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El. knyga: From the Basic Homotopy Lemma to the Classification of $C^*$-algebras

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Lin explores the structure of C*-algebras and their homomorphisms, exploring both the mathematical phenomenon and practical applications. After an an overview of the Elliott program; he covers homomorphisms from subhomogeneous C*-algebras to finite dimensional C*-algebras, the stable version of the basic homotopy lemma, a concrete version of the Bott map, a version of the basic homotopy lemma in finite dimensional C*-algebras, the notation of gTR(A)<=1, the classification of C*-algebras with gTR(A)<=1, the basic homotopy lemma in C*-algebras with gTR(A)<=1, a theorem concerning how to lift KK-elements to homomorphisms, the notation of asymptotic unitary equivalence, and some current developments in the Elliott program (without full proofs). Annotation ©2017 Ringgold, Inc., Portland, OR (protoview.com)
Preface v
Chapter 0 An overview of the Elliott program
1(6)
Chapter 1 An introduction to the Basic Homotopy Lemma
7(14)
1.1 A taste of the Basic Homotopy Lemma
7(2)
1.2 Some easy applications
9(6)
1.3 The Voiculescu example and an Exel-Loring invariant
15(3)
1.4 Exercises
18(3)
Chapter 2 Maps to finite dimensional C*-algebras
21(30)
2.1 Preliminaries
21(7)
2.2 Homogeneous C*-algebras
28(7)
2.3 Subhomogeneous C*-algebras
35(13)
2.4 Exercises
48(3)
Chapter 3 Stable Homotopy Lemmas
51(22)
3.1 Stable uniqueness theorems
51(5)
3.2 Stable homotopy
56(4)
3.3 The Bott map revisited
60(10)
3.4 Exercises
70(3)
Chapter 4 The Basic Homotopy Lemma, finite dimensional cases
73(36)
4.1 Two uniqueness theorems
73(7)
4.2 Real rank zero and quasidiagonal extensions
80(3)
4.3 Almost multiplicative maps
83(6)
4.4 Basic Homotopy Lemma in finite dimensional C*-algebras
89(7)
4.5 An existence theorem for Bott maps
96(11)
4.6 Exercises
107(2)
Chapter 5 C*-algebras of generalized tracial rank one
109(48)
5.1 Elliott-Thomsen building blocks
109(8)
5.2 A uniqueness theorem for subhomogeneous C*-algebras
117(12)
5.3 C*-algebras with gTR(A) ≤ 1
129(10)
5.4 A uniqueness theorem for C*-algebras with gTR(A) ≤ 1
139(10)
5.5 The range theorem and the isomorphism theorem
149(5)
5.6 Exercises
154(3)
Chapter 6 More Basic Homotopy Lemmas
157(32)
6.1 Pairs of almost commuting unitaries
157(4)
6.2 More existence theorems related to the Bott map
161(7)
6.3 The Basic Homotopy Lemma in B0
168(6)
6.4 KK-lifting
174(13)
6.5 Exercises
187(2)
Chapter 7 Asymptotic unitary equivalence
189(26)
7.1 The rotation map
189(5)
7.2 Introduction of asymptotic unitary equivalence
194(4)
7.3 A uniqueness theorem for asymptotic unitary equivalence
198(7)
7.4 The range of asymptotic unitary equivalence classes
205(7)
7.5 Exercises
212(3)
Chapter 8 Classification of simple C*-algebras of finite rank
215(18)
8.1 Strong asymptotic unitary equivalence
215(2)
8.2 The Jiang-Su algebra and Winter's deformation
217(8)
8.3 The classification theorem
225(3)
8.4 Non-unital cases
228(5)
Bibliography 233(6)
Index 239
Huaxin Lin, East China Normal University, Shanghai, China, and University of Oregon, Eugene, OR.
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