Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0818.46076
Connes, Alain
Noncommutative geometry. Transl. from the French by Sterling Berberian. (English)
[B] San Diego, CA: Academic Press. xiii, 661 p. \$ 59.95 (1994). ISBN 0-12-185860-X/pbk

This is the English version of A. Connes' popular book first published in French in 1990 (see my review Zbl 0745.46067). Since it is more than a mere translation we have to supplement the previous review accordingly. Some of the material has been rearranged and a lot more details and new (also unpublished) results have been included in this new edition.\par After a general introduction which summarizes the content of the book, there are four chapters dealing essentially with noncommutative measure theory, topology, and differential topology. The next chapter is on operator algebras. Here the only notable change to the original version is a section about Hecke algebras and a statistical theory of prime numbers which is based on joint work with {\it J. B. Bost} [C. R. Acad. Sci., Paris, Sér. I 315, 279-284 (1992; Zbl 0781.46045)]. The final chapter is devoted to differential geometry and to Connes' most original application of noncommutative geometry to particle physics.\par Let us be more specific. In the first chapter the author starts from the Heisenberg picture of quantum mechanics and motivates the modular theory of operator algebras. After displaying the classification of von Neumann algebras he gives examples for the various types of factors in terms of von Neumann algebras associated with foliations. Then he states the index theorem for longitudinal elliptic operators on foliations with transverse measure. This was his first major result in noncommutative analysis obtained in 1978 [in: Algèbres d'opérateurs, Lect. Notes Math. 725, 19-143 (1979; Zbl 0412.46053)]. For a fuller treatment see also {\it C. C. Moore} and {\it C. Schochet} [`Global analysis on foliated spaces', Springer-Verlag, Berlin (1988; Zbl 0648.58034)].\par The longitudinal index theorem is also one of the highlights of noncommutative topology as developed in Chapter II. The essential construction which encodes the information about noncommutative spaces is that of a groupoid. This occurs in the guise of the tangent groupoid of a manifold and of the holonomy groupoid of a foliation. The $K$-theory of the associated $C\sp*$-algebras renders a unifying framework for the appropriate index theorems which, besides the classical Atiyah-Singer index theorem for which a new short proof is given, also includes the index theorems for covering spaces and for homogeneous spaces related to the dual spaces of discrete groups and Lie groups, respectively. There are several technical appendices providing the proper background in e.g. $C\sp*$-modules and crossed products. The most important one explains $E$-theory [the bivariant semi-exact theory introduced by the author and {\it N. Higson}, C. R. Acad. Sci. Paris, Sér. I 311, 101-106 (1990; Zbl 0717.46062), that extends Kasparov's KK-theory], and contains complete proofs of the main properties of the intersection product in this setting.\par In Chapter III cyclic cohomology is expounded. In order to translate the $K$-theoretical formulation of index theorems into cohomological language there is a need for de Rham (co)homology in the noncommutative setting. The analogue of de Rham homology is provided by cyclic cohomology which is dealt with in great detail (including proofs). The cyclic cohomology of several smooth commutative and noncommutative spaces (e.g. compact smooth manifolds, group rings of discrete groups and crossed products with discrete groups) is given explicitly. Then using connections and curvature operators, as in the theory of characteristic classes, the pairing of cyclic cohomology with $K$-theory is established and is used to state the higher index theorem for covering spaces and the cohomological version of the longitudinal index theorem. Applications are given for the first to Novikov's conjecture concerning higher signatures [see the author and {\it H. Moscovici}, Topology 29, 345-388 (1990; Zbl 0759.58047)] and for the second to obstructions to leafwise positive scalar curvature on foliations [see the author in: Geometric methods in operator algebras (Kyoto 1983), Pitman Res. Notes Math. 123, 52-144 (1986; Zbl 0647.46054)] using the transverse fundamental class of a foliation.\par The main theme of Chapter IV (and of noncommutative geometry) is the construction of the Chern character in $K$-homology. The underlying idea is lent from physics where commutators of classical observables are replaced by commutators of operators. The role of the Hamilton operator is taken here by a Fredholm module given by a selfadjoint involution on Hilbert space and by a representation of the algebra of ``observables''. In this chapter one finds a lot of examples and (partly new and unpublished) applications ranging from fractal sets to the quantum Hall effect which cannot be summarized in a review.\par Strictly speaking, noncommutative differential geometry is introduced in Chapter VI. Here the author shows how the metric aspects of Riemannian geometry can be described using Dirac operators. This new concept allows for the extension of the Yang-Mills action of the curvature of a connection to the noncommutative case. The computation of this action in several special examples leads to a new interpretation of the Glashow- Weinberg-Salam model in particle physics and finally gives an improved model (the standard $U(1)\times SU(2)\times SU(3)$ model) incorporating quarks and strong interactions. These applications to particle physics (partly obtained in joint work with J. Lott) build on the paper mentioned at the end of the previous review and have been published before in [New symmetry principles in quantum field theory (ed. J. Fröhlich et al.), Plenum Press, New York, 1992 (see also the review in M.R. 93m:58011)].\par This book is a masterpiece of mathematical exposition. One cannot put it better than in the words of V. F. R. Jones: ``A milestone for mathematics. Connes has created a theory that embraces most aspects of `classical' mathematics and sets us out on a long and exciting voyage into the world of noncommutative mathematics''.
[H.Schröder (Dortmund)]

MSC 2000:: *46-02 Research monographs (functional analysis)
58-02 Research monographs (global analysis)
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L80 K-theory and operator algebras
58J20 Index theory and related fixed point theorems
58J22 Exotic index theories

Keywords: noncommutative measure theory, topology, and differential topology; Hecke algebras; particle physics; Heisenberg picture of quantum mechanics; modular theory of operator algebras; classification of von Neumann algebras; factors; von Neumann algebras associated with foliations; index theorem for longitudinal elliptic operators on foliations with transverse measure; tangent groupoid of a manifold; holonomy groupoid of a foliation; Atiyah-Singer index theorem; $E$-theory; bivariant semi-exact theory; cyclic cohomology; de Rham (co)homology in the noncommutative setting; connections; curvature operators; Novikov's conjecture; Chern character; Fredholm module given by a selfadjoint involution on Hilbert space; fractal; quantum Hall effect; Dirac operators; Yang-Mills action; curvature of a connection; Glashow-Weinberg-Salam model in particle physics

Citations: Zbl 0745.46067; Zbl 0781.46045; Zbl 0412.46053; Zbl 0648.58034; Zbl 0717.46062; Zbl 0759.58047; Zbl 0647.46054

Cited in: Zbl 1260.58002 Zbl 1256.19003 Zbl 1256.46042 Zbl 1251.58010 Zbl 1248.58015 Zbl 1241.58012 Zbl 1219.58002 Zbl 1242.46072 Zbl 1232.58012 Zbl 1213.81209 Zbl 1201.58002 Zbl 1197.17015 Zbl 1193.81003 Zbl 1210.58006 Zbl 1204.58016 Zbl 1191.53059 Zbl 1189.46023 Zbl 1188.19004 Zbl 1188.46045 Zbl 1182.58005 Zbl 1171.19003 Zbl 1161.58006 Zbl 1147.58021 Zbl 1176.11024 Zbl 1151.46053 Zbl 1133.58010 Zbl 1120.46044 Zbl 1117.58012 Zbl 1112.45006 Zbl 1109.15020 Zbl 1106.58014 Zbl 1082.58005 Zbl 1081.58005 Zbl 1077.22006 Zbl 1076.58003 Zbl 1116.46022 Zbl 1071.58008 Zbl 1064.01018 Zbl 1058.19002 Zbl 1043.46048 Zbl 1043.46049 Zbl 1013.46056 Zbl 1011.19004 Zbl 1008.58008 Zbl 0999.58004 Zbl 1018.58011 Zbl 1003.46041 Zbl 0989.81128 Zbl 0987.57007 Zbl 0958.46039 Zbl 0987.53035 Zbl 0939.35202 Zbl 0936.46002 Zbl 0932.35171 Zbl 0951.16003 Zbl 0906.43009 Zbl 0895.18006 Zbl 0993.46032 Zbl 0942.46044 Zbl 0924.58002 Zbl 0911.53060 Zbl 0908.46041 Zbl 0902.19002 Zbl 0899.58006 Zbl 0898.46053 Zbl 0921.46066 Zbl 0881.58009 Zbl 0868.58076 Zbl 0868.58009 Zbl 0849.46049 Zbl 1042.46515 Zbl 0933.46069 Zbl 0871.58008 Zbl 0827.19001

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]