Course  Materials  

Ktheory of operator algebras
In mathematics it is always important to classify objects in question, which is usually done by assigning to them some recognizable invariants. The Kfunctor gives a very powerful method of this kind that associates abelian groups to various mathematical objects, mostly algebras. In particular this approach is very fruitful in the context of algebras of linear transformations of Hilbert spaces or, more specifically, C*algebras. The Ktheory of operator algebras, developed in the last thirty years, is now a cornerstone of Connes. Noncommutative geometry, being deeply interrelated with other aspects of the latter theory (cyclic homology, index theorems, foliations). Via a theorem by Gelfand and Naimark, which identifies commutative C*algebras with locally compact Hausdorff spaces, it also includes a version of Ktheory for classical spaces, going far beyond the classical situation in many respects. Indeed, it allows for a meaningful treatment of certain singular spaces (e.g., spaces of leaves of a foliation) that have a trivial topology in the usual sense. It also leads to classification results for certain C*algebras. Moreover, since C*algebras are the natural language for quantum physics, Ktheory also may give invariants of physical systems (e.g., in the quantum Hall effect). The objective of this lecture course is to present the most elementary aspects of the Ktheory of C*algebras, providing an entry to more advanced topics in noncommutative geometry to be considered in forthcoming courses. Concretely, the contents of the lectures is as follows:

Ktheory of operator algebras
Rainer Matthes,
Wojciech Szymański


Download  Last update  
19.07.2005 

Table of contents  


Graph C*algebras
Rainer Matthes
Notes by P. Witkowski


Download  Last update  
30.07.2005 

Table of contents  

Exam  Exam questions  

The exam was on 20th January 2005. It consisted of the written part (six exercises) and oral part. In the oral part each student had to answer two questions: easy one and difficult one (chosen from the two difficult questions). Five students (on the graduate and undergraduate level) passed the exam. 
Written part Oral part 