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Preface Chapter 1. Introduction to C*-Algebras 1. Definition and examples 2. Abelian C*-algebras and the Functional Calculus 3. The positive elements in a C*-algebra 4. Approximate identities 5. Ideals in a C*-algebra 6. Representations of a C*-algebra 7. Positive linear functionals and the GNS construction Chapter 2. Normal Operators 8. Some topologies on B(H) 9. Spectral measures 10. The Spectral Theorem 11. Star-cyclic normal operators 12. The commutant 13. Von Neumann algebras 14. Abelian von Neumann algebras 15. The functional calculus for normal operators Chapter 3. Compact Operators 16. C*-algebras of compact operators 17. Ideals of operators 18. Trace class and Hilbert-Schmidt operators 19. The dual spaces of the compact operators and the trace class 20. The weak-star topology 21. Inflation and the topologies Chapter 4. Some Non-Normal Operators 22. Algebras and lattices 23. Isometries 24. Unilateral and bilateral shifts 25. Some results on Hardy spaces 26. The functional calculus for the unilateral shift 27. Weighted shifts 28. The Volterra operator 29. Bergman operators 30. Subnormal operators 31. Essentially normal operators Chapter 5. More on C*-Algebras 32. Irreducible representations 33. Positive maps 34. Completely positive maps 35. An application: Spectral sets and the Sz.-Nagy DilationTheorem 36. Quasicentral approximate identitites Chapter 6. Compact Perturbations 37. Behavior of the spectrum under a compact perturbation 38. Bp perturbations of hermitian operators 39. The Weyl-von Neumann-Berg Theorem 40. Voiculescu's Theorem 41. Approximately equivalent representations 42. Some applications Chapter 7. Introduction to Von Neumann Algebras 43. Elementary properties and examples 44. The Kaplansky Density Theorem 45. The Pedersen Up-Down Theorem 46. Normal homomorphisms and ideals 47. Equivalence of projections 48. Classification of projections 49. Properties of projections 50. The structure of Type I algebras 51. The classification of Type I algebras 52. Operator-valued measurable functions 53. Some structure theory for continuous algebras 54. Weak-star continuous linear functionals revisited 55. The center-valued trace Chapter 8. Reflexivity 56. Fundamentals and examples 57. Reflexive operators on finite dimensional spaces 58. Hyperreflexive subspaces 59. Reflexivity and duality 60. Hypereflexive von Neumann algebras 61. Some examples of operators Bibliography Index List of Symbols