Operator algebras II: Graph C*-Algebras

Graph C*-algebras are a class of C*-algebras constructed from directed graphs. They form a bridge between algebra, analysis, and combinatorics. Their origins trace back to the 1980s with the work of Cuntz and Krieger, who introduced what are now called Cuntz–Krieger algebras. These arose in the study of Markov processes and symbolic dynamics.

In the 1990s, it was realized that Cuntz–Krieger algebras could be seen as special cases of a broader class: graph C*-algebras. Since then, a rich field of research has developed, touching on dynamical systems, K-theory, Morita equivalence, groupoid theory, and more.

Graph C*-algebras are not only concrete and accessible but also offer deep insights into structural questions: When is a C*-algebra simple? What does its ideal structure look like? How can such algebras be classified? These questions can often be tackled explicitly in the graph-theoretic setting.

Lecture and exercise sessions

Lecturer: Prof. Dr. Moritz Weber

Assistant: Jonas Metzinger

---

Lecture time: Wednesdays from 12 noon s.t. to 1.30 p.m. in Lecture Hall 4, E2.4

Exercise sessions: Every two weeks on Wednesdays from 10 a.m. to 12 p.m., starting on 3 December in Seminar Room 5, E2.4

Contents

• Definition of Cuntz-Krieger families, graph C*-algebras
• Gauge actions, Uniqueness theorems for graph C*-algebras
• Simplicity and ideal structure
• Geometric classification: Moves on simple graph C*-algebras

Further content will be included in the exercises.

Language

The course will be taught in English unless all participants speak German.

Prerequisites

Knowledge about C*-algebras and universal C*-algebras as in the ISEM 24 lecture notes is required.

Exercise sessions

There will be exercise sessions every two weeks, starting in the 8-th week of lecture time.

The exercise sessions will be structured as a small seminar. At the beginning of the semester, you will be given a selection of topics to choose from. You will then prepare these in pairs and give together a presentation (60 to 90 minutes) on them.
In the week before (!) the presentation, you will meet with Jonas to discuss the presentation.

There will be a sheet for each topic, which will tell you what to focus on.  Keep in mind that some topics can seem very technical, so start asking questions right at the beginning of your preparation.

 

Date	Topic	Talk is given by
03. December	Hypergraph C*-algebras	Hendrik and Alex
17. December	Cuntz-Pimsner algebras	Marie and Celine
14. January	Classification of graph C*-algebras via K-theory	Benjamin (and Jonas)
21. January	Morita Equivalence	Elias and Luc
28. January	Higher rank graphs	Cuma and Gajanan
04. February	Additional lecture (?)

 

Exam and admission requirements

There will be an oral exam. You can obtain 4.5 CP. To be admitted, you must attend the exercises regularly and have given a talk.

Your presentation topic will also be a small part of the oral exam.

Literature

Any reader who is insufficiently curious to have wondered about this should instead be asking: Why am I reading this book at all?
(E.C. Lance in "Hilbert C*-Modules: A toolkit for operator algebraists")

Basics on C*-algebras and universal C*-algebras:

• ISEM 24 lecture notes
• Bruce Blackadar, Operator algebras. Theory of C*-algebras and von Neumann algebras, 2006.
• Kenneth Davidson, C*-algebras by example, 1996.
• Gerard Murphy, C*-algebras and operator theory, 1990.

The lecture notes by Ian Raeburn are used as the main source for the lecture:

• Iain Raeburn. "Graph algebras.", 2004
• Alex Kumjian, David Pask, Iain Raeburn, Jean Renault. "Graphs, Groupoids, and Cuntz–Krieger Algebras.", 1996
• Iain Raeburn, Dana P. Williams. "Morita equivalence and continuous-trace C*-algebras", 1998
• E.C. Lance. "Hilbert C*-Modules: A toolkit for operator algebraists", 1995
• Mark Tomforde. Slides "Graph C*-algebras as Cuntz-Pimsner algebras", 2016 Houston
• Aidan Sims. "k-graphs", 

Additional literature on Geometric Classification and Moves on Graph C*-algebras (arXiv Links):

• Sørensen. "Geometric classification of simple graph algebras.", 2012 (last revision)
• Eilers, Restorff, Ruiz, Sørensen. "The complete classification of unital graph C*-algebras: Geometric and strong", 2019 (last revision)
• Teresa Bates, David Pask. "Flow equivalence of graph algebras", 2003 (last revision)
• Tyrone Crisp, Daniel Gow. "Contractible subgraphs and Morita equivalence of graph C*-algebras", 2004
• Søren Eilers. Slides "The complete classification of unital graph C*-algebras", 2016 Oberwolfach