C*-代数例析

C*-代数在20世纪70年代得到了极大复兴，这缘于Brown、Douglas和Fillmore在C*-代数扩张中引入了拓扑方法，以及Elliott使用K-理论为AF代数提供了一个有用分类。这些结果成为一系列用于分析具体C*-代数出色的新工具之开端。本书通过详细分析几种重要的C*-代数类，介绍了该主题的基础知识，可作为研究生的入门级教材。过去二十年来，算子代数的发展正是基于对这些特殊类的仔细研究。尽管有很多关于C*-代数和算子代数的著作，但这是第一本试图解释研究人员用来检验其假设的真实例子的书籍。本书内容包括AF代数、Bunce-Deddens和Cuntz代数、Toeplitz代数、无理旋转代数、群C*-代数、离散叉积、AbelC*-代数(谱理论和近似酉等价)及其扩展。书中还介绍了该主题的许多现代概念和结果，例如实秩零代数、拓扑稳定秩、拟对角性和各种新结构。C*-代数涉及数学的许多其他领域，例如群表示、动力系统、数学物理、K-理论和拓扑。书中提供了各种帮助学生了解这些联系的示例。具备扎实泛函分析功底的研究生可以阅读本书，这可以帮助他们为阅读许多当前文献做好准备。本书内容相当完备，作者在必要时提供了其他领域的结果。本书可供对泛函分析、C*-代数和算子理论感兴趣的数学与物理方向的研究生和科研人员阅读参考。

Preface
Ⅰ.The Basics of C*-algebras
Ⅰ.1 Definitions
Ⅰ.2 Banach Algebra Basics
Ⅰ.3 Commutative C*-algebras
Ⅰ.4 Positive Elements
Ⅰ.5 Ideals, Quotients and Homomorphisms
Ⅰ.6 Weak Topologies
Ⅰ.7 The Density Theorems
Ⅰ.8 Some Operator Theory
Ⅰ.9 Representations of C*-algebras
Ⅰ.10 C*-algebras of Compact Operators
Exercises
Ⅱ.Normal Operators and Abelian C*-algebras
Ⅱ.1 Spectral Theory
Ⅱ.2 The Lo Functional Calculus
Ⅱ.3 Multiplicity Theory
Ⅱ.4 The Weyl-von Neumann-Berg Theorem
Ⅱ.5 Voiculescu's Theorem
Exercises
Ⅲ.AF C*-algebras
Ⅲ.1 Finite Dimensional C*-algebras
Ⅲ.2 AF Algebras
Ⅲ.3 Perturbations
Ⅲ.4 Ideals and Quotients
Ⅲ.5 Examples
Ⅲ.6 Extensions
Exercises
Ⅳ.K-theory for AF C*-algebras
Ⅳ.1 Idempotents
Ⅳ.2 Ko
Ⅳ.3 Dimension Groups
Ⅳ.4 Elliott's Theorem
Ⅳ.5 Applications
Ⅳ.6 Riesz groups
Ⅳ.7 The Effros-Handelman-Shen Theorem
Ⅳ.8 Blackadar's Simple Unital Projectionless C*-algebra
Exercises
Ⅴ.C*-Algebras of Isometries
Ⅴ.1 Toeplitz Operators
Ⅴ.2 Isometries
Ⅴ.3 Bunce-Deddens Algebras
Ⅴ.4 Cuntz Algebras
Ⅴ.5 Simple Infinite C*-algebras
Ⅴ.6 Classification of Cuntz Algebras
Ⅴ.7 Real Rank Zero
Exercises
Ⅵ.Irrational Rotation Algebras
Ⅵ.1 The algebras Aθ
Ⅵ.2 Projections in Aθ
Ⅵ.3 An AF algebra
Ⅵ.4 Berg's technique
Ⅵ.5 Imbedding Aθ into 2lθ
Exercises
Ⅶ.Group C*-Algebras
Ⅶ.1 Group Representations
Ⅶ.2 Amenability
Ⅶ.3 Primitive Ideals
Ⅶ.4 A Crystallographic Group
Ⅶ.5 The Discrete Heisenberg Group
Ⅶ.6 The Free Group
Ⅶ.7 The Reduced C*-algebra of the Free Group
Ⅶ.8 C*τ (F2) is Projectionless
Exercises
Ⅷ.Discrete Crossed Products
Ⅷ.1 Crossed Products
Ⅷ.2 Crossed Products by Z
Ⅷ.3 Minimal Dynamical Systems
Ⅷ.4 Odometers
Ⅷ.5 K-theory of Crossed Products
Ⅷ.6 AF Subalgebras of Crossed Products
Ⅷ.7 Crossed Product subalgebras of AF Algebras
Ⅷ.8 Topological Stable Rank
Ⅷ.9 An Order 2 Automorphism
Exercises
Ⅸ.Brow n-Douglas-Fillmore Theory
Ⅸ.1 Extensions
Ⅸ.2 An Addition and Zero Element for Ext(X)
Ⅸ.3 Some Special Cases
Ⅸ.4 Positive maps
Ⅸ.5 Ext(X) is a group
Ⅸ.6 First Topological Properties
Ⅸ.7 Ext for planar sets
Ⅸ.8 Quasidiagonality
Ⅸ.9 Homotopy Invariance
Ⅸ.10 The Mayer-Vietoris Sequence
Ⅸ.11 Examples
Exercises
References
Index