Fundamentals of the Theory of Operator Algebras. V1 (eBook)

Front Cover 1
Fundamentals of the Theory of Operator Algebras 4
Copyright Page 5
Contents 6
Preface 8
Contents of Volume II 14
Chapter 1. Linear Spaces 18
1.1. Algebraic results 18
1.2. Linear topological spaces 29
1.3. Weak topologies 45
1.4. Extreme points 48
1.5. Normed spaces 52
1.6. Linear functionals on normed spaces 60
1.7. Some examples of Banach spaces 65
1.8. Linear operators acting on Banach spaces 76
1.9. Exercises 82
Chapter 2. Basics of Hilbert Space and Linear Operators 92
2.1. Inner products on linear spaces 92
2.2. Orthogonality 102
2.3. The weak topology 114
2.4. Linear operators 116
2.5. The lattice of projections 126
2.6. Constructions with Hilbert spaces 137
2.7. Unbounded linear operators 171
2.8. Exercises 178
Chapter 3. Banach Algebras 190
3.1. Basics 190
3.2. The spectrum 195
3.3. The holomorphic function calculus 219
3.4. The Banach algebra C(X) 227
3.5. Exercises 240
Chapter 4. Elementary C*-Algebra Theory 253
4.1. Basics 253
4.2. Order structure 261
4.3. Positive linear functionals 272
4.4. Abelian algebras 286
4.5. States and representations 292
4.6. Exercises 302
Chapter 5. Elementary von Neumann Algebra Theory 321
5.1. The weak- and strong-operator topologies 321
5.2. Spectral theory for bounded operators 326
5.3. Two fundamental approximation theorems 342
5.4. Irreducible algebras—an application 347
5.5. Projection techniques and constructs 349
5.6. Unbounded operators and abelian von Neumann algebras 357
5.7. Exercises 387
Bibliography 401
Index of Notation 404
Index 408
Pure and Applied Mathematics 416