TABLE OF CONTENTS INTRODUCTION Chapter 1 EQUIVALENCE RELATIONS AND CLASSIFICATION THEORY 1. Quasi-equivalence 2. Von Neumann classification of operators, in operator theoretic terms 3. The bridge between operators and operator algebras k. Multiplicity theory for factor operators 5. Weak containment and weak equivalence for operators 6. Smooth operators 7. The envelope of an operator 11 18 28 3^ 37 in Chapter 2 THE SPECTRUM AND QUASI-SPECTRUM OF AN OPERATOR 1. The * - strong operator topology and its associated Borel structure ^ 2. The concrete dual of an operator 62 3. The spectrum and quasi-spectrum of an operator 66 h. Some spectral theory 73 5. The topology on the spectrum 83 Chapter 3 DECOMPOSITION THEORY 1. The central decomposition 2. The central measure lattice 3. Multiplicity and the central decomposition 91 107 118 Chapter h SPECTRAL MULTIPLICITY THEORY AND THE UNITARY EQUIVALENCE PROBLEM FOR OPERATORS 129 Chapter 5 APPLICATIONS, EXAMPLES Am OPEN QUESTIONS 1. Smooth operators 2. Normal operators and spectral measures 3. Binormal operators h. Trinormal operators 5. Quasinormal operators 6. Direct sum decompositions 7. Some research proposals 152 156 159 166 172 177 179 REFERENCES NOTATION INDEX INDEX 193 200 202 iii
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