An introduction to noncommutative topology

On this page you can find the lecture notes for a mini-course on Noncommutative topology at the autumn school on Deformations and Rigidity in Algebra, Geometry and Analysis at Würzburg University, October 7-11, 2019.

Warning: this is a work in progress! As such, the notes are far from being final.If you find any inconsistencies or mistakes, please let me know!

Lecture Plan:

  1. Trading spaces for algebras: commutative unital C*-algebras, spectral theory for Banach algberas.
  2. Trading spaces for algebras: the Gel'fand transform and the Gelfand Naimark theorem.
  3. Representations of C*-algebras and the GNS construction.
  4. Hilbert C*-modules and C*-correspondences.
  5. Self-Morita equivalences, line bundles, and the Picard group. Cuntz--Pimsner algebras of self-Morita equivalences and circle bundles.

Much inspiration for the first half of this mini-course is drawn from the course C*-algebras and Representation Theory, that I delivered together with Tyrone Crisp in 2017-2018.