C [asterisk]-algebras and operator theory

This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required. Front cover......Page 1 Title page......Page 3 Date-line......Page 4 Dedication......Page 5 Contents......Page 7 Preface......Page 9 1.1. Banach Algebras......Page 11 1.2. The Spectrum and the Spectral Radius......Page 15 1.3. The Gelfand Representation......Page 23 1.4. Compact and Fredholm Operators......Page 28 Exercises......Page 40 Addenda......Page 44 2.1. C$^\ast$-Algebras......Page 45 2.2. Positive Elements of C$^\ast$-Algebras......Page 54 2.3. Operators and Sesquilinear Forms......Page 58 2.4. Compact Hilbert Space Operators......Page 63 2.5. The Spectral Theorem......Page 76 Exercises......Page 83 Addenda......Page 85 3.1. Ideals in C$^\ast$-Algebras......Page 87 3.2. Hereditary C$^\ast$-Subalgebras......Page 93 3.3. Positive Linear Functionals......Page 97 3.4. The Gelfand-Naimark Representation......Page 103 3.5. Toeplitz Operators......Page 106 Exercises......Page 117 Addenda......Page 120 4.1. The Double Commutant Theorem......Page 122 4.2. The Weak and Ultraweak Topologies......Page 134 4.3. The Kaplansky Density Theorem......Page 139 4.4. Abelian Von Neumann Algebras......Page 143 Exercises......Page 146 Addenda......Page 148 5.1. Irreducible Representations and Pure States......Page 150 5.2. The Transitivity Theorem......Page 159 5.3. Left Ideals of C$^\ast$-Algebras......Page 163 5.4. Primitive Ideals......Page 166 5.5. Extensions and Restrictions of Representations......Page 172 5.6. Liminal and Postliminal C$^\ast$-Algebras......Page 177 Exercises......Page 181 Addenda......Page 182 6.1. Direct Limits of C$^\ast$-Algebras......Page 183 6.2. Uniformly Hyperfinite Algebras......Page 188 6.3. Tensor Products of C$^\ast$-Algebras......Page 194 6.4. Minimality of the Spatial C$^\ast$-Norm......Page 206 6.5. Nuclear C$^\ast$-Algebras and Short Exact Sequences......Page 220 Exercises......Page 223 Addenda......Page 226 7.1. Elements of K-Theory......Page 227 7.2. The K-Theory of AF-Algebras......Page 231 7.3. Three Fundamental Results in K-Theory......Page 239 7.4. Stability......Page 251 7.5. Bott Periodicity......Page 255 Exercises......Page 272 Addenda......Page 274 Appendix......Page 277 Notes......Page 287 References......Page 289 Notation Index......Page 291 Subject Index......Page 293 Back cover......Page 297

This book introduces the reader, graduate student, and non-specialist alike to a lively and important area of mathematics. By its careful and detailed presentation, the book enables the reader to approach the contemporary literature with confidence. A plentiful number of exercises and the choice of topics, which reflect current research interests, distinguish this book from other texts. In addition to the basic theorems of operator theory, including the spectral theorem, the Gefland-Naimark theorem, the double communtant theorem, and the Kaplanski density theorem, some major topics covered by this text are: K-theory, tensor products, and representation theory of C*-algebras.

"This book introduces the reader, graduate student, and non-specialist alike to a lively and important area of mathematics. By its careful and detailed presentation, the book enable the reder to approach the contemporary literature with confidence ... In addition to the basic theorems of operator theory, including the spoectral theorem, the Gelfand-Naimark theorem, the double commutant theorem, and the Kaplanski density theorem, some major topics covered by this text are: K-theory, tensor products and representation theory of C*-algebras."--BOOK COVER This book constitutes a first- or second-year graduate course in operator theory. It is a field that affects other areas of mathematics and physics, such as algebraic topology, differential geometry and quantum mechanics.