内容

目次

1. General Theory.- 1.1. Definitions of C*-Algebras and W*-Algebras.- 1.2. Commutative C*-Algebras.- 1.3. Stonean Spaces.- 1.4. Positive Elements of a C*-Algebra.- 1.5. Positive Linear Functionals on a C*-Algebra.- 1.6. Extreme Points in the Unit Sphere of a C*-Algebra.- 1.7. The Weak Topology on a W*-Algebra.- 1.8. Various Topologies on a W*-Algebra.- 1.9. Kaplansky’s Density Theorem.- 1.10. Ideals in a W*-Algebra.- 1.11. Spectral Resolution of Self-Adjoint Elements in a W*-Algebra.- 1.12. The Polar Decomposition of Elements of a W*-Algebra.- 1.13. Linear Functionals on a W*-Algebra.- 1.14. Polar Decomposition of Linear Functionals on a W*-Algebra.- 1.15. Concrete C*-Algebras and W*-Algebras.- 1.16. The Representation Theorems for C*-Algebras and W*- Algebras.- 1.17. The Second Dual of a C*-Algebra.- 1.18. Commutative W*-Algebras.- 1.19. The C*-Algebra C(?) of all Compact Linear Operators on a Hilbert Space ?.- 1.20. The Commutation Theorem of von Neumann.- 1.21. *-Representations of C*-Algebras, 1.- 1.22. Tensor Products of C*-Algebras and W*-Algebras.- 1.23. The Inductive Limit and Infinite Tensor Product of C*- Algebras.- 1.24. Radon-Nikodym Theorems in W*-Algebras.- 2. Classification of W*-Algebras.- 2.1. Equivalence of Projections and the Comparability Theorem.- 2.2. Classification of W*-Algebras.- 2.3. Type I W*-Algebras.- 2.4. Finite W*-Algebras.- 2.5. Traces and Criterions of Types.- 2.6. Types of Tensor Products of W*-Algebras.- 2.7. *-Representations of C*-Algebras and W*-Algebras, 2.- 2.8. The Commutation Theorem of Tensor Products.- 2.9. Spatial Isomorphisms of W*-Algebras.- 3. Decomposition Theory.- 3.1. Decompositions of States (Non-Separable Cases).- 3.2. Reduction Theory (Space-Free).- 3.3. Direct Integral of Hilbert Spaces.- 3.4. Decomposition of States (Separable Cases).- 3.5. Central Decomposition of States (Separable Cases).- 4. Special Topics.- 4.1. Derivations and Automorphisms of C*-Algebras and W*-Algebras.- 4.2. Examples of Factors, 1 (General Construction).- 4.3. Examples of Factors, 2 (Uncountable Families of Types II 1, II? and III.- 4.4. Examples of Factors, 3 (Other Results and Problems).- 4.5. Global W*-Algebras (Non-Factors).- 4.6. Type I C*-Algebras.- 4.7. On a Stone-Weierstrass Theorem for C*-Algebras.- List of Symbols.