Biography of Shôichirô Sakai

Shôichirô 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.

Every chapter of the book contains illuminating remarks and open problems and references for other details or topics not discussed.

The style of the book is fluent.

It represents an important contribution to the literature on C*-algebras and through results, problems, a stimulating book for researchers."

Revue Romaine de Mathématiques

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.