算子理论教程 影印版PDF电子书下载

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Chapter 1.Introduction to C＊-Algebras 1

1.Definition and examples 1

2.Abelian C＊-algebras and the Functional Calculus 7

3.The positive elements in a C＊-algebra 12

4.Approximate identities 17

5.Ideals in a C＊-algebra 21

6.Representations of a C＊-algebra 24

7.Positive linear functionals and the GNS construction 29

Chapter 2.Normal Operators 37

8.Some topologies on B（H） 37

9.Spectral measures 41

10.The Spectral Theorem 47

11.Star-cyclic normal operators 51

12.The commutant 55

13.Von Neumann algebras 60

14.Abelian von Neumann algebras 62

15.The functional calculus for normal operators 65

Chapter 3.Compact Operators 71

16.C＊-algebras of compact operators 71

17.Ideals of operators 82

18.Trace class and Hilbert-Schmidt operators 86

19.The dual spaces of the compact operators and the trace class 93

20.The weak-star topology 95

21.Inflation and the topologies 99

Chapter 4.Some Non-Normal Operators 105

22.Algebras and lattices 105

23.Isometries 111

24.Unilateral and bilateral shifts 118

25.Some results on Hardy spaces 126

26.The functional calculus for the unilateral shift 132

27.Weighted shifts 136

28.The Volterra operator 143

29.Bergman operators 147

30.Subnormal operators 157

31.Essentially normal operators 170

Chapter 5.More on C＊-Algebras 181

32.Irreducible representations 181

33.Positive maps 187

34.Completely positive maps 190

35.An application： Spectral sets and the Sz.-Nagy Dilation Theorem 198

36.Quasicentral approximate identitites 204

Chapter 6.Compact Perturbations 207

37.Behavior of the spectrum under a compact perturbation 207

38.Bp perturbations of hermitian operators 211

39.The Weyl-von Neumann-Berg Theorem 214

40.Voiculescu’s Theorem 220

41.Approximately equivalent representations 229

42.Some approcations 236

Chapter 7.Introduction to Von Neumann Algebras 241

43.Elementary properties and examples 242

44.The Kaplansky Density Theorem 250

45.The Pedersen Up-Down Theorem 253

46.Normal homomorphisms and ideals 258

47.Equivalence of projections 265

48.Classification of projections 270

49.Properties of projections 278

50.The structure of Type I algebras 282

51.The classification of Type I algebras 289

52.Operator-valued measurable functions 294

53.Some structure theory for continuous algebras 301

54.Weak-star continuous linear functionals revisited 305

55.The center-valued trace 311

Chapter 8.Reflexivity 319

56.Fundamentals and examples 319

57.Reflexive operators on finite dimensional spaces 323

58.Hyperreflexive subspaces 327

59.Reflexivity and duality 335

60.Hypereflexive von Neumann algebras 342

61.Some examples of operators 348

Bibliography 355

Index 367

List of Symbols 371

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