## Math 520 Homepage for Fall 2000

## Operator Algebras

## Course Description and Syllabus

Let H be a Hilbert space. The set B(H) of bounded operators from H to H has simultaneously atopological structure(it has a topology defined by the operator norm, as well as several different kinds of weak topologies) and analgebraic structure(it is a ring, and in fact a ring with involution --- the involution is provided by the adjoint operation). The interaction of these two kinds of structure is what defines the subject of Operator Algebras. We will consider subsets of B(H) which are simultaneously subrings (thus respecting the algebraic structure) and closed subspaces (thus respecting the topological structure). Depending on what topology we use, we obtain the notions ofC*-algebraorvon Neumann algebra. The theories of these objects are the noncommutative counterparts of topology and of measure theory, respectively.

The course will give an introduction based on examples to the theory of C*-algebras (with some von Neumann algebra examples included too). My own interest in these objects arises from geometry and topology; but there are also connections to quantum physics (von Neumann's original motivation), dynamical systems, and representation theory, as well as a rich internal structure. K-theory and K-homology will not be discussed in detail in the course; we will focus on the more analytic aspects of the theory.

The course will be based on the book

C*-Algebras by Example, by Ken Davidson. I hope to cover material from chapters I, III, V, VI, VII and VIII.

Homework will be assigned every two weeks.

The course meets MWF 11.15 - 12.05 in 104 Osmond. I will hold an office hour after the course on Friday, but you should feel free to contact me at any time.

I plan to provide printed lecture notes. The notes are available here.