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Basic noncommutative geometry. (English) Zbl 1210.58006

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-061-6/pbk). xvi, 223 p. (2009).
There are already a few textbooks on noncommutative geometry [J. M. Gracia-Bondia, J. C. Várilly and H. Figueroa, Elements of noncommutative geometry. Birkhäuser Advanced Texts. Boston, MA: Birkhäuser (2001; Zbl 0958.46039); J. C. Várilly, An introduction to noncommutative geometry. EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society Publishing House (2006; Zbl 1097.58004); H. Moriyoshi and T. Natsume, Operator algebras and geometry. Translations of Mathematical Monographs 237. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1163.46001); G. Landi, An introduction to noncommutative spaces and their geometries. Lecture Notes in Physics. New Series m: Monographs. m51. Berlin: Springer (1997; Zbl 0909.46060); and J. Cuntz, R. Meyer and J. M. Rosenberg, Topological and bivariant K-theory. Oberwolfach Seminars 36. Basel: Birkhäuser (2007; Zbl 1139.19001)] besides Connes’ [A. Connes and M. Marcolli, Noncommutative geometry, quantum fields and motives. Hindustan Book Agency, New Delhi (2008; Zbl 1159.58004); and A. Connes, Noncommutative geometry. San Diego, CA: Academic Press (1994; Zbl 0818.46076)].
This book, consisting of 4 chapters, is a truely introductory textbook on noncommutative geometry. Now I will explain the contents of the book.
To understand the basic ideas of noncommutative geometry one should perhaps first grasp the idea of noncommutative spaces, which is based upon a duality or correspondence between algebra and geometry. Therefore, Chapter 1 of the book is devoted to some famous examples of the duality successfully finding their place in noncommutative geometry, say,
1.
the famous theorem of Gelfand and Naimark in operator algebra claiming that the information about a compact Hausdorff space is completely encoded in the commutative \(C^{\ast}\)-algebra of continuous complex-valued functions on that space;
2.
the duality between affine schemes and commutative rings in algebraic geometry, the basic idea of which goes back to David Hilbert’s celebrated Nullstellensatz and
3.
the Serre-Swan theorem concerning the duality between the category of vector bundles over a compact Hausdorff space (the category of algebraic vector bundles over an affine algebraic variety, resp.) and that of finitely generated projective modules over the algebra of continuous functions (that of finite projective modules over the coordinate ring of the variety, resp.) [cf. R. G. Swan, Trans. Am. Math. Soc. 105, 264–277 (1962; Zbl 0109.41601); and J.-P. Serre, Algèbre Théorie Nombres, Sem. P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot 11 (1957/58), No. 23 (1958; Zbl 0132.41202)],
and to some other examples failing to find their place in noncommutative geometry, say,
4.
the duality between the category of sets and the category of complete atomic Boolean algebras; and
5.
the duality between the category of compact Riemann surfaces and the category of algebraic function fields.
The final section of Chapter 1 is concerned with Hopf algebras as the noncommutative counterpart of the classical notion of group.
Most of the noncommutative spaces that are currently in use in noncommutative geometry are constructed by the method of noncommutaive quotients which was pioneered by A. Connes in [Noncommutative geometry. Paris: InterEditions (1990; Zbl 0745.46067); and loc. cit.]. Since groupoids and groupoid algebras provide a unified approach, Chapter 2 begins with the definition of a groupoid (Section 2.1), which is followed by groupoid algebras in Section 2.2. Since Morita theory [K. Morita, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83–142 (1958; Zbl 0080.25702)] finds its natural extension in the context of \(C^{\ast}\)-algebras, Sections 2.3 and 2.4 are devoted to Morita equivalence and its extension to \(C^{\ast}\)-algebras. The final section of Chapter 2 describes the method of noncommutaive quotients described above.
Cyclic cohomology was announced by Alain Connes in 1981 at a conference in Oberwolfach as a noncommutative analogue of the Chern character. His basic ideas can be seen in [A. Connes, C. R. Acad. Sci., Paris, Sér. I 296, 953–958 (1983; Zbl 0534.18009)]. This theory plays the role of the de Rham cohomology of currents on smooth manifolds. It has to be stressed that to define a noncommutative de Rham theory for noncommutative algebras is by no means trivial, while to extend the topological \(K\)-theory to noncommutative Banach algebras is almost straightforward. Chapter 3 is devoted to cyclic cohomology, its relation with Hochschild cohomology through Connes’ long exact sequence and spectral sequence, and its relation with de Rham cohomology.
As is well known, the classical commutative Chern character relates the \(K\)-theory of a space to its ordinary cohomology. In noncommutative geometry, there is also a very important dual \(K\)-homology theory built out of abstract elliptic operators on the noncommutative space, or rather a single theory called \(KK\)-theory combining \(K\)-theory and \(K\)-homology in the sense of [G. G. Kasparov, Math. USSR, Izv. 16, 513–572 (1981; Zbl 0464.46054)]. Chapter 4 studies the noncommutative analogues of Chern character maps for both \(K\)-theory and \(K\)-homology, with values in cyclic homology and cyclic cohomology, respectively, culminating in a beautiful index formula of Connes.
The book is accompanied by four appendices on the Gelfand-Naimark theorem, the abstract index theory, projective modules and equivalence of categories.

MSC:

58B34 Noncommutative geometry (à la Connes)
16S38 Rings arising from noncommutative algebraic geometry
46L87 Noncommutative differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
16T05 Hopf algebras and their applications
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