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Noncommutative geometry. (Géométrie non commutative.) (French) Zbl 0745.46067

Paris: InterEditions. 240 p. (1990).
With this book the foremost proponent of noncommutative geometry, in fact its inventor or rather discoverer as he would perhaps object, tries to popularize the basic concepts of noncommutative geometry and some of its applications. It is addressed to mathematicians and physicists who want to delve into the subject without going through all the technical details. But it is also worth reading for people already working in this field, since here they will find the underlying ideas arising from the author’s view of the interaction of mathematics and physics.
The book is composed of different articles originating from previously delivered talks which were however not completely independent. Therefore, one meets one and the same topic such as cyclic cohomology several times, but seen under different aspects.
The introductory chapter 1 is based on the author’s inaugural lecture at the Collège de France and gives a general outline of noncommutativity. It starts with the precursor of noncommutative geometry. Heisenberg’s quantization of classical mechanics which replaces the phase space of classical physics by quantum space or, more precisely, the commuting classical observables, i.e. functions in a function space, by noncommuting operators in an operator algebra put on solid ground by the work of von Neumann. Next the author recalls how non Neumann algebras provide a noncommutative integration theory and how \(C^*\)-algebras help conceive such nonspaces like the space of leaves of a foliation or the space of irreducible representations (mod equivalency) of a nonabelian group as a noncommutative topological space. Choosing a dense subalgebra and abstract elliptic operators these can moreover be endowed with differentiable or Riemannian metric structures which allows to perform cohomological calculations. Finally, this new concept of space reenters physics in new models for particles.
Chapter 2 is an extended version of a talk given at a conference in honour of John von Neumann [Proc. Sympos. Pure Math. 50 (1990; Zbl 0707.46053)]. It is emphasized that even a commutative space is in the first instance not commutative at all. Thinking of a space as given by local coordinate patches which are linked by suitable changes of frames commutativity comes in only after glueing these local charts, i.e., after an identification. Now a noncommutative space might be trivial if these identifications are made and so is better described by an algebra that incorporates the transition maps between the various local data. A simple example is given by a system of two one-pointed charts describing one and the same point. To build in the information that is lost by simply identifying them one better chooses a \(2\times 2\)-matrix algebra where the single charts are represented by the diagonal entries and the coupling by the off-diagonal terms.
Elaborating on this basic example the author has introduced for instance the foliation \(C^*\)-algebra associated to the space of leaves of a foliation and has used the group \(C^*\)-algebra of a nonabelian group to represent the “dual” group. From this point of view the Penrose tilings appear as a zero-dimensional noncommutative space. One of the central problems for commutative spaces, e.g. topological or differentiable manifolds, is to classify them up to homeomorphisms or diffeomorphisms or at least up to homotopy type. This is done using the apparatus of algebraic topology, i.e., homology and cohomology theories. In the case of noncommutative spaces given by a \(C^*\)-algebra the simplest cohomology to define is \(K\)-theory and leads to a first rough classification if the corresponding \(K\)-groups can be computed explicitly. To obtain numerical invariants in the commutative case one is supplied with a fundamental tool, the Chern character relating \(K\)-theory and cohomology (or \(K\)-homology and homology in the dual situation), which leads to characteristic numbers by pairing with homology classes (or with cohomology classes). An auxiliary differentiable structure can then be used (in case of a differentiable manifold) to make this pairing explicit: evaluating cohomology classes on a fundamental cycle is done by choosing a representing differential form and integration over the manifold, or in functional analytic terms applying a current to the differential form. Note that this pairing lies also at the heart of the index theorem in its cohomological version or in its abstract form as a pairing with a cycle in \(K\)-homology.
To this end the author introduced the notion of differential forms, by specifying a dense subalgebra of a given \(C^*\)-algebra to have a differentiable structure at hand, and the appropriate pendant to (co)homology, i.e., cyclic (co)homology. He also succeeded in establishing index theorems in this setting with some remarkable applications: the proof of Kadison’s conjecture that certain group \(C^*\)-algebras do not contain any nontrivial idempotent and (in joint work with H. Moscovici) a proof of Novikov’s conjecture concerning higher signatures for a wide class of groups.
Chapter 3 grew out of lectures given at a conference on operator theory in 1980 [Proc. Sympos. Pure Math. 38 (1982; Zbl 0503.46043)], and deals with the author’s work on the classification of von Neumann algebra factors that earned him the Fields medal in 1982. Here he begins with Murray’s and von Neumann’s fundamental papers, covers the Gelfand-Naimark theory, noncommutative integration, weights, and the theory of factors, and ends up with Jones’s classification of subfactors which in turn led to new knot invariants and recently had a strong impact on string theory.
The next two chapters are primarily devoted to applications in physics. In chapter 4 the author presents the theoretic model proposed by J. Bellissard to explain the integer jumps in the quantum Hall effect, and the final chapter — almost identical with “Essays on Physics and Noncommutative Geometry” [In: Interfaces of Mathematics and Particle Physics, Oxford, Clarendon Press (1990)] — develops new models in particle physics, in particular, a double sheeted space-time which is in accordance with the Weinberg-Salem model of electroweak interaction. Although these models only use commutative algebras the metric structure is imposed by using noncommutative concepts, more precisely, a \(K\)-cycle given by the Dirac operator. Note that subsequent joint work of the author with J. Lott has led to refined particle models that include quarks [In: Recent Advances in Field Theory, Annecy 1990, Nucl. Physics B, Proc. Suppl. 18 B (1991)].

MSC:

46L87 Noncommutative differential geometry
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46L85 Noncommutative topology
81T70 Quantization in field theory; cohomological methods
81S10 Geometry and quantization, symplectic methods
58J22 Exotic index theories on manifolds
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L37 Subfactors and their classification
19K56 Index theory
19D55 \(K\)-theory and homology; cyclic homology and cohomology
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