Spring 2021 MATH 208 001 LEC

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 through the library website.You may need to use campus computers to authenticate yourself to gain access.  

Description Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.

Office Hours  TBA

Recommended Text  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 can freely download this book through the librart website.

Comments   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 from the web. 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.)

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 Letter grade. 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.

Course Webpage Link on math.berkeley.edu/~rieffel