Topological K-theory of C*-algebras for the Working Mathematician – closure (Lectures 5,6 and 7)

The mini course in K-theory given by Haim (Claude) Schochet here at the Technion continued as planned until its end, with lectures 5,6 and 7 following the first four lectures. The topics of these lectures were

Lecture 5 – Kasparov’s KK-theory

Lecture 6 – Foliated spaces and C*-algebras of foliated spaces

Lecture 7 – Applications.

As Haim told us, each of these topics could be a one semester course. The scope and speed were such that a detailed account was impossible for me to produce. However, I will still like to record here the fact that this course ended, since I wrote summaries of the first four lectures and someone may find these and look for the rest of the notes. I cannot write such notes because it takes a master of this field like Schochet to give a brief and colourful overview; an amateur like me will only make a mess.

In the last three lectures, we learned that there is something called KK-theory, which is at once both a generalisation of K-theory and of K-homology (see this survey article by Nigel Higson), we learned that there is a geometrical object called a foliated space (or foliated manifold, see wiki article), we learned that with a foliated space one may associated a groupoid C*-algebra (see this survey by Debord and Lescure), and finally, we were told that all of this can be used to prove an index theorem for foliated spaces (the whole story can be found in the book by Moore and Schochet).

I am somewhat of a mathematical frog (or maybe a mathematical chicken would be a better description of what I am), and I cannot take much from such speedy talks except motivation and inspiration. Motivation and inspiration are important, but you have to be there to get them. I have not much to pass on.

Topological K-theory of C*-algebras for the Working Mathematician – Lecture 4 (K-homology and Brown-Douglas-Fillmore)

My notes of Haim’s Schochet’s fourth lecture in this series is here below.

It is impossible to start without mention that Alexander Grothendieck passed away last week. Grothendieck is considered by many as one of the greatest mathematicians of 20th century, and his contributions affect the material in this lecture series in at least two significant ways. As we mentioned, a first version of K-theory was developed by Grothendieck opening the door for topological K-theory (which, in turn, opened the door for K-theory of C*-algebras). Grothendieck also developed the theory of nuclear topological vector spaces and tensor products of topological vector spaces, a theory that has influenced the development of the concepts of nuclearity and tensor product which are central to contemporary C*-algebra theory. 

Topological K-theory of C*-algebras for the Working Mathematician – Lecture 3 (Topological K-theory and three big theorems)

Here is a write up of the third lecture. (Here are links to the first and second ones.) I want to stress that although Haim is giving me a lot of support in preparing these notes (thanks!), any mistakes you find here are my own.

In this lecture we briefly heard about the origin of K-theory (topological K-theory) and then we learned about three theorems (of Connes, Pimsner-Voiculescu and Schochet) describing how to compute the K-theory of various C*-algebras constructed from given C*-algebras in a given way.

Topological K-theory of C*-algebras for the Working Mathematician – Lecture 2 (Definitions and core examples)

This is a write-up of the second lecture in the course given by Haim Schochet. For the first lecture and explanations, see the previous post.

I will very soon figure out how to put various references online and post links to that, too.

Topological K-theory of C*-algebras for the Working Mathematician – Lecture 1

Claude (Haim) Schochet is spending this semester at the Technion, and he kindly agreed to give a series of lectures on K-theory. This mini-course is called “Topological K-theory of C*-algebras for the Working Mathematician”.

There will be seven lectures (they take place in Amado 814, Mondays 11:00-12:30):

1. A crash course in C*-algebras.
2. K-theory by axioms and core examples.
3. K-theory strengths and limitations.
4. Payoffs in functional analysis: elliptic operators on compact spaces, essentially normal and Toeplitz operators.
5. Payoffs in algebraic topology: bivariant K-theory by axioms, core examples, and the UCT.
6. Modelling of groups, groupoids, and foliations.
7. Payoffs in geometry: Atiyah-Singer and Connes index theorems.

Since the pace will be really fast and the scope very broad, I plan to write up some of the notes I take, to help myself keep track of these lectures. When I write I will probably introduce some mistakes, and this is completely my fault. I will also probably not be able to hold myself from making some silly remarks, for which only I am responsible.

I also hope that these notes I post may help someone who has missed one or several of the talks make up and come to the next one.

The first talk took place last Monday. To be honest I wasn’t 100% on my guard since I heard such crash courses so many times, I was sure that I’ve heard it all before but very soon I was in territory which is not so familiar to me (The title “crash course” was justified!). Maybe I will make up some of the things I write, or imagine that I heard them.

(The next lectures will be on stuff that is more advances and I will take better notes, and hopefully provide a more faithful representation of the actual lecture).

I will refer in short to the following references:

1. Pedersen – C*-algebras and their automorphism groups.

2. Brown and Ozawa – C*-algebras and finite dimensional approximation.

3. Davidson – C*-algebras by example.

4. Dixmier – C*algebras

5. Blackadar – K-theory for operator algebras