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.