Math 208: C*-algebras

Spring 2018

Instructor: M. Rieffel

Lectures: TTH 12:300-2:00, Room 5 Evans

Course Control Number: 39250

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

Office Hours: TBA

Prerequisites: Because Math 206 was not offered this past Fall, the prerequisite for this course will be Math 202A-B or equivalent. (In fact, it will be reasonable to take Math 208 concurrently with Math 202B if one studies ahead of time pages 65-83 and 95-107 in the book "Real and Functional Analysis" by S. Lang.) The consequence is that during the first several weeks of the course we will develop the theory of commutative C*-algebras, a topic usually covered in Math 206. (So we will not be able to cover as much advanced material at the end of the course.)

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).

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 is available on the web as a free download. Of course much has happened since that book was written, but it is still a very good guide to a 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 several problem sets. 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.

(most recent update: 10/05/2017)