Functional Analysis and Operator Algebras

Description

This book offers a comprehensive introduction to various aspects of functional analysis and operator algebras. In Part I, readers will find the foundational material suitable for a one-semester course on functional analysis and linear operators. Additionally, Part I includes enrichment topics that provide flexibility for instructors. Part II covers the fundamentals of Banach algebras and C*-algebras, followed by more advanced material on C* and von Neumann algebras. This section is suitable for use in graduate courses, with instructors having the option to select specific topics. Part III explores a range of important topics in operator theory and operator algebras. These include $H^p$ spaces, isometries and Toeplitz operators, nest algebras, dilation theory, applications to various classes of nonself-adjoint operator algebras, and noncommutative convexity and Choquet theory. This material is suitable for graduate courses and learning seminars, offering instructors flexibility in selecting topics. Dedication Preface Contents Part I Functional Analysis Chapter 1. Set Theory and Topology 1.1. Orders on Sets 1.2. The Axiom of Choice 1.3. Ordinals 1.4. Topological spaces 1.5. Nets 1.6. Continuity 1.7. Compactness 1.8. Weak Topologies 1.9. Compact Hausdorff Spaces 1.10. Locally compact spaces Exercises for Chapter 1 Notes on Chapter 1 Chapter 2. Banach Spaces 2.1. Examples 2.2. Constructions of Banach Spaces 2.3. Hilbert spaces 2.4. Category Theorems 2.5. Fourier series 2.6. The Hahn–Banach Theorems 2.7. Complemented Subspaces Exercises for Chapter 2 Notes on Chapter 2 Chapter 3. LCTVSs and Weak Topologies 3.1. Locally convex topological vector spaces 3.2. Schwartz Space and Distributions 3.3. Geometric Hahn–Banach Theorems 3.4. Compactness in Weak Topologies 3.5. Extreme points 3.6. The Krein–Smulian Theorem 3.7. Weakly Compact Sets 3.8. Fixed Point Theorems 3.9. Haar Measure on Compact Topological Groups Exercises for Chapter 3 Notes on Chapter 3 Chapter 4. Linear Operators 4.1. Adjoint Operators 4.2. The Hilbert Space Adjoint 4.3. Invertible Operators and the Spectrum 4.4. Analyticity and the Resolvent Exercises for Chapter 4 Notes on Chapter 4 Chapter 5. Compact Operators 5.1. Compact Operators 5.2. Structure of Compact Operators 5.3. Fredholm Operators 5.4. Normal Operators 5.5. Invariant Subspaces Exercises for Chapter 5 Notes on Chapter 5 Part II Banach and C*-algebras Chapter 6. Banach Algebras 6.1. Banach algebra basics 6.2. Spectrum 6.3. Riemann integration in Banach space 6.4. Riesz–Dunford functional calculus Exercises for Chapter 6 Notes on Chapter 6 Chapter 7. Commutative Banach Algebras 7.1. Multiplicative linear functionals 7.2. The Non-unital case 7.3. Semisimplicity and automatic continuity 7.4. L1(G) 7.5. More about L1(G) 7.6. Commutative C*-algebras 7.7. Weakly Almost Periodic Functions 7.8. The Shilov Boundary Exercises for Chapter 7 Notes on Chapter 7 Chapter 8. Noncommutative Banach algebras 8.1. Representation Theory 8.2. The radical 8.3. Jacobson Density Theorem 8.4. Automatic Continuity 8.5. Cohen Factorization Theorem Exercises for Chapter 8 Notes on Chapter 8 Chapter 9. C*-algebras 9.1. Positive elements 9.2. Ideals and Quotients 9.3. Von Neumann Algebras 9.4. Density Theorems 9.5. Transitivity 9.6. States 9.7. Constructing Representations 9.8. Representations and Ideals 9.9. The Spectral Theorem 9.10. Direct Limits 9.11. Cuntz Algebras 9.12. Group C*-algebras 9.13. Amenability 9.14. Crossed Products Exercises for Chapter 9 Notes on Chapter 9 Chapter 10. Von Neumann Algebras 10.1. Trace Class Operators 10.2. The Weak* Topology and Normality 10.3. Normal Homomorphisms 10.4. Abelian von Neumann Algebras 10.5. The Universal Representation 10.6. Abstract von Neumann Algebras 10.7. Types for von Neumann Algebras 10.8. Tensor products 10.9. Type I von Neumann Algebras 10.10. II1 von Neumann Algebras 10.11. More about Equivalence of Projections 10.12. The Centre-Valued Trace 10.13. Infinite von Neumann Algebras 10.14. The Group Measure Space Construction Exercises for Chapter 10 Notes on Chapter 10 Part III Operator Theory Chapter 11. Hardy Spaces 11.1. Harmonic Functions 11.2. Hardy spaces Hp 11.3. Consequences of the Jensen Formula 11.4. Inner and Outer Functions 11.5. The Disc Algebra A(mathbbD) 11.6. The Maximal Ideal Space of Hinfty 11.7. More about Hinfty Exercises for Chapter 11 Notes on Chapter 11 Chapter 12. Isometries and Toeplitz Operators 12.1. Isometries 12.2. Invariant Subspaces for Shifts 12.3. Toeplitz Operators 12.4. Analytic Toeplitz Operators 12.5. Isometries are Reflexive 12.6. Invertibility of Toeplitz Operators 12.7. The algebra Hinfty+ C Exercises for Chapter 12 Notes on Chapter 12 Chapter 13. Nest Algebras 13.1. Nests: definitions and examples 13.2. Compact Operators in Nest Algebras 13.3. The Distance Formula 13.4. Isomorphisms of Nest Algebras 13.5. The Similarity Theorem 13.6. Continuous Nests 13.7. The general case 13.8. Consequences of the Similarity Theorem 13.9. The Distance to a MASA 13.10. Perturbations 13.11. Interpolation Theorem 13.12. Connectedness of the Invertibles Exercises for Chapter 13 Notes on Chapter 13 Chapter 14. Dilation Theory 14.1. Dilation of a Contraction 14.2. The Hinfty Functional Calculus 14.3. Commuting Pairs of Operators 14.4. Nevanlinna–Pick Interpolation 14.5. Counterexamples for more variables 14.6. Completely Positive Maps 14.7. Dilation and Extension of CP Maps 14.8. General Dilation Theory 14.9. The minimal C*-algebra 14.10. Boundary Representations 14.11. Injective Envelopes 14.12. Completely Bounded Maps 14.13. Some Similarity Problems 14.14. Abstract Operator Systems Exercises for Chapter 14 Notes on Chapter 14 Chapter 15. Nonselfadjoint Operator Algebras 15.1. Semicrossed Product Basics 15.2. Injective C*-dynamical Systems 15.3. Recovering the Dynamics 15.4. Row Contractions 15.5. Noncommutative Analytic Toeplitz Algebra 15.6. Invariant Subspaces of mathfrakLd 15.7. Free Semigroup Algebras 15.8. Ideals of mathfrakLd 15.9. Reproducing Kernel Hilbert Spaces 15.10. H2d as a RKHS 15.11. Commuting Row Contractions 15.12. The Complete Pick Property Exercises for Chapter 15 Notes on Chapter 15 Chapter 16. Noncommutative Convexity 16.1. Affine and Convex Functions 16.2. The Choquet Theorem 16.3. Orders on Measures 16.4. NC Convexity 16.5. The NC Hahn–Banach Theorem 16.6. Categorical Duality 16.7. The NC Krein–Milman Theorem 16.8. NC Functions 16.9. The Maximal C*-algebra 16.10. Continuity and C(K) 16.11. The Takesaki–Bichteler Theorem 16.12. NC states on C(K) 16.13. Convex NC Functions 16.14. UCP Maps of Convex NC Functions 16.15. Orders on CP Maps 16.16. Maximal Dilations and Extreme Points 16.17. The Pedersen–Baire Envelope 16.18. The NC Choquet–Bishop–de Leeuw Theorem Exercises for Chapter 16 Notes on Chapter 16 Bibliography