San Diego, CA: Academic Press. xiii, 661 p. $ 59.95 (1994).
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 Some of the material has been rearranged and a lot more details and new (also unpublished) results have been included in this new edition.
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 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.
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 C. C. Moore and C. Schochet [‘Global analysis on foliated spaces’, Springer-Verlag, Berlin (1988; Zbl 0648.58034)].
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 -theory of the associated -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. -modules and crossed products. The most important one explains -theory [the bivariant semi-exact theory introduced by the author and 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.
In Chapter III cyclic cohomology is expounded. In order to translate the -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 -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 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.
The main theme of Chapter IV (and of noncommutative geometry) is the construction of the Chern character in -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.
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 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)].
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”.