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60 pages PDF, year 2007

Abstract:

An excellent presentation of the noncommutative index theory and the equivariant KK-theory, described by the following exact citation from the author's presentation: "In noncommutative geometry, a C*-algebra is regarded as an algebra of continuous functions on a certain 'noncommutative space'. There are two natural topological invariants of such spaces. On the one hand, K-theory of C*-algebras extends the Atiyah-Hirzebruch K-theory of topological spaces. On the other hand, K-homology arises from the study of analysis on the noncommutative space. These two theories, which are dual to each other, find a natural unification in the KK-theory of Kasparov, which arises from a deep understanding of the Atiyah-Singer index theory. The first part of this course will provide the necessary background, the definition and the fundamental properties of KK-theory. In particular, we shall cover the basic properties of Fredholm modules, and their characters, which are given by natural cyclic cocycles first defined by Alain Connes" CONTENTS: Introduction to KK-theory p. 4 Motivation and background p.4 4 Definition . . . . . . . . . . . . . . 5 Properties . . . . . . . . . . . . 6 Applications and further development . . . . .. 7 1 C*-algebras p.9 1.1 Definitions . . . . . . . . . . . 9 1.2 Examples . . . . . . . . . . . . . . . 9 1.3 Gelfand transform . . . . . . . . . 13 2 K-theory 16 2.1 Definitions . . . . . . . . 16 2.2 Unitizations and multiplier algebras . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Stabilization . . . . . . . . . 18 2.4 Higher K-theory . . . . . . 18 2.5 Excision and relative K-theory . . . . . . 19 2.6 Products p. 21 2.7 Bott periodicity 21 2.8 Cuntza's proof of Bott periodicity . . 23 2.9 The Mayer-Vietoris sequence . . 24 3 Hilbert modules 25 3.1 Definitions 25 3.2 Kasparov stabilization theorem . 29 3.3 Morita equivalence . 29 3.4 Tensor products of Hilbert modules . 30 4 Fredholm modules and Kasparovâs K-homology 32 4.1 Fredholm modules 32 4.2 Commutator conditions . 35 4.3 Quantised calculus of one variable 37 4.4 Quantised differential calculus . 37 4.5 Closed graded trace. 38 4.6 Index pairing formula . 39 4.7 Kasparov's K--homology. (Free online download of PDF lecture notes at: http://toknotes.mimuw.edu.pl/sem6/files/Brodzki_ekk.pdf )

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Open download of PDF at: http://toknotes.mimuw.edu.pl/sem6/files/Brodzki_ekk.pdf

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