# zbMATH — the first resource for mathematics

Non-commutative differential geometry. (English) Zbl 0592.46056
In this article, Connes lays the groundwork for a theory of noncommutative differential geometry, i.e. differential geometry for noncommutative algebras generalizing the commutative algebra $${\mathcal C}^{\infty}(M)$$ of smooth functions on a compact manifold. The idea of doing topology of noncommutative ”topological spaces”, i.e. $$C^*$$- algebras, is not new and has, in fact, found a very satisfactory realization via $$K$$–theory and notably Kasparov’s $$KK$$–theory for $$C^*$$- algebras. To do differential geometry, one would like to have an analogue of de Rham (co)homology and characteristic classes. This generalization, the ”cyclic cohomology” is obtained by Connes in the following way.
Let $${\mathfrak A}$$ be a complex algebra. The differential envelope $$\Omega$$ $${\mathfrak A}$$ consists of all linear combinations of abstract (not necessarily antisymmetric) ”differential forms” $$x_ 0dx_ 1...dx_ n$$ over $${\mathfrak A}$$ (i.e. $$x_ i\in {\mathfrak A})$$ where $$d(xy)=xdy+d(x)y$$, x,y$$\in {\mathfrak A}$$. In other words $$\Omega$$ $${\mathfrak A}$$ is the universal algebra containing $${\mathfrak A}$$ and admitting a linear map $$d: \Omega{\mathfrak A}\to \Omega{\mathfrak A}$$ such that $$d(xy)=xdy+dx y$$, $$(x,y\in {\mathfrak A})$$ and $$d^ 2=0$$. Let $$\Omega^ n{\mathfrak A}$$ be the space of all such forms of degree $$n$$. Every trace $$T: \Omega^ n{\mathfrak A}\to {\mathbb C}$$ ($$T$$ is a trace if $$T(\omega\omega')= T(\omega'\omega)$$ for $$\omega \in \Omega^ k{\mathfrak A}$$, $$\omega'\in \Omega^{\ell}{\mathfrak A}$$, $$k+\ell =n)$$ which is in addition closed, i.e. $$T(d\omega)=0$$, $$\forall \omega$$, gives rise to an $$(n+1)$$-linear functional $$f(x_ 0,...,x_ n)=T(x_ 0dx_ 1...dx_ n)$$ which has the following properties
(1) $$f(x_ 1,...,x_ n,x_ 0)=(-1)^ nf(x_ 0,...,x_ n)$$
(2) $$bf=0$$ where $$b$$ is the Hochschild boundary operator
$$bf(x_ 0,...,x_{n+1})=f(x_ 0x_ 1,...,x_{n+1})-f(x_ 0,x_ 1x_ 2, ...,x_{n+1})+...+(-1)^{n+1}f(x_{n+1}x_ 0,...,x_ n).$$
Let $$C^ n_{\lambda}({\mathfrak A})$$ be the space of all $$f$$ satisfying (1). Then $$bC^ n_{\lambda}\subset C_{\lambda}^{n+1}$$ so that one obtains a subcomplex $$(C^ n_{\lambda},b)$$ of the Hochschild complex on $$(C^ n({\mathfrak A},{\mathfrak A}^*),b)$$. The cohomology of this complex is the ”cyclic cohomology” of $${\mathfrak A}$$- denoted by $$H^ n_{\lambda}({\mathfrak A}).$$
The introduction of this concept is motivated by Ext-theory and the Chern-character in $$K$$–homology. The construction of the Chern character in turn has its roots in the work of Helton and Howe and Kasparov’s $$KK$$–theory. An element of Kasparov’s $$K$$–homology group $$K^ 0({\mathfrak A})$$ is described by a pair ($$\phi$$,$${\bar \phi}$$) of *-homomorphisms of $${\mathfrak A}$$ into $${\mathcal B}(H)$$, $$H$$ a Hilbert space, such that $$\phi$$ (x)-$${\bar \phi}$$(x) is a compact operator for all $$x\in {\mathfrak A}$$. If now ($$\phi$$,$${\bar \phi}$$) is even such that $$q(x)=\phi (x)-{\bar \phi}(x)$$ is in the Schatten class $${\mathcal C}^{p+1}(H)$$ for all $$x\in {\mathfrak A}$$, one can define $$f(x_ 0,...,x_ p)=\text{Tr}(q(x_ 0)...q(x_ p))$$ where $$\text{Tr}$$ is the usual trace $${\mathcal C}^ 1(H)\to {\mathbb{C}}$$. One easily checks that $$f\in C^ p_{\lambda}({\mathfrak A})$$ if p is even and one can define the Chern character ch by $$ch_ p((\phi,{\bar \phi}))=[f]\in H^ p_{\lambda}({\mathfrak A})$$. This construction gives rise naturally to an operator $$S: H^ p_{\lambda}\to H_{\lambda}^{p+2}$$ which has the property that $$ch_{p+2}((\phi,{\bar \phi}))$$ (which is also defined since $${\mathcal C}^{p+1}\subset {\mathcal C}^{p+3})$$ equals S ch$${}_ p((\phi,{\bar \phi})).$$
One can now define $$H^{even}({\mathfrak A})=\lim_{\to}(H^ 0_{\lambda}({\mathfrak A})\to^{S}H^ 2_{\lambda \quad}({\mathfrak A})\to^{S}...)$$ and $$H^ p({\mathfrak A})$$ the image of $$H^ p_{\lambda}({\mathfrak A})$$ in $$H^{even}$$ divided by the image of $$H_{\lambda}^{p-2}({\mathfrak A})$$ (of course, the odd case is treated similarly). Connes shows that for $${\mathfrak A}={\mathcal C}^{\infty}(M)$$, $$M$$ a smooth compact manifold, one finds that $$H^ n({\mathfrak A})$$ is equal to the de Rham homology group $$H_ n(M,{\mathbb C})$$. He also establishes a long exact sequence $...\to H^{n+1}({\mathfrak A},{\mathfrak A}^*)\to H^ n_{\lambda}({\mathfrak A})\to^{S}H_{\lambda}^{n+2}({\mathfrak A})\to H^{n+2}({\mathfrak A},{\mathfrak A}^*)\to...$ connecting cyclic cohomology with Hochschild cohomology, and uses this sequence for instance to compute $$H^*({\mathfrak A})$$ for the canonical dense subalgebra of the ”irrational rotation algebra”.
The article contains, in addition, a wealth of information which is impossible to describe in a brief review. With cyclic cohomology we dispose of a completely new, unexpected and powerful tool opening many new roads in non-commutative topology, homological algebra, algebraic $$K$$–theory and probably also classical differential geometry.
Reviewer: J. Cuntz

##### MSC:
 46L87 Noncommutative differential geometry 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory 46L85 Noncommutative topology 58B32 Geometry of quantum groups 58B34 Noncommutative geometry (à la Connes) 18G35 Chain complexes (category-theoretic aspects), dg categories 18G60 Other (co)homology theories (MSC2010) 58A12 de Rham theory in global analysis 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 19D55 $$K$$-theory and homology; cyclic homology and cohomology 19K35 Kasparov theory ($$KK$$-theory) 19K56 Index theory 58J20 Index theory and related fixed-point theorems on manifolds 58J22 Exotic index theories on manifolds
Full Text:
##### References:
  W. Arveson, The harmonic analysis of automorphism groups, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), part I, 199–269.  M. F. Atiyah, Transversally elliptic operators and compact groups,Lecture Notes in Math. 401, Berlin-New York, Springer (1974). · Zbl 0297.58009  M. F. Atiyah, Global theory of elliptic operators,Proc. Internat. Conf. on functional analysis and related topics, Tokyo, Univ. of Tokyo Press (1970), 21–29. · Zbl 0193.43601  M. F. Atiyah, K-theory, Benjamin (1967).  M. F. Atiyah andI. Singer, The index of elliptic operators IV,Ann. of Math. 93 (1971), 119–138. · Zbl 0212.28603 · doi:10.2307/1970756  S. Baaj etP. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les C* modules Hilbertiens,C. r. Acad. Sci. Paris, Série I,296 (1983), 875–878.  P. Baum andA. Connes,Geometric K-theory for Lie groups and Foliations, Preprint I.H.E.S., 1982.  P. Baum andA. Connes,Leafwise homotopy equivalence and rational Pontrjagin classes, Preprint I.H.E.S., 1983.  P. Baum andR. Douglas, K-homology and index theory, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), part I, 117–173. · Zbl 0532.55004  O. Bratteli, Inductive limits of finite dimensional C*-algebras,Trans. Am. Math. Soc. 171 (1972), 195–234. · Zbl 0264.46057  L. G. Brown, R. Douglas andP. A. Fillmore, Extensions of C*-algebras and K-homology,Ann. of Math. (2)105 (1977), 265–324. · Zbl 0376.46036 · doi:10.2307/1970999  R. Carey andJ. D. Pincus, Almost commuting algebras, K-theory and operator algebras,Lecture Notes in Math. 575, Berlin-New York, Springer (1977). · Zbl 0358.46031  H. Cartan andS. Eilenberg,Homological algebra, Princeton University Press (1956). · Zbl 0075.24305  A. Connes, The von Neumann algebra of a foliation,Lecture Notes in Physics 80 (1978), 145–151, Berlin-New York, Springer. · Zbl 0433.46056  A. Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs,Lecture Notes in Math. 725, Berlin-New York, Springer (1979).  A. Connes, A Survey of foliations and operator algebras, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), Part I, 521–628. · Zbl 0531.57023  A. Connes, Classification des facteurs, Operator algebras and applications,Proc. Symposia Pure Math. 38 (1982), Part II, 43–109.  A. Connes andG. Skandalis, The longitudinal index theorem for foliations,Publ. R.I.M.S., Kyoto,20 (1984), 1139–1183. · Zbl 0575.58030 · doi:10.2977/prims/1195180375  A. Connes, C* algèbres et géométrie différentielle,C.r. Acad. Sci. Paris, Série I,290 (1980), 599–604.  A Connes,Cyclic cohomology and the transverse fundamental class of a foliation, Preprint I.H E.S. M/84/7 (1984). · Zbl 0647.46054  A. Connes, Spectral sequence and homology of currents for operator algebras. Math. Forschungsinstitut Oberwolfach Tagungsbericht 42/81,Funktionalanalysis und C*-Algebren, 27-9/3-10-1981.  J. Cuntz andW. Krieger, A class of C*-algebras and topological Markov chains,Invent. Math. 56 (1980), 251–268. · Zbl 0434.46045 · doi:10.1007/BF01390048  J. Cuntz, K-theoretic amenability for discrete groups,J. Reine Angew. Math. 344 (1983), 180–195. · Zbl 0511.46066 · doi:10.1515/crll.1983.344.180  R. Douglas, C*-algebra extensions and K-homology,Annals of Math. Studies 95, Princeton University Press, 1980.  R. Douglas andD. Voiculescu, On the smoothness of sphere extensions,J. Operator Theory 6 (1) (1981), 103. · Zbl 0501.46055  E. G. Effros, D. E Handelman andC. L. Shen, Dimension groups and their affine representations,Amer. J. Math. 102 (1980), 385–407. · Zbl 0457.46047 · doi:10.2307/2374244  G. Elliott, On the classification of inductive limits of sequences of semi-simple finite dimensional algebras,J. Alg. 38 (1976), 29–44. · Zbl 0323.46063 · doi:10.1016/0021-8693(76)90242-8  E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah Singer index theorem,Commun. Math. Physics 92 (1983), 163–178. · Zbl 0543.58026 · doi:10.1007/BF01210843  A. Grothendieck, Produits tensoriels topologiques,Memoirs Am. Math. Soc. 16 (1955). · Zbl 0123.30301  J. Helton andR. Howe, Integral operators, commutators, traces, index and homology,Proc. of Conf. on operator theory, Lecture Notes in Math. 345, Berlin-New York, Springer (1973). · Zbl 0268.47054  J. Helton andR. Howe, Traces of commutators of integral operators,Acta Math. 135 (1975), 271–305. · Zbl 0332.47010 · doi:10.1007/BF02392022  M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,Publ. Math. I.H.E.S. 49 (1979).  G. Hochschild, B. Kostant andA. Rosenberg, Differential forms on regular affine algebras,Trans. Am. Math. Soc. 102 (1962), 383–408. · Zbl 0102.27701 · doi:10.1090/S0002-9947-1962-0142598-8  L. Hörmander, The Weyl calculus of pseudodifferential operators,Comm. Pure Appl. Math. 32 (1979), 359–443. · Zbl 0396.47029 · doi:10.1002/cpa.3160320304  B. Johnson, Cohomology in Banach algebras,Memoirs Am. Math. Soc. 127 (1972).  B. Johnson, Introduction to cohomology in Banach algebras, inAlgebras in Analysis, Ed. Williamson, New York, Academic Press (1975), 84–99.  P. Julg andA. Valette, K-moyennabilité pour les groupes opérant sur les arbres,C. r. Acad. Sci. Paris, Série I,296 (1983), 977–980. · Zbl 0537.46055  D. S. Kahn, J. Kaminker andC. Schochet, Generalized homology theories on compact metric spaces,Michigan Math. J. 24 (1977), 203–224. · Zbl 0384.55001 · doi:10.1307/mmj/1029001885  M. Karoubi, Connexions, courbures et classes caractéristiques en K-théorie algébrique,Canadian Math. Soc. Proc., Vol. 2, part I (1982), 19–27.  M. Karoubi, K-theory. An introduction,Grundlehren der Math., Bd.226 (1978), Springer Verlag. · Zbl 0382.55002  M. Karoubi etO. Villamayor, K-théorie algébrique et K-théorie topologique I.,Math. Scand. 28 (1971), 265–307. · Zbl 0231.18018  G. Kasparov, K-functor and extensions of C*-algebras,Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571–636.  G. Kasparov, K-theory, group C*-algebras and higher signatures, Conspectus, Chernogolovka (1983). · Zbl 0526.22007  G. Kasparov,Lorentz groups: K-theory of unitary representations and crossed products, preprint, Chernogolovka, 1983. · Zbl 0526.22007  B. Kostant, Graded manifolds, graded Lie theory and prequantization,Lecture Notes in Math. 570, Berlin-New York, Springer (1975). · Zbl 0358.53024  J. L. Loday andD. Quillen, Cyclic homology and the Lie algebra of matrices,C. r. Acad. Sci. Paris, Série I,296 (1983), 295–297.  S. Mac Lane,Homology, Berlin-New York, Springer (1975).  J. Milnor, Introduction to algebraic K-theory,Annals of Math. Studies,72, Princeton Univ. Press. · Zbl 0237.18005  J. Milnor andD. Stasheff, Characteristic classes,Annals of Math. Studies 76, Princeton Univ. Press. · Zbl 1079.57504  A. S. Miščenko, Infinite dimensional representations of discrete groups and higher signatures,Math. USSR Izv. 8 (1974), 85–112. · Zbl 0299.57010 · doi:10.1070/IM1974v008n01ABEH002097  G. Pedersen, C*-algebras and their automorphism groups, New York, Academic Press (1979). · Zbl 0416.46043  M. Penington, K-theory and C*-algebras of Lie groups and Foliations, D. Phil. thesis, Oxford, Michaelmas, Term., 1983. · Zbl 0542.22013  M. Penington andR. Plymen, The Dirac operator and the principal series for complex semi-simple Lie groups,J. Funct. Analysis 53 (1983), 269–286. · Zbl 0542.22013 · doi:10.1016/0022-1236(83)90035-6  M. Pimsner andD. Voiculescu, Exact sequences for K-groups and Ext groups of certain cross-product C*-algebras,J. of operator theory 4 (1980), 93–118. · Zbl 0474.46059  M. Pimsner andD. Voiculescu, Imbedding the irrational rotation C* algebra into an AF algebra,J. of operator theory 4 (1980), 201–211. · Zbl 0525.46031  M. Pimsner andD. Voiculescu, K groups of reduced crossed products by free groups,J. operator theory 8 (1) (1982), 131–156. · Zbl 0533.46045  M. Reed andB. Simon,Fourier Analysis, Self adjointness, New York, Academic Press (1975). · Zbl 0308.47002  M. Rieffel, C*-algebras associated with irrational rotations,Pac. J. of Math. 95 (2) (1981), 415–429. · Zbl 0499.46039  J. Rosenberg, C*-algebras, positive scalar curvature and the Novikov conjecture,Publ. Math. I.H.E.S. 58 (1984), 409–424. · Zbl 0526.53044  W. Rudin,Real and complex analysis, New York, McGraw Hill (1966). · Zbl 0142.01701  I. Segal, Quantized differential forms,Topology 7 (1968), 147–172. · Zbl 0162.40602 · doi:10.1016/0040-9383(68)90024-4  I. Segal, Quantization of the de Rham complex,Global Analysis, Proc. Symp. Pure Math. 16 (1970), 205–210. · Zbl 0214.49001  B. Simon, Trace ideals and their applications,London Math. Soc. Lecture Notes 35, Cambridge Univ. Press (1979). · Zbl 0423.47001  I. M. Singer, Some remarks on operator theory and index theory,Lecture Notes in Math. 575 (1977), 128–138, New York, Springer. · Zbl 0444.47040  J. L. Taylor, Topological invariants of the maximal ideal space of a Banach algebra,Advances in Math. 19 (1976), 149–206. · Zbl 0323.46058 · doi:10.1016/0001-8708(76)90061-X  A. M. Torpe, K-theory for the leaf space of foliations by Reeb components,J. Funct. Analysis 61 (1985), 15–71. · Zbl 0594.46062 · doi:10.1016/0022-1236(85)90038-2  B. L. Tsigan, Homology of matrix Lie algebras over rings and Hochschild homology,Uspekhi Math. Nauk. 38 (1983), 217–218.  A. Valette, K-Theory for the reduced C*-algebra of semisimple Lie groups with real rank one,Quarterly J. of Math., Oxford, Série 2,35 (1984), 334–359.  A. Wasserman, Une démonstration de la conjecture de Connes-Kasparov, to appear inC. r. Acad. Sci. Paris.  A. Weil, Elliptic functions according to Eisenstein and Kronecker,Erg. der Math. vol. 88, Berlin-New York, Springer (1976). · Zbl 0318.33004  R. Wood, Banach algebras and Bott periodicity,Topology 4 (1965–1966), 371–389. · Zbl 0163.36702 · doi:10.1016/0040-9383(66)90035-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.