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
CHAPTER 2: VON NEUMANN ALGEBRAS
CHAPTER 3: APPROXIMATELY FINITE DIMENSIONAL C*-ALGEBRAS
CHAPTER 4: TENSOR PRODUCTS OF C*-ALGEBRAS AND NUCLEARITY
CHAPTER 5: CROSSED PRODUCTS OF C*-ALGEBRAS
REFERENCES
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
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
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
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
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