Noncommutative Analysis

Month: December, 2014

The isomorphism problem for complete Pick algebras: a survey

Guy Salomon and I recently finished preparing a survey article for the Proceedings of IWOTA 2014. The talk I gave at the conference was an overview of the current state of this problem, so it made sense to prepare a contribution that did the same. I discussed this at length in a previous post. I think that the current survey is currently the best overview of the subject, and also contains some modest improvements and corrections to what appears in the literature. Here is the link: The isomorphism problem for complete Pick algebras: a survey.

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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.