Theory of Operator AlgebrasⅠ Introduction Chapter Ⅰ Fundamentals of Banach Algebras and C*-Algebras 0. Introduction 1. Banach Algebras 2. Spectrum and Functional Calculu 3. Gelfand Representation of Abelian Banach Algebras 4. Spectrum and Functional Calculus in C*-Algebra 5. Continuity of Homomorphisms 6. Positive Cones of C*-Algebras 7. Approximate Identities in C*-Algebras 8. Quotient Algebras of C*-Algebras 9. Representations and Positive Linear Functionals 10. Extreme Points of the Unit Ball of a C*-Algebra 11. Finite Dimensional C*-Algebras Notes Exercises ChapterⅡ Topologies and Density Theorems in Operator Algebras 0. Introduction 1. Banach Spaces of Operators on a Hilbert Space 2. Locally Convex Topologies in (.) 3. The Double Commutation Theorem of J. yon Neumann 4. Density Theorems Notes Chapter Ⅲ Conjugate Spaces 0. Introduction 1. Abelian Operator Algebras 2. The Universal Enveloping von Neumann Algebra of a C-Algebra 3. W*-Algebras 4. The Polar Decomposition and the Absolute Value of Functionals 5. Topological Properties of the Conjugate Space 6. Semicontinuity in the Universal Enveloping von Neumann Algebra* Notes Chapter Ⅳ Tensor Products of Operator Algebras and Direct Integrals 0. Introduction 1. Tensor Product of Hilbert Spaces and Operators 2. Tensor Products of Banach Spaces 3. Completely Positive Maps 4. Tensor Products of C*-Algebras 5. Tensor Products of W*-Algebras Notes 6. Integral Representations of States 7. Representation of LE(F,/.μ) (0+) , LI(F,μ) (0+) m*,, and L(F,μ)(0+) u 8. Direct Integral of Hilbert Spaces, Representations, and von Neumann Algebras Notes Chapter Ⅴ Type, s of von Neumann Algebras and Traces 0. Introduction 1. Projections and Types of von Neumann Algebras 2. Traces on von Neumann Algebras Notes 3. Multiplicity of a yon Neumann Algebra on a Hilbert Space 4. Ergodic Type Theorem for von Neumann Algebras* 5. Normality of Separable Representations* 6. The Borel Spaces of von Neumann Algebras 7. Construction of Factors of Type II and Type Ⅲ Notes Appendix Polish Spaces and Standard Borel Spaces Bibliography Monographs Papers Notation Index Subject Index
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