An Introduction to C*-Algebras and Noncommutative Geometry

Preface
Contents
1 An Introduction to C*-Algebras
	1.1 The Definition of C*-Algebra
	1.2 The C*-Algebra of Bounded Operators on a Hilbert Space
	1.3 Group C*-Algebras
	1.4 C*-Algebras of the Integers and the Circle
	1.5 C*-Algebras of Finite Groups
	1.6 The Compact Operators
	1.7 Inductive Limits of C*-Algebras
	1.8 Crossed Product C*-Algebras
	1.9 The C*-Algebra and Fourier Transform for the Group of Real Numbers
2 An Introduction to Index Theory and Noncommutative Geometry
	2.1 Trace-Class and Schatten Classes of Compact Operators
	2.2 The Toeplitz Algebra
	2.3 The Toeplitz Trace Theorem
	2.4 The Calkin Algebra
	2.5 General Properties of the Fredholm Index
	2.6 The Toeplitz Index Theorem
3 Spectral Theory and Representations
	3.1 Spectrum in a Banach Algebra
	3.2 The Holomorphic Functional Calculus
	3.3 Characters and Gelfand's Theorem
	3.4 Gelfand's Theorem
	3.5 Functional Calculus, Isospectral Subalgebras
4 Positivity, Representations, Tensor Products, and Ideals in C*-Algebras
	4.1 Positivity and States
	4.2 The GNS Theorem
	4.3 Generalities About Completions of *-Algebras
	4.4 Ideals and Quotients of C*-Algebras
	4.5 Tensor Products of C*-Algebras
	4.6 Structure of Crossed Products by Proper Actions of Discrete Groups
5 Module Theory of C*-Algebras
	5.1 Vector Bundles
	5.2 Finitely Generated Projective (f.g.p.) Modules and Vector Bundles: Swan's Theorem
	5.3 Multiplier Algebras
	5.4 Hilbert Modules
	5.5 Operators on Hilbert Modules, Tensor Products, and Applications
6 Morita Equivalence
	6.1 Morita Equivalence
	6.2 Finitely Generated Projective Modules and Morita Correspondences
	6.3 Morita Correspondences from Equivariant Vector Bundles
	6.4 Morita Correspondences and Representations of Finite Groups
	6.5 Remarks on Compact Noncommutative Spaces
	6.6 Morita Correspondences Between Irrational Tori
	6.7 Morita Equivalence and Asymptotics in Hyperbolic Geometry
7 Topological K-Theory and Clifford Algebras
	7.1 The Definition of K-Theory, the K-Theory of the Circle
	7.2 Vector Bundles on Smooth Manifolds
	7.3 Functoriality and Homotopy-Invariance
	7.4 K-Theory for Noncompact Spaces, Higher K-groups
	7.5 The Long Exact Sequence of a Pair
	7.6 Bott Periodicity, the 6-Term Exact Sequence
	7.7 Spin Geometry and Clifford Algebras
	7.8 Representation Theory of Clifford Algebras
	7.9 Clifford Algebras and K-Theory: The Bott-Shapiro Theorem
	7.10 K-Orientations and the Thom Isomorphism Theorem
8 K-Theory for C*-Algebras
	8.1 Basic Definitions of C*-Algebra K-Theory
	8.2 Morita Invariance and Applications
	8.3 Higher K-Theory, Loops and Unitaries
	8.4 The Long Exact Sequence
	8.5 The Long Exact Sequence
	8.6 Examples of the Connecting Homomorphism
	8.7 The External Product Operation on K-Theory
	8.8 The Bott Periodicity Theorem
	8.9 Some Orbifold K-Theory Computations
9 The Index Theorem of Atiyah and Singer
	9.1 Differential Operators on Euclidean Space
	9.2 Differential Operators on Manifolds
	9.3 Analytic Aspects of Elliptic Operators
	9.4 Dirac Operators
	9.5 Bounded Transforms of Dirac Operators
	9.6 The K-Theory Index Theorem(s)
10 K-Homology and Noncommutative Geometry
	10.1 Fredholm Modules and Their Pairing with K-Theory
	10.2 Cyclic Cohomology
	10.3 Finitely Summable Fredholm Modules and Connes Character Formula
	10.4 Fredholm Modules from Boundary Actions of Hyperbolic Groups
	10.5 Fredholm Modules from Extensions
	10.6 Spectral Cycles and Fredholm Modules
	10.7 The Heat Equation Proof of the Atiyah–Singer Index Theorem
	10.8 The Atiyah–Singer and Connes–Moscovici Local Index Theorems
	10.9 Zeta Functions and the Local Index Theorem for the Circle
	10.10 Heisenberg Spectral Cycles and Irrational Rotation
	10.11 The Harmonic Oscillator Residue Trace
	10.12 The Local Index Formula for the Heisenberg Cycles
11 An Introduction to KK-Theory
	11.1 Basic Definitions of KK
	11.2 Category Structure of KK
	11.3 The Axiomatic Approach to the Kasparov Product
	11.4 The Bott Periodicity Theorem in KK-Theory
	11.5 Equivariant Bott Periodicity and the K-Theory of Crossed Products
Bibliography