Spring Semester 2003, Introduction to Operator Algebras
(Math 394B, Section 01)



Instructor: Dietmar Bisch
Lecture: TuTh, 9:30am-10:45am, SC 1210
First lecture: Thursday, January 9.
Office: SC 1405, (615) 322-1999
Office hours: TuTh 10:45am-11:30am & by appt.
Mailbox: SC 1326


Prerequisites: Linear algebra. Basic real and functional analysis, i.e. the part of analysis which is usually covered in a first year graduate course in analysis. The spectral theorem for operators on Hilbert space and functional calculus (will be reviewed in the first week).

Recommended Literature: There will be no textbook. The following books are recommended reading:

1) Kehe Zhu, An Introduction to Operator Algebras, CRC Press, 1993.
2) K. Davidson, C*-algebras by example, American Math. Soc., Fields Institute Monograph No. 6, 1996.
3) G. Pedersen, Analysis Now, Springer Verlag, GTM 118, 1988, 2nd edition.
Further references to other textbooks and research articles will be given throughout the course.

Syllabus: The main objects of this course are C*-algebras and von Neumann algebras, which are certain algebras of bounded operators on a Hilbert space.

Every abelian C*-algebra with unit is an algebra of continuous functions on a compact Hausdorff space and two such algebras are isomorphic iff the underlying spaces are homeomorphic. Noncommutative C*-algebras can therefore be viewed as algebras of ``functions'' on ``noncommutative spaces'' and noncommutative algebraic topology (called K-theory or KK-theory) can be developed to study these spaces. Von Neumann algebras are noncommutative measure spaces and ideas from ergodic theory and probability theory play an important role in studying these objects. The idea of replacing a space by a ``quantum space'', i.e. a naturally associated operator algebra, is at the basis of Alain Connes' noncommutative geometry, a mathematical theory, which has already had many deep applications to theoretical physics and geometry.

This course is an introduction to the theory of operator algebras and I will start with a discussion of the basic properties and constructions of a C*-algebra. Then I plan to discuss many concrete examples such as the non-commutative tori, which have come up recently in the context of string theory. General ideas and operator algebras concepts will be explained via these concrete examples.

The prerequisite for this course is the material covered in a first year real analysis course at the graduate level. In particular a basic knowledge of the elementary principles of functional analysis will be assumed.

Grading: There will be no exams. The course grade will be based on attendance, homework problems and a short presentation that each enrolled student will give during the lectures. Topics for these presentations will be distributed at the beginning of the course.