Theory of Operator Algebras I

I Fundamentals of Banach Algebras and C-Algebras.- 0. Introduction.- 1. Banach Algebras.- 2. Spectrum and Functional Calculus.- 3. Gelfand Representation of Abelian Banach Algebras.- 4. Spectrum and Functional Calculus in C-Algebras.- 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 Functional.- 10. Extreme Points of the Unit Ball of a C-Algebra.- 11. Finite Dimensional C-Algebras.- Notes.- Exercises.- II 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. von Neumann.- 4. Density Theorems.- Notes.- 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 Valueof Functionals.- 5. Topological Properties of the Conjugate Space.- 6. Semicontinuity in the Universal Enveloping von Neumann Algebra.- Notes.- 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 L2(?,?) ? ?, L1(?,?) ?y? , and L(?,?) ?? ?.- 8. Direct Integral of Hubert Spaces, Representations, and von Neumann Algebras.- Notes.- V Types 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 von 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. Constructionof Factors