My C*-algebra notes (from the Fall 2015 course) are now on *AMS Open Math Notes*. You can find them here.

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My C*-algebra notes (from the Fall 2015 course) are now on *AMS Open Math Notes*. You can find them here.

Here is the follow-up lecture (second of two) on coarse index theory. I tried to bear in mind that the conferees in Germany had heard quite a few presumably much more detailed presentations in between by lectures 1 and 2, so I attempted to give a fairly “big picture” overview. I had prepared to talk about several examples that I didn’t have time to discuss, so you will find some slides at the end of the presentation below that were not talked about in the video.

Here’s the video of Lecture 2:

And here is the link to the corresponding slides. Hope you find the presentation helpful and enjoyable!

Recently Matt Wiersma from Waterloo spoke in our seminar about some of his work related to “exotic group C*-algebras”. A more detailed account is on the arXiv. I thought I would try to write up some of what I learned (probably, as usual, this is the most elementary points, but it was new to me).

What is an exotic group C*-algebra? It is a completion of the group algebra which is different from the two standard examples (maximal and reduced) that we describe in C*-algebra courses. Oversimplifying, we might make an analogy with *compactifications* of a locally compact Hausdorff space. There is always a minimal one (one-point compactification) and a maximal (Stone-Cech), but there are also plenty of other things in between. Analogously, in the case where a group \(\Gamma\) is non amenable, one might imagine that there should be many other C*-completions of \({\mathbb C}\Gamma\) lying between the maximal and the reduced C*-algebras. (Whether, in fact, there exists *any* group for which \({\mathbb C}\Gamma\) has *exactly two* distinct completions appears to be an open question.)

In the previous post, I talked about the second law of thermodynamics: there can be do thermodynamic transformation whose overall effect is to move heat from a cooler body to a hotter one. Since the *reverse* of such a transformation (moving heat from a hotter body to a cooler one) happens naturally by conduction, the second law naturally contains an element of *irreversibility* which it is natural to expect is expressed by an *inequality*. The quantity to which this applies is the famous **entropy**.

The first law of thermodynamics says that heat is a form of energy. There is a lot of heat about! For instance, the amount of heat energy it would take to change the temperature of the world’s oceans by one degree is about \(6 \times 10^{24}\) joules. That is four orders of magnitude greater than the world’s annual energy consumption! So, if we could somehow how to figure out how to extract one degree’s worth of heat energy from the oceans, we could power the world for ten thousand years! Continue reading