This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-aigebras become GCR.

Chapter 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

Chapter 2 Multiplicity Theory

2.1.From type I to multiplicity-free

2.2.Commutative C*-algebras and normal operators

2.3.An application: type !von Neumann algebras

2.4.GCR algebras are type I

Chapter 3 Borel Structures

3.1.Polish spaces

3.2.Borel sets and analytic sets

3.3.Borei spaces

3.4.Cross sections

Chapter 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

Bibliography

Index