# Thinking about “maximal Roe algebras”

One of the things that has happened in coarse geometry while I was busy being department chair is a bunch of papers about “maximal Roe algebras” (some references at the end). Of course these are objects that I feel I ought to understand, so I spent some time trying to figure out the basics.

Let $$X$$ be a bounded geometry uniformly discrete metric space.  (Something like bounded geometry seems to be necessary, for a reason that I’ll explain below.)  We know how to form the translation algebra $${\mathbb C}[X]$$ (the *-algebra of finite-propagation matrices on $$X$$ ), and this has an obvious representation (the regular representation) on $$\ell^2(X)$$.  Then the usual version of the (uniform) Roe algebra is just the C*-algebra obtained by completing $${\mathbb C}[X]$$ in this representation.  Because it involves only the regular representation we may call this the reduced Roe algebra (in analogy to the group case). Continue reading

# C*-algebras, foliations and K-theory

In 1980, Alain Connes gave a course entitled “C*-algebras, foliations and K-theory”. Jean Renault was a student in the course at that time and took notes, and photocopies of his meticulously handwritten manuscript have been passed around generations of students. I must have acquired mine some time around 1988.

The notes describe projective modules, Morita equivalence, K-theory, non-unital algebras and multipliers, quasi-isomorphisms, smooth subalgebras and “holomorphic closure”, Bott periodicity, crossed products, the Thom isomorphism for crossed products, and the beginnings of noncommutative geometry. Its fascinating to see how early some of these ideas were germinating, and what they looked like at that early stage.

In our seminar last year we assigned graduate students to read and lecture on various parts of the manuscript. This led to a rough English translation, which I’m now in the process of tidying up. I hope to post a more polished version to the arXiv before too long. I’m grateful to Alain and Jean for encouraging this project.

You can find scans (not too legible) of the lecture notes here.