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El. knyga: Introduction to C*-Algebras and Noncommutative Geometry

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This is the first textbook on C*-algebra theory with a view toward Noncommutative Geometry. Moreover, it fills a gap in the literature, providing a clear and accessible account of the geometric picture of K-theory and its relation to the C*-algebraic picture. The text can be used as the basis for a graduate level or a capstone course with the goal being to bring a relative novice up to speed on the basic ideas while offering a glimpse at some of the more advanced topics of the subject.  Coverage includes C*-algebra theory, K-theory, K-homology, Index theory and Connes Noncommuntative Riemannian geometry. 





Aimed at graduate level students, the book is also a valuable resource for mathematicians who wish to deepen their understanding of noncommutative geometry  and algebraic K-theory. A wide range of important examples are introduced at the beginning of the book. There are lots of excellent exercises and any student working through these will benefit significantly. Prerequisites include  a basic knowledge of algebra, analysis, and a bit of functional analysis.  As the book progresses, a little more mathematical maturity is required as the text discusses smooth manifolds, some differential geometry and elliptic operator theory, and K-theory. The text is largely self-contained though occasionally the reader may opt to consult more specialized material to further deepen their understanding of certain details.

Recenzijos

This book serves as a textbook that leads readers from a beginner's course on C*-algebra, through the Atiyah-Singer index theorem, to advanced topics in noncommutative geometry, such as cyclic cohomology and Kasparov's KK-theory. ... Each section is accompanied by a number of problems, and a notable advantage of the book is the extensive collection of examples scattered throughout the text. Some of them serve as a pretext for introducing some interesting advanced topics of noncommutative geometry. (Vladimir Manuilov, Mathematical Reviews, April, 2025)

An introduction to C*-algebras.- An Introduction to Index Theory and Noncommutative Geometry.- Spectral Theory and Representation.- Positivity, Representations, Tensor Products and Ideals in C*-algebras.- Module theory of C*-algebras.- Morita Equivalence.- Topological K-theory and Clifford Algebras.- K-theory for C*algebras.- The Index Theorem of Atiyah and Singer.- K-homology and Noncommutative Geometry.- An Introduction to KK-theory.- Bibliography.

Heath Emerson is Professor of Mathematics at the University of Victoria, British Columbia. His research interests include noncommutative geometry, equivariant K-theory, and the BaumConnes conjecture.
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