An Introduction to K-Theory for C*-Algebras ( London Mathematical Society Student Texts )

Publication series ：London Mathematical Society Student Texts

Author： M. Rørdam; F. Larsen; N. Laustsen

Publisher： Cambridge University Press‎

Publication year： 2000

E-ISBN: 9780511826030

P-ISBN(Paperback): 9780521783347

Subject： O154.3 Algebraic K - theory.

Keyword：概率论与数理统计

Language： ENG

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An Introduction to K-Theory for C*-Algebras

Description

Over the last 25 years K-theory has become an integrated part of the study of C*-algebras. This book gives an elementary introduction to this interesting and rapidly growing area of mathematics. Fundamental to K-theory is the association of a pair of Abelian groups, K0(A) and K1(A), to each C*-algebra A. These groups reflect the properties of A in many ways. This book covers the basic properties of the functors K0 and K1 and their interrelationship. Applications of the theory include Elliott's classification theorem for AF-algebras, and it is shown that each pair of countable Abelian groups arises as the K-groups of some C*-algebra. The theory is well illustrated with 120 exercises and examples, making the book ideal for beginning graduate students working in functional analysis, especially operator algebras, and for researchers from other areas of mathematics who want to learn about this subject.

Chapter

Chapter 1 C*-Algebra Theory

1.1 C*-algebras and *-homomorphisms

1.2 Spectral theory

1.3 Matrix algebras

1.4 Exercises

Chapter 2 Projections and Unitary Elements

2.4 Exercises

2.1 Homotopy classes of unitary elements

2.2 Equivalence of projections

2.3 Semigroups of projections

Chapter 3 The K0-Group of a Unital C*-Algebra

3.1 Definition of the K0-group of a unital

3.2 Functoriality of K0

3.3 Examples

3.4 Exercises

Chapter 4 The Functor K0

4.1 Definition and functoriality of K0

4.2 The standard picture of the group K0(A)

4.3 Half and split exactness and stability of K0

4.4 Exercises

Chapter 5 The Ordered Abelian Group K0(A)

5.3 Exercises

5.1 The ordered K0-group of stably finite C*-algebras

5.2* States on K0(A) and traces on A

Chapter 6 Inductive Limit C*-Algebras

6.5 Exercises

6.1 Products and sums of C*-algebras

6.2 Inductive limits

6.3 Continuity of K0

6.4* Stabilized C*-algebras

Chapter 7 Classification of AF-Algebras

7.1 Finite dimensional C*-algebras

7.2 AF-algebras

7.3 Elliott's classification theorem

7.4* UHF-algebras

7.5 Exercises

Chapter 8 The Functor K1

8.4 Exercises

8.1 Definition of the K1-group

8.2 Functoriality of K1

8.3* K1-groups and determinants

Chapter 9 The Index Map

9.1 Definition of the index map

9.2 The index map and partial isometries

9.3 An exact sequence of K-groups

9.4* Fredholm operators and Fredholm index

9.5 Exercises

Chapter 10 The Higher K-Functors

10.3 Exercises

10.1 The isomorphism between K1(A) and K0(SA)

10.2 The long exact sequence in -RT-theory

Chapter 11 Bott Periodicity

11.1 The Bott map

11.2 The proof of Bott periodicity

11.3 Applications of Bott periodicity

11.4* Homotopy groups and X-theory

11.5* The holomorphic function calculus

11.6 Exercises

Chapter 12 The Six-Term Exact Sequence

12.1 The exponential map and the six-term exact sequence

12.2 An explicit description of the exponential map

12.3 Exercises

Chapter 13 Inductive Limits of Dimension

13.1 Dimension drop algebras

13.2 Realizing countable Abelian groups as K -groups

13.3 Exercises

References

Table of X-groups

Index of symbols

General index

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