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An Introduction to K-Theory for C*-Algebras : London Mathematical Society Student Texts, 49 - M. Rordam

An Introduction to K-Theory for C*-Algebras

London Mathematical Society Student Texts, 49

Paperback Published: 15th September 2000
ISBN: 9780521789448
Number Of Pages: 256

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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.

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'The textbook is a nice introduction to the subject preparing the ground for the study of more advanced texts.' H. Schroeder, Zentralblatt fur Mathematik

Prefacep. ix
C*-Algebra Theoryp. 1
C*-algebras and *-homomorphismsp. 1
Spectral theoryp. 5
Matrix algebrasp. 9
Exercisesp. 10
Projections and Unitary Elementsp. 15
Homotopy classes of unitary elementsp. 15
Equivalence of projectionsp. 21
Semigroups of projectionsp. 28
Exercisesp. 30
The K[subscript 0]-Group of a Unital C*-Algebrap. 35
Definition of the K[subscript 0]-group of a unital C*-algebrap. 35
Functoriality of K[subscript 0]p. 41
Examplesp. 46
Exercisesp. 54
The Functor K[subscript 0]p. 59
Definition and functoriality of K[subscript 0]p. 59
The standard picture of the group K[subscript 0](A)p. 62
Half and split exactness and stability of K[subscript 0]p. 66
Exercisesp. 70
The Ordered Abelian Group K[subscript 0](A)p. 77
The ordered K[subscript 0]-group of stably finite C*-algebrasp. 77
States on K[subscript 0](A) and traces on Ap. 82
Exercisesp. 83
Inductive Limit C*-Algebrasp. 89
Products and sums of C*-algebrasp. 89
Inductive limitsp. 91
Continuity of K[subscript 0]p. 97
Stabilized C*-algebrasp. 101
Exercisesp. 104
Classification of AF-Algebrasp. 109
Finite dimensional C*-algebrasp. 109
AF-algebrasp. 113
Elliott's classification theoremp. 118
UHF-algebrasp. 125
Exercisesp. 130
The Functor K[subscript 1]p. 133
Definition of the K[subscript 1]-groupp. 133
Functoriality of K[subscript 1]p. 138
K[subscript 1]-groups and determinantsp. 144
Exercisesp. 147
The Index Mapp. 153
Definition of the index mapp. 154
The index map and partial isometriesp. 158
An exact sequence of K-groupsp. 163
Fredholm operators and Fredholm indexp. 165
Exercisesp. 169
The Higher K-Functorsp. 175
The isomorphism between K[subscript 1](A) and K[subscript 0](SA)p. 175
The long exact sequence in K-theoryp. 178
Exercisesp. 182
Bott Periodicityp. 185
The Bott mapp. 185
The proof of Bott periodicityp. 187
Applications of Bott periodicityp. 199
Homotopy groups and K-theoryp. 201
The holomorphic function calculusp. 204
Exercisesp. 206
The Six-Term Exact Sequencep. 209
The exponential map and the six-term exact sequencep. 209
An explicit description of the exponential mapp. 211
Exercisesp. 215
Inductive Limits of Dimension Drop Algebrasp. 219
Dimension drop algebrasp. 219
Countable Abelian groups as K-groupsp. 222
Exercisesp. 228
Referencesp. 231
Table of K-groupsp. 234
Index of symbolsp. 236
General indexp. 239
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521789448
ISBN-10: 0521789443
Series: London Mathematical Society Student Texts, 49
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 256
Published: 15th September 2000
Country of Publication: GB
Dimensions (cm): 21.59 x 14.61  x 1.27
Weight (kg): 0.39

Earn 126 Qantas Points
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