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. Language: English