(in Polish) Algebry operatorów dające się widzieć II 1000-1M20AOW2
Graph C*-algebras proved to be tremendously successful in studying the K-theory of operator
algebras. They are currently at the research frontier of noncommutative topology enjoying a
substantial ongoing research output. The goal of this lecture course is to explain the fundamentals
of path algebras and Leavitt path algebras so as to build from scratch and in a systematic way the
knowledge of graph C*-algebras.
The course begins with the introduction of the path algebra of a directed graph (quiver), which is
defined as the linear span of all finite paths in the graph with the multiplication given by the
composition of paths. Thus the number of finite paths in a graph is the dimension of its path
algebra. Next, a key step is introduce the Cuntz-Krieger relations in the path algebra of the ghost
extension of a graph - they define the Leavitt path algebra of the graph as the quotient of the
path algebra of the extended graph by the ideal generated by the Cuntz-Krieger relations. Taking
the ground field of the Leavitt path algebra of a graph to be the field of complex numbers, and
defining an involution in terms of the extended graph, we obtain a complex *-algebra. Now, we can
define graph C*-algebras as the universal enveloping C*-algebras of Leavitt path algebras. Here key
results to be explained concern representations on a Hilbert space and the ideal structure of graph
C*-algebras.
The course culminates with applications in noncommutative topology. First, we prove that, by
equipping graphs with Leavitt morphisms, the assignment of graph C*-algebras to graphs becomes a
contravariant functor into the category of C*-algebras and *-homomorphisms. Then we show when this
contravariant functor turns pushouts of graphs into pullbacks of graph C*-algebras. All this is
exemplified by plethora of natural examples rooted in classical topology.
Type of course
Learning outcomes
Acquiring a working knowledge of graph C*-algebras allowing one to start research in this area of
mathematics. Depending on the level of involvement, this course might lead either to a Master
thesis or a PhD dissertation.
Assessment criteria
regular attendance or an oral exam
Bibliography
1. Graph Algebras, Piotr M. Hajac, Mariusz Tobolski, arxiv 1912.05136.
2. Leavitt Path Algebras, Gene Abrams, Pere Ara, Mercedes Siles Molina.
3. Algebras and Representation Theory, Karin Erdmann, Thorsten Holm.
4. C*-algebras and Operator Theory, Gerard J. Murphy.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: