Notes on Operator Algebras
by Seung-Hyeok Kye
RIM-GARC Lecture Notes Ser. No. 7, Seoul National University, 1993, pp.179

This is the collection of notes which have been distributed during the lectures on operator algebras in the academic year 1992. The primary purpose of the lecture was to help the students to catch up current topics in C*-algebras. It has been assumed that they have backgrounds on abstract measure theory and elementary functional analysis. After a brief introduction to the general theory of C*-algebras and von Neumann algebras, we plunge into concrete examples of C*-algebras such as AF algebras, free group C*-algebras, irrational rotation algebras and Cuntz algebras, which have been studied during the seventies and early eighties. Through the discussion, we introduce the notions of tensor product, crossed product and K-theory for C*-algebras. We refer to the Introduction at the beginning of each Chapter for more detailed contents of this note.

The author would like to express his deep gratitude to all participants of the lecture. Special thanks are due to Professors Sa-Ge Lee and Sung Je Cho who attended the latter part of the lecture. Their criticisms and encouragements were indispensable to prepare this note.

February 1993

CHAPTER 1: C*-ALGEBRAS AND THEIR REPRESENTATIONS
1.1. Definition and Examples
1.2. Spectrum and Function Calculus
1.3. Commutative C*-algebras
1.4. Order Structures of C*-algebras
1.5. Representations of C*-algebras
1.6. Pure States and Irreducible Representations

CHAPTER 2: VON NEUMANN ALGEBRAS
2.1. Spectral Resolution and Double Commutant Theorem
2.2. Preduals of von Neumann Algebras
2.3. Type Classification of Factors
2.4. Factors Arising from Discrete Groups
2.5. Factors Arising from Ergodic Theory

CHAPTER 3: APPROXIMATELY FINITE DIMENSIONAL C*-ALGEBRAS
3.1. Bratteli Diagrams for AF C*-algebras
3.2. Ideals and Representations of AF algebras
3.3. $K_0$-groups for C*-algebras
3.4. $K_0$-groups of AF algebras
3.5. Classification of AF algebras

CHAPTER 4: TENSOR PRODUCTS OF C*-ALGEBRAS AND NUCLEARITY
4.1. Minimal and Maximal Tensor Products of C*-algebras
4.2. Completely Positive Linear Maps
4.3. Approximation Properties for C*-algebras
4.4. Injective von Neumann Algebras
4.5. Amenable Groups
4.6. Group C*-algebras of the Free Groups
4.7. Exact C*-algebras

CHAPTER 5: CROSSED PRODUCTS OF C*-ALGEBRAS
5.1. Full and Reduced Crossed Products of C*-algebras
5.2. Irrational Rotation Algebras
5.3. C*-algebras Generated by Isometries
5.4. K-theory for C*-algebras
5.5. K-theory for Crossed Products of C*-algebras

REFERENCES