Foundations of von Neumann algebras
Build Hilbert-space representations from positive functionals, compare norm and operator closures, and classify commutative and finite-dimensional approximation models. The course includes complete functional and complex analysis prerequisites, C*-algebra calculus and quotients, normal duality, projection types, dilation, state decompositions and measurable selection tools.
- Hahn–Banach, Baire and the basic theorems on Banach spaces
- Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian
- Hilbert spaces and compact operators
- Cauchy's theorem for cycles and its consequences
- Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory
- Order, local units and quotients of C*-algebras
- Building representations from positive functionals
- The spectral theorem for bounded self-adjoint operators
- Compact and trace-class operators, the predual of B(H), and the operator topologies
- The double commutant theorem
- Kaplansky's density theorem and its consequences
- Completely positive maps
- The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras
- Borel models, measurable inversion and Polish group quotients
- Spectral estimates and Borel selection: two reading routes
- Commutative operator algebras: measure, order and duality
- Projections and types of von Neumann algebras
- Polar decomposition of functionals and weak compactness in preduals
- Integral representations of states
- Finite-dimensional approximations and AF-algebra classification
Supporting chapters
- Function algebras and uniform approximation
- Polish spaces and the Effros Borel structure
- Local measurability and Radon integration
- Group integration, recovery and regular faithfulness
- Covariant integration and recovery with nonunital coefficients
Proofs and dependencies · Source editions · Sources and attribution
Download the complete reader and editable sources · Group representations · Covariant representations