Introduction to von Neumann Algebras
This graduate-level course offers a rigorous introduction to the theory of von Neumann algebras, a central topic in functional analysis, operator algebras, and mathematical physics. Named after the mathematician John von Neumann, these algebras arise naturally in the study of quantum mechanics, ergodic theory, and representation theory. The course will emphasize both the abstract structural theory of von Neumann algebras and their profound applications in diverse fields.
We will explore foundational concepts such as weak operator topology, factors, modular theory, and classification theorems, while gaining familiarity with advanced tools like the Tomita-Takesaki theory and Connes' contributions. The course will also highlight connections to modern topics, including subfactor theory and quantum information theory.
We will explore foundational concepts such as weak operator topology, factors, modular theory, and classification theorems, while gaining familiarity with advanced tools like the Tomita-Takesaki theory and Connes' contributions. The course will also highlight connections to modern topics, including subfactor theory and quantum information theory.
Lecturer
Date
19th March ~ 4th June, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | A3-4-301 | ZOOM A | 388 528 9728 | BIMSA |
Prerequisite
Functional Analysis
Syllabus
1. Preliminaries
1.1 Hilbert spaces, bounded operators, and the weak/strong operator topologies
1.2 States
2. von Neumann Algebra Basics
2.1 von Neumann’s bicommutant theorem and Kaplansky density theorem
2.2 Projections, classification of abelian von Neumann algebras
3. Factors and Classification
3.1 Type I, II, and III factors; hyperfinite II_1 factors
3.2 The Murray-von Neumann dimension theory
4. The crossed product construction
5. Unbounded operators and spectral theory
6. Tomita-Takesaki theory: modular automorphism group, Connes’ cocycle
7. Connes' theory of type II factors
8. Subfactor theory and Jones’ index
1.1 Hilbert spaces, bounded operators, and the weak/strong operator topologies
1.2 States
2. von Neumann Algebra Basics
2.1 von Neumann’s bicommutant theorem and Kaplansky density theorem
2.2 Projections, classification of abelian von Neumann algebras
3. Factors and Classification
3.1 Type I, II, and III factors; hyperfinite II_1 factors
3.2 The Murray-von Neumann dimension theory
4. The crossed product construction
5. Unbounded operators and spectral theory
6. Tomita-Takesaki theory: modular automorphism group, Connes’ cocycle
7. Connes' theory of type II factors
8. Subfactor theory and Jones’ index
Reference
1. Von Neumann algebras, V. F. R. Jones, 2010
2. R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras, Vol. I, II,
Graduate Studies in Mathematics, Vol. 15, 21, American Mathematical Society, 1997.
2. R. V. Kadison, J. R. Ringrose, Fundamentals of the theory of operator algebras, Vol. I, II,
Graduate Studies in Mathematics, Vol. 15, 21, American Mathematical Society, 1997.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
No
Notes Public
No
Language
Chinese