Table of Contents

1 Fundamentals.- 1.1. Operators and C*-algebras.- 1.2. Two density theorems.- 1.3. Ideals, quotients, and representations.- 1.4. C*-algebras of compact operators.- 1.5. CCR and GCR algebras.- 1.6. States and the GNS construction.- 1.7. The existence of representations.- 1.8. Order and approximate units.- 2 Multiplicity Theory.- 2.1. From type I to multiplicity-free.- 2.2. Commutative C*-algebras and normal operators.- 2.3. An application: type I von Neumann algebras.- 2.4. GCR algebras are type I.- 3 Borel Structures.- 3.1. Polish spaces.- 3.2. Borel sets and analytic sets.- 3.3. Borel spaces.- 3.4. Cross sections.- 4 From Commutative Algebras to GCR Algebras.- 4.1. The spectrum of a C*-algebra.- 4.2. Decomposable operator algebras.- 4.3. Representations of GCR algebras.

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