Leavitt path algebras and graph C*-algebras

Graphs are combinatorial objects that sit at the core of mathematical intuition. They appear in numerous situations all throughout Mathematics and have often constituted a source of inspiration for researchers. A striking instance of this can be found within the classes of graph C*-algebras and of Leavitt path algebras. These are classes of algebras over fields that emanate from different sources in the history yet quite possibly have a common future.

Let E be a graph, i.e. a collection of vertices and edges that connect them. Very roughly, the process by which a C*-algebra is associated to E consists of decorating the vertices with orthogonal projections on a Hilbert space H and the edges by suitable operators. The ensuing C*-subalgebra of the bounded linear operators B(H) is then the graph C*-algebra C*(E). The Leavitt path algebras, denoted L(E), are the algebraic siblings of the aforementioned graph C*-algebras and are constructed over an arbitrary field (whereas here C*-algebras will always be over the complex numbers). Both classes of algebras, L(E) and C*(E), share a beautiful interplay between highly visual properties of the graph and algebraic/analytical properties of the corresponding underlying graphs.

The aim of the Research School is to provide students with the basic as well as more advanced notions of both theories, to show some of the connections between them, to explore several of the generalizations, and to give a glimpse at the state-of-the-art in the ongoing research carried out within these fields.