**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

1.4. Order Structures of

1.5. Representations of

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

3.2. Ideals and Representations of

3.3. $K_0$-groups for

3.4. $K_0$-groups of

3.5. Classification of

**CHAPTER 4: TENSOR PRODUCTS OF C*-ALGEBRAS AND NUCLEARITY**

4.1. Minimal and Maximal Tensor Products of

4.2. Completely Positive Linear Maps

4.3. Approximation Properties for

4.4. Injective von Neumann Algebras

4.5. Amenable Groups

4.6. Group

4.7. Exact

**CHAPTER 5: CROSSED PRODUCTS OF C*-ALGEBRAS**

5.1. Full and Reduced Crossed Products of

5.2. Irrational Rotation Algebras

5.3.

5.4.

5.5.

**REFERENCES**