Elementary Theory

PrefaceContents of Volume IIChapter 1. Linear Spaces 1.1. Algebraic Results 1.2. Linear Topological Spaces 1.3. Weak Topologies 1.4. Extreme Points 1.5. Normed Spaces 1.6. Linear Functionals on Normed Spaces 1.7. Some Examples of Banach Spaces 1.8. Linear Operators Acting on Banach Spaces 1.9. ExercisesChapter 2. Basics of Hilbert Space and Linear Operators 2.1. Inner Products on Linear Spaces 2.2. Orthogonality 2.3. The Weak Topology 2.4. Linear Operators General Theory Classes of Operators 2.5. The Lattice of Projections 2.6. Constructions with Hilbert Spaces Subspaces Direct Sums Tensor Products and the Hilbert-Schmidt Class Matrix Representations 2.7. Unbounded Linear Operators 2.8. ExercisesChapter 3. Banach Algebras 3.1. Basics 3.2. The Spectrum The Banach Algebra L1(R) and Fourier Analysis 3.3. The Holomorphic Function Calculus Holomorphic Functions The Holomorphic Function Calculus 3.4. The Banach Algebra C(X) 3.5. ExercisesChapter 4. Elementary C*-Algebra Theory 4.1. Basics 4.2. Order Structure 4.3. Positive Linear Functionals 4.4. Abelian Algebras 4.5. States and Representations 4.6. ExercisesChapter 5. Elementary von Neumann Algebra Theory 5.1. The Weak- and Strong-Operator Topologies 5.2. Spectral Theory for Bounded Operators 5.3. Two Fundamental Approximation Theorems 5.4. Irreducible Algebras—An Application 5.5. Projection Techniques and Constructs Central Carriers Some Constructions Cyclicity, Separation, and Countable Decomposability 5.6. Unbounded Operators and Abelian Von Neumann Algebras 5.7. ExercisesBibliographyIndex of NotationIndex