C*-Algebras and W*-Algebras

Biography of Shôichirô SakaiShôichirô Sakai was born in 1928 in Kanuma, Japan. He received his B. A. in mathematics in 1953 and his doctorate in 1961 from Tohoku University, Sendai. He was a research assistant there (195-1960), then a faculty member of Waseda University, Tokyo (1960-1964). After 2 years on the faculty of the University of Pennsylvania, he became a professor in 1966 and stayed there till 1979 when he returned to Japan to a chair at Nihon University, Tokyo. Prof. Sakai has also held visiting positions at Yale University (1962-1964) and at MIT (1967-1968). His main interests are in operator algebras, functional analysis and mathematical physics.

From the reviews: "This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert." (Math. Reviews) "In theory, this book can be read by a well-trained third-year graduate student - but the reader had better have a great deal of mathematical sophistication. The specialist in this and allied areas will find the wealth of recent results and new approaches throughout the text especially rewarding." (American Scientist) "The title of this book at once suggests comparison with the two volumes of Dixmier and the fact that one can seriously make this comparison indicates that it is a far more substantial work than others on this subject which have recently appeared"(BLMSoc)

• 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 ofStates (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.