**Spring 2023**

**Instructor:** M. Rieffel

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

**Course Control Number:** 25496

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

**Office Hours:** M 10:45-11:45; W 2:15-3:15; F 9:45-10:45

**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
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
physics, but by now it also has strong interactions with most 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 ("non-commutative 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:
Here are the links**

(most recent update: 3/11/2023)