Fall Semester -- 2001

Graduate Course Description

Course Title: C*-Algebras
Unique Number(s): M391C (56405) 
Time/Location of Lecture: (New Time and Place)MW 3:00-4:30 pm (occasional meetings Fri. 3-4:30) / RLM 11.176
Instructor: Professor Haskell Rosenthal

Brief description:

C*-algebras are fundamental mathematical objects underlying quantum mechanics and lots of research in modern functional analysis. The aim of this course is to present the basic theory, and also to build intuition with concrete fundamental examples.


Brief Outline: 1. Trace class operators and trace duality. Includes the structure of the algebras of complex n x n matrices and the compact operators, singular numbers and the Weyl inequality.


2. Banach algebras, spectrum and spectral radius, Riesz theory of compact operators and the holomorphic operational calculus, a brief treatment of the spectral theorem for self adjoint operators.


3. Abstract and concrete C*-algebras; ideals, approximate units, quotients, positivity, representations, the GNS construction, the enveloping algebra.

4. Nuclear C*-algebras-the CAR or Fermion algebra, type I C*-algebras, continuous fields of C*-algebras.

5. von Neumann algebras-the double commutant theorem, the Kaplansky density theorem, normal states and the predual, the type II_1 hyperfinite injective factor, and a brief discussion of Connes' profound work on injective von Neumann algebras.

6. Tensor products of C*-algebras, Kirchberg's remarkable theory of exact C*-algebras; amenability.

If time permits,we will also give a brief introduction to K-theory and AF algebras.

Many exercises and problems will be given. To receive an A, students should turn in 25% to 50% of these, depending on how many are suggested.

Prerequisite: Some acquaintance with basic functional analysis would be helpful, but is not required.

References:(to be placed on reserve in the PMA Library)