Math 208: C*-algebras

Spring 2022

Instructor: M. Rieffel

Lectures: MWF 12:10-1:00 in Evans 31

Course Control Number: 29165

Office: 811 Evans, e-mail: rieffel at math.berkeley.edu

Office Hours: TBA.

Prerequisites: The basic theory of bounded operators on Hilbert space and of Banach algebras, especially commutative ones. Math 206 is more than sufficient. Self-study of sections 3.1-2, 4.1-4 of "Analysis Now" by G. K. Pedersen would be sufficient. It is my understanding that through an agreement between UC and the publisher, the Pedersen text can be downloaded at no cost at here. You may need to use campus computers to authenticate yourself to gain access.

Recommended Reading: None of the available textbooks follows closely the path that I will take through the material. The closest is probably: "C*-algebras by Example", K. R. Davidson, Fields Institute Monographs, A. M. S. I strongly recommend this text for its wealth of examples (and attractive exposition). UCB students may be able to freely download this book by searching for it at Davidson C*-Algebras by Example .

Syllabus: The theory of operator algebras grew out of the needs of quantum mechanics, but by now it also has strong interactions with many other areas of mathematics. Operator algebras are very profitably viewed as "non-commutative (algebras "of functions" on) spaces", thus "quantum spaces". As a rough outline, we will first develop the basic facts about C*-algebras ("non-commutative locally compact spaces"), and examine a number of interesting examples. We will then briefly look at "non-commutative differential geometry". Finally, time permitting, we will glance at "non-commutative vector bundles" and K-theory ("noncommutative algebraic topology") . But I will not assume any prior knowledge of algebraic topology or differential geometry, and we are unlikely to have time to go into these last topics in any depth. (For a vast panorama of the applications I strongly recommend Alain Connes' 1994 book "Noncommutative Geometry", which can be freely downloaded at connes . Of course much has happened since that book was written, but it is still a very good guide to the very large variety of applications.)

I will discuss a variety of examples, drawn from dynamical systems, group representations and mathematical physics. But I will somewhat emphasize examples which go in the directions of my current research interests, which involve certain mathematical issues which arise in string theory and related parts of high-energy physics. Thus one thread that will run through the course will be to see what the various concepts look like for quantum tori, which are the most accessible interesting non-commutative differentiable manifolds.

In spite of what is written above, the style of my lectures will be to give motivational discussion and complete proofs for the central topics, rather than just a rapid survey of a large amount of material.

Grading: I plan to assign problem sets roughly every other week. Grades for the course will be based on the work done on these. But students who would like a different arrangement are very welcome to discuss this with me. There will be no final examination.

Problem sets:

(most recent update: 10/17/2021)