Operator Algebras

Contents

Link to Notes

This lecture will give an introduction to the theory of C*- and von Neumann algebras, including some examples where they arise in mathematical physics. It is suited for Bachelor or Master students in mathematics, mathematical physics, physics or teacher students with some previous basic knowledge of functional analysis.

Prospective topics:

• Banach algebras and their spectral theory
• commutative Banach and C*-algebras, representation theorem of Gelfand-Naimark
• functional calculus
• states and representations, GNS construction
• Gelfand-Naimark, non-commutative version
• introduction to von Neumann algebras, bicommutant theorem
• applications in quantum statistical physics: quantum dynamical systems and KMS states, CAR and CCR algebras

The topics especially at the end can be adjusted according to the interests and previous knowledge of the audience.

Some Literature

• Lecture Notes of Siegried Echterhoff (English, German)
• Strung: An introduction to C*-algebras and the classification programme (English)
• Notes of Daniel Lenz (only German), use the same password as for my notes
• Bratelli, Robinson: Operator Algebras and Quantum Statistical Mechanics 1 and 2
• Murphy: C*-Algebras and Operator Theory
• Moretti: Spectral Theory and Quantum Mechanics, With an Introduction to the Algebraic Formulation

Prerequisites

Linear algebra and analysis, basic knowledge in functional analysis (Banach/Hilbert spaces, bounded linear operators, general topology)

Exercises

There is a weekly exercise session and exercise sheets where 50% of the points are required to be admitted to the exam.

Exam

oral or written exam, depending on the number of participants

If you are interested in this course, please register on URM.