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. 

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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.