(in Polish) Algebry operatorów dające się widzieć

elective monographs

Graphs: finite paths in oriented graphs, hereditary and saturated subsets of vertices, admissible subgraphs, morphisms and Leavitt morphisms of graphs. Graph algebras: path algebras, Leavitt path algebras, graph C*-algebras, from pushouts of graphs to pullbacks of graph C*-algebras.

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 oriented graphs (quivers) so as to build from scratch and in a systematic way the knowledge of graph C*-algebras.

The course begins by answering basic questions about paths in graphs. In particular, we show how to combine a number of terminating algorithms to prove a theorem about the amount of finite paths in finite graphs without loops. Then come definitions of hereditary and saturated subsets of graph vertices, followed by the concept of admissible subgraphs. We instantiate the idea of admissibility by proving that, if the intersection of two graphs exists and is an admissible subgraph of both graphs, then also the union graph exists and both graphs are its admissible subgraphs. We end our study of graphs by introducing the standard morphisms of graphs, and showing how they evolve into the Leavitt morphisms of graphs.

The path algebra of a graph 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.

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.

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.

regular attendance or an oral exam.