This book is an introduction to C^*-algebras and their representations on Hilbert spaces. The presentation is as simple and concrete as possible; the book is written for a second-year graduate student who is familiar with the basic results of functional analysis, measure theory and Hilbert spaces. The author does not aim for great generality, but confines himself to the best-known and also to the most important parts of the theory and the applications.

Inhalt
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.