An_Introduction_to_Operator_Algebras-Marcoux.pdf

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AnIntroductiontoOperatorAlgebras
LaurentW.Marcoux
March30,2005
Preface
These notes were designed as lecture notes for a first course in Operator
Algebras. The student is assumed to have already taken a first course in
Linear Analysis. In particular, they are assumed to already know the Hahn-
Banach Theorem, the Open Mapping Theorem, etc. A list of those results
which will be used in the sequel is included in the second section of the first
chapter.
March 30, 2005
i
Contents
Preface
i
Chapter 1. A Brief Review of Banach Space Theory
1
1. Definitions and examples
1
2. The main theorems
4
Chapter 2. Banach Algebras
7
1. Basic theory
7
2. The functional calculus
19
3. The spectrum
30
Notes for Chapter Two
37
Chapter 3. Operator Algebras
41
1. The algebra of Banach space operators
41
2. The Fredholm Alternative
51
3. The algebra of Hilbert space operators
60
4. The spectral theorem for compact normal operators
66
5. Fredholm theory in Hilbert space
74
Notes for Chapter Three
80
Chapter 4. Abelian Banach Algebras
83
1. The Gelfand Transform
83
2. The radical
89
Chapter 5. C*-Algebras
99
1. Definitions and Basic Theory.
99
2. Elements of C -algebras.
108
3. Ideals in C -algebras.
117
4. Linear Functionals and States on C -algebras.
125
5. The GNS Construction.
134
Chapter 6. Von Neumann algebras
139
1. Introduction
139
2. The spectral theorem for normal operators.
145
Appendix A: The essential spectrum
155
155
iii
3. Examples
91

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