C*代数入门
2009-8
世界图书出版公司
艾文森
106
This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basicideas, as simply and concretely as we could. So whenever it is convenient （and it usually is）, Hilbert spaces become separable and C*-algebras become GCR. Thispractice probably creates an impression that nothing of value is known about other*-algebras. Of course that is not true. But insofar as representations are con-cerned, we can point to the empirical fact that to this day no one has given aconcrete parametric description of even the in educible representations of any*-algebra which is not GCR. Indeed, there is metamathematical evidence whichstrongly suggests that no one ever will （see the discussion at the end of Section3.4）. Occasionally, when the idea behind the proof of a general theorem is exposedvery clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. Iand 2.2. （it would be desirable but not necessary to include Section I. 1 as well）,whereas someone interested in BoreL structures could read Chapter 3 separately.Chapter I could he used as a bare-bones introduction to C*-algebras. Sections 2. Iand 2.3 together contain the basic structure theory for type I yon Neumannalgebras, and are also largely independent of the rest of the book. The level of exposition should be appropriate for a second year:graduate studentwho is familiar with the basic results of functional analysis, measure theory, andHilhert space. For example, we assume the reader knows the Hahn - Banachtheorem, Alaoglus theorem, the Klein-Milman theorem, the spectral theoremfor normal operators, and the elementary theory of commutative Banach algebras.
This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basicideas, as simply and concretely as we could. So whenever it is convenient （and it usually is）, Hilbert spaces become separable and C*-algebras become GCR. Thispractice probably creates an impression that nothing of value is known about other*-algebras. Of course that is not true. But insofar as representations are con-cerned, we can point to the empirical fact that to this day no one has given aconcrete parametric description of even the in educible representations of any*-algebra which is not GCR. Indeed, there is metamathematical evidence whichstrongly suggests that no one ever will （see the discussion at the end of Section3.4）. Occasionally, when the idea behind the proof of a general theorem is exposedvery clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. Iand 2.2. （it would be desirable but not necessary to include Section I. 1 as well）,whereas someone interested in BoreL structures could read
Chapter 1 Fundamentals 1.1.Operators and C*-algebras 1.2.Two density theorems 1.3.Ideals, quotients, and representations 1.4.C*-algebras of compact operators 1.5.CCR and GCR algebras 1.6.States and the GNS construction 1.7.The existence of representations 1.8.Order and approximate unitsChapter 2 Multiplicity Theory 2.1.From type I to multiplicity-free 2.2.Commutative C*-algebras and normal operators 2.3.An application: type !von Neumann algebras 2.4.GCR algebras are type IChapter 3 Borel Structures 3.1.Polish spaces 3.2.Borel sets and analytic sets 3.3.Borei spaces 3.4.Cross sectionsChapter 4 From Commutative Algebras to GCR Algebras 4.1.The spectrum of a C*-algebra 4.2.Decomposable operator algebras 4.3.Representations of GCR algebrasBibliographyIndex
点名要买的，应该不错