io, Paul Deheuvels and Sara van de Geer, Lectures on Empirical ProcessesIskander A. Taimanov, Lectures on Differential GeometryMartin J. Mohlenkamp, María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for YouStanley E. Payne and Joseph A. Thas, Finite Generalized QuadranglesMa

5、soud KhalkhaliBasic Noncommutative GeometryAuthor:Masoud Khalkhali Department of MathematicsThe University of Western Ontario London, Ontario N6A 5B7Canada: masouduwo.ca2010 Mathematics Subject Classification: 58-02; 58B34Key words: Noncommutative space, noncommutative quotient, groupoid, cyclic coh

6、omology, ConnesChern character, index formulaISBN 978-3-03719-061-6The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,and theed bibliographic data are available on the Internet at.This work is subject to copyright.s are, whether the whole or part of

7、the material isconcerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permissionof the copyright owner must be obtained.© 2009 European Mathemat

8、ical SocietyContact address:European Mathematical Society Publishing House Seminar for Applied MathematicsETH-Zentrum FLI C4 CH-8092 Zürich SwitzerlandPhone: +41 (0)44 632 34 36: Homepage:Typeset using the authors TEX files: I. Zimmermann, FreiburgPrinted on acid-paper produced f

9、rom chlorine-pulp. TCF °°Printed in Germany9 8 7 6 5 4 3 2 1For Azadeh and SamanContents1Examples of algebra-geometry correspondences11152124252627Locally compact spaces and commutative C -algebras . . . . . . . .1.7Vector bundles, finite projective modules, and idempoten

10、ts . . . . . .Affine varieties and finitely generated commutative reduced algebras . Affine schemes and commutative rings . . . . . . . . . . . . . . . . .Compact Riemann surfaces and algebraic function fields . . . . . . .Sets and Boolean algebras . . . . . . . . . . . . . . . . . . . . . . .From g

11、roups to Hopf algebras and quantum groups . . . . . . . . . .2Noncommutative quotients4343486069758Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Groupoid algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .Morita equivalence . . . . . . . . .

12、 . . . . . . . . . . . . . . . . . .Morita equivalence for C -algebras . . . . . . . . . . . . . . . . . .Noncommutative quotients . . . . . . . . . . . . . . . . . . . . . . .Sources of noncommutative spaces . . . . . . . . . . . . . . . . . . .3Cyclic cohomology838591981081111201321361391443.13.23

13、.3.10Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . .Hochschild cohomology as a derived functor . . . . . . . . . . . . . .Deformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . .Topological algebras . . . . . . . . . . . . . . . . . . . . . .

14、. . . .Examples: Hochschild (co)homology . . . . . . . . . . . . . . . . .Cyclic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . .Connes long exact sequence . . . . . . . . . . . . . . . . . . . . . .Connes spectral sequence . . . . . . . . . . . . . . . . . . . . . . .Cyclic modules

15、. . . . . . . . . . . . . . . . . . . . . . . . . . . . .Examples: cyclic cohomology . . . . . . . . . . . . . . . . . . . . .4ConnesChern character150150163180184.4ConnesChern character in K-theory . . . . . . . . . . . . . . . . . .ConnesChern character in K-homology . . . . . . . . . .

16、. . . . . .Algebras stable under holomorphic functional calculus . . . . . . . .A final word: basic noncommutative geometry in a nutshell . . . . . .viiiIntroductionAppendices186AGelfandNaimark theoremsA.1 Gelfands theory of commutative Banach algebras . . . . . . . . . . .A.2 States and the GNS con

17、struction . . . . . . . . . . . . . . . . . . . .186186190BCompact operators, Fredholm operators, and abstract index theory197CProjective modules204DEquivalence of categories206Bibliography209Index219IntroductionOne of the major advances of science in the 20th century was the discovery of a math- em

18、atical formulation of quantum mechanics by Heisenberg in 1925 94.1 From a mathematical point of view, this transition from classical mechanics to quantum me- chanics amounts to, among other things, passing from the commutative algebra of classical observables to the noncommutative algebra of quantum

19、 mechanical observ- ables. To understand this better we recall that in classical mechanics an observable of a system (e.g. energy, position, momentum, etc.) is a function on a manifold called the phase space of the system. Classical observables can therefore be multiplied in a pointwise manner and t

20、his multiplication is obviously commutative. Immediately after Heisenbergs work, ensuing papers by Dirac 67 and BornHeisenbergJordan 16, it clear that a quantum mechanical observable is a (selfadjoint) linear operatoron a Hilbert space, called the state space of the system. These operators can again

21、 be multiplied with composition as their multiplication, but this operation is not necessarily commutative any longer.2 In fact Heisenbergs commutation relationh12 ipq qp Dshows that position and momentum operators do not commute and this in turn can be shown to be responsible for the celebrated unc

22、ertainty principle of Heisenberg. Thus, to get a more accurate description of nature one is more or less forced to replace the commutative algebra of functions on a space by the noncommutative algebra of operators on a Hilbert space.A little more than fifty years after these developments Alain Conne

23、s realized that a similar procedure can in fact be applied to areas of mathematics where the classical notions of space (e.g. measure space, locally compact space, or smooth space) lose its applicability and relevance 37, 35, 36, 39. The inadequacy of the classical notion of space manifests itself f

24、or example when one deals with highly singular “bad quotients”: spaces such as the quotient of a nice space by the ergodic action of a group,or the space of leaves of a foliation in teric case, to give just two examples. In allthese examples the quotient space is typically ill-behaved, even as a top

25、ological space.For instance it may fail to be even Hausdorff, or have enough open sets, let alone being a reasonably smooth space. The unitary dual of a discrete group, except when the group is abelian or almost abelian, is another example of an ill-behaved space.1A rival proposal which, by the Ston

26、evon Neumann uniqueness theorem, turned out to be essentially equivalent to Heisenbergs was arrived at shortly afterwards by Schrödinger 161. It is however Heisenbergs matrix mechanics that directly and most naturally relates to noncommutative geometry.2Strictly speaking selfadjoint operators d

27、o not form an algebra since they are not d under multipli- cation. By an algebra of observables we therefore mean the algebra that they generate.xIntroductionOne of Connes key observations is that in all these situations one can define a noncommutative algebra through a universal method which we cal

28、l the noncommutative quotient construction that captures most of the information hidden in these unwieldy quotients. Examples of this noncommutative quotient construction include the crossed product by an action of a group, or in general by an action of a groupoid. In general the noncommutative quot

29、ient is the groupoid algebra of a topological groupoid.This new notion of geometry, which is generally known as noncommutative geome- try, is a rapidly growing new area of mathematics that interacts with and contributes to many disciplines in mathematics and physics. Examples of such interactions an

30、d con- tributions include: the theory of operator algebras, index theory of elliptic operators,algebraic and differential topology, number theory, the Standard Mof elementaryparticles, the quantum Hall effect, renormalization in quantum field theory, and string theory.To understand the basic ideas o

31、f noncommutative geometry one should perhaps first come to grips with the idea of a noncommutative space. What is a noncommutativespace? The answer to this question is based on one of the most profound ideas in mathematics, namely a duality or correspondence between algebra and geometry,3according t

32、o which every concept or statement in Algebra corresponds to, and can be equally formulated by, a similar concept and statement in Geometry.On a physiological level this correspondence is perhaps related to a division in the human brain: one computes and manipulates symbols with the left hemisphere

33、of the brain and one visualizes things with the right hemisphere. Computations evolve in time and have a temporal character, while visualization is instant and immediate. It was for a good reason that Hamilton, one of the creators of modern algebraic methods, called his approach to algebra, e.g. to

34、complex numbers and quaternions, the science of pure time 92.We emphasize that the algebra-geometry correspondence is by no means a new observation or a new trend in mathematics. On the contrary, this duality has always existed and has been utilized in mathematics and its applications very often. Th

35、e ear- liest example is perhaps the use of numbers in counting. It is, however, the case that throughout history each new generation of mathematicians has found new ways of formulating this principle and at the same time broadening its scope. Just to mention a few highlights of this rich history we

36、quote Descartes (analytic geometry), Hilbert (affine varieties and commutative algebras), GelfandNaimark (locally compact spaces and commutative C -algebras), and Grothendieck (affine schemes and commutative3For a modern and very broad point of view on this duality,to the one adopted in this book,th

37、e first section of Shafarevichs book 164 as well as Cartiers article 31.Algebra ! GeometryxiIntroductionrings). A key idea here is the well-known relation between a space and the commu- tative algebra of functions on that space. More precisely, there is a duality between certain categories of geomet

38、ric spaces and the corresponding categories of algebras representing those spaces. Noncommutative geometry builds on, and vastly extends, this fundamental duality between classical geometry and commutative algebras.For example, by a celebrated theorem of Gelfand and Naimark 82, one knows that the in

39、formation about a compact Hausdorff space is fully encoded in the algebra of continuous complex-valued functions on that space. The space itself can be recoveredas the space ofal ideals of the algebra. Algebras that appear in this way arecommutative C -algebras. This is a remarkable theorem since it

40、 tells us that any natural construction that involves compact spaces and continuous maps between themhas a purely algebraic reformulation, and vice-versa any statement about commutative C -algebras and C -algebraic maps between them has a purely geometric-topological meaning.Thus one can think of th

41、e category of not necessarily commutative C -algebras as the dual of an, otherwise undefined, category of noncommutative locally compact spaces. What makes this a successful proposal is, first of all, a rich supply of examples and, secondly, the possibility of extending many of the topological and g

42、eometric invariants to this new class of spaces and applications thereof.Noncommutative geometry has as its special case, in faits limiting case, classi-cal geometry, but geometry expressed in algebraic terms. In some respect this should be compared with the celebrated correspondence principle in qu

43、antum mechanics whereclassical mechanics appears as a limit of quantum mechanics for large quantum num- bers or small values of Plancks constant. Before describing the tools needed to study noncommutative spaces let us first briefly recall a couple of other examples from a long list of results in ma

44、thematics that put in duality certain categories of geometric objects with a corresponding category of algebraic objects.To wit, Hilberts Nullstellensatz states that the category of affine algebraic vari-eties over an algebraicallyd field is equivalent to the opposite of the category offinitely gene

45、rated commutative algebras without nilpotent elements (so-called reduced algebras). This is a perfect analogue of the GelfandNaimark theorem in the world ofalgebraic geometry. Similarly, Swans (resp. Serres) theorem states that the category of vector bundles over a compact Hausdorff space (resp. ove

46、r an affine algebraic vari- ety) X is equivalent to the category of finitely generated projective modules over the algebra of continuous functions (resp. the algebra of regular functions) on X.A pervasive idea in noncommutative geometry is to treat certain classes of noncom- mutative algebras as non

47、commutative spaces and to try to extend tools of geometry, topology, and analysis to this new setting. It should be emphasized, however, that, as a rule, this extension is hardly straightforward and most of the times involves sur- prises and new phenomena. For example, the theory of the flow of weig

48、hts and the corresponding modular automorphism group in von Neumann algebras 41 has no counterpart in classical measure theory, though the theory of von Neumann algebras is generally regarded as noncommutative measure theory. Similarly, as we shall see inxiiIntroductionChapters 3 and 4 of this book,

49、 the extension of de Rham (co)homology of manifolds to cyclic (co)homology for noncommutative algebras was not straightforward and needed some highly non-trivial considerations. As a matter of fact, de Rham cohomology can be defined in an algebraic way and therefore can be extended to all commutativ

50、e alge- bras and to all schemes. This extension, however, heavily depends on exterior products of the module of Kähler differentials and on the fact that one works with commutative algebras. In the remainder of this introduction we focus on topological invariants that have proved very useful in

51、 noncommutative geometry.Of all topological invariants for spaces, topological K-theory has the most straight- forward extension to the noncommutative realm. Recall that topological K-theory classifies vector bundles on a topological space. Motivated by the above-mentioned SerreSwan theorem, it is n

52、atural to define, for a not necessarily commutative ring A, K0.A/ as the group defined by the semigroup of isomorphism classes of finite projec- tive A-modules. Provided that A is a Banach algebra, the definition of K1.A/ follows the same pattern as for spaces, and the main theorem of topological K-

53、theory, the Bott periodicity theorem, extends to all Banach algebras 14.The situation was much less clear for K-homology, a dual of K-theory. By the work of Atiyah 6, BrownDouglasFillmore 22, and Kasparov 106, one can say, roughly speaking, that K-homology cycles on a space X are represented by abst

54、ract elliptic operators on X and, whereas K-theory classifies the vector bundles on X , K- homology classifies the abstract elliptic operators on X . The pairing between K-theoryand K-homology takes the form hD ; E iD index.DE /, the Fredholm index of theelliptic operator D with coefficients in the

55、vector bundle E. Now one good thing about this way of formulating K-homology is that it almost immediately extends to noncommutative C -algebras. The two theories are unified in a single theory called KK-theory, due to Kasparov 106.Cyclic cohomology was discovered by Connes in 1981 36, 39 as the rig

56、ht non- commutative analogue of the de Rham homology of currents and as a receptacle for a noncommutative Chern character map from K-theory and K-homology. One of the main motivations was transverse index theory on foliated spaces. Cyclic cohomology can be used to identify the K-theoretic index of t

57、ransversally elliptic operators which lie in the K-theory of the noncommutative algebra of the foliation. The formalism of cyclic cohomology and noncommutative Chern character maps form an indispensable part of noncommutative geometry. A very interesting recent development in cyclic co- homology theory is the Hopf cyclic cohomology of Hopf algebras and Hopf (co)module (co)algebras in general. Motivated by the original work in 54, 55 this theory has now been extended in 89, 90.The following “dictionary” illustrates noncommutative analogues of some of the c