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Chapter 1．Spectral Theory and Banach Algebras 1

1.1．Origins of Spectral Theory 1

1.2．The Spectrum of an Operator 5

1.3．Banach Algebras:Examples 7

1.4．The Regular Representation 11

1.5．The General Linear Group of A 14

1.6．Spectrum of an Element of a Banach Algebra 16

1.7．Spectral Radius 18

1.8．Ideals and Quotients 21

1.9．Commutative Banach Algebras 25

1.10．Examples:C(X)and the Wiener Algebra 27

1.11．Spectral Permanence Theorem 31

1.12．Brief on the Analytic Functional Calculus 33

Chapter 2．Operators on Hilbert Space 39

2.1．Operators and Their C*-Algebras 39

2.2．Commutative C*-Algebras 46

2.3．Continuous Functions of Normal Operators 50

2.4．The Spectral Theorem and Diagonalization 52

2.5．Representations of Banach *-Algebras 57

2.6．Borel Functions of Normal Operators 59

2.7．Spectral Measures 64

2.8．Compact Operators 68

2.9．Adjoining a Unit to a C*-Algebra 75

2.10．Quotients of C*-Algebras 78

Chapter 3．Asymptotics:Compact Perturbations and Fredholm Theory 83

3.1．The Calkin Algebra 83

3.2．Riesz Theory of Compact Operators 86

3.3．Fredholm Operators 92

3.4．The Fredholm Index 95

Chapter 4．Methods and Applications 101

4.1．Maximal Abelian von Neumann Algebras 102

4.2．Toeplitz Matrices and Toeplitz Operators 106

4.3．The Toeplitz C*-Algebra 110

4.4．Index Theorem for Continuous Symbols 114

4.5．Some H2 Function Theory 118

4.6．Spectra of Toeplitz Operators with Continuous Symbol 120

4.7．States and the GNS Construction 122

4.8．Existence of States:The Gelfand-Naimark Theorem 126

Bibliography 131

Index 133