# Centre International de Mathématiques Pures et Appliquées

## Partenaires       Accueil > Écoles de recherche > Ecoles de recherche 2015 > Liste chronologique des écoles de recherche 2015 > Leavitt path algebras and graph C*-algebras

## Leavitt path algebras and graph C*-algebras

English Version

CIMPA-TURKEY research school

Nesin Mathematics Village, Sirince, Selcuk, Izmir, June 29-July 12, 2015

Report by Muge Kanuni

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. Muge Kanuni (Düzce University, Duzce, Turkey) mugekanuni@duzce.edu.tr Gonzalo Aranda Pino (University of Málaga, Málaga, Spain) g.aranda@uma.es

### Scientific Committee

• Pere Ara - Universitat Autònoma de Barcelona, Spain
• Astrid an Huef - University of Otago, Dunedin, New Zealand
• Mercedes Siles Molina - Universidad de Málaga, Málaga, Spain
• Efim Zelmanov - University of California, San Diego, CA, U.S.A.

### Local Organizing Committee

• Songül Esin, Istanbul, Turkey
• Müge Kanuni, Düzce University, Duzce, Turkey
• Ayten Koç, Istanbul Kültür University, Istanbul, Turkey
• Aslı Can Korkmaz, Nesin Mathematics Village, Izmir, Turkey
• Serkan Sütlü, Yeditepe University, Istanbul, Turkey

### Scientific Program

All the courses and talks of the Research School will be in English.

Mini Courses

1. Gene Abrams (5-hour course)
Title : Introduction to Leavitt path algebras
Abstract : In this portion of the course we will give an introduction to Leavitt path algebras. First we will look at the history of the subject, starting with the fundamental article by W.G. Leavitt in 1962, in which the first examples of these algebras arise. Then we will define the more general Leavitt path algebras : these are algebras, denoted L(E), formed from a directed graph E and a field K. We then give a number of examples of familiar algebras which are isomorphic to Leavitt path algebras. We follow this discussion by giving some of the early, fundamental results in the subject. These include the criteria on the graph E so that the algebra L(E) is simple, purely infinite simple, and finite dimensional.

2. Iain Raeburn (5-hour course)
Title : Introduction to C*-algebras with a view to graph algebras
Abstract : This series of 5 lectures will provide an overview of the subject in which the examples and selection of topics is guided by the needs of basic graph-algebra theory. They will cover the Gelfand-Naimark theorem for commutative algebras and its applications, especially to C*-algebras generated by unitary elements. There will then be a discussion of the C*-algebra of bounded operators and the noncommutative Gelfand-Naimark theorem. This will be followed by a discussion of special families of operators, including projections, isometries and partial isometries.

3. Mercedes Siles Molina (5-hour course)
Title : Structure theory of Leavitt path algebras
Abstract : The description of ideals and graded ideals of Leavitt path algebras will be the starting point of this course.
We will continue by giving the structure of the ideals generated by the line points, by the vertices in cycles without exits, and by vertices in extreme cycles. These correspond to the socle, which is the locally artinian part of the Leavitt path algebra, to the locally noetherian, and to the purely infinite simple sides of the Leavitt path algebras.
Primeness, primitivity and other general ring-theoretic results will be studied.

Title : Introduction to graph C*-algebras
Abstract : This series of lectures will provide an overview of the subject in which the examples and theory developed will be guided by the emerging interface between graph C*-algebras and Leavitt path-algebras. Topics will begin with the basic definitions, and uniqueness theorems for graph C*-algebras. Then we will discuss, with examples, the structure of graph C*-algebras given in terms of their connectivity, with topics such as : simplicity, ideal structure and pure infiniteness. Finally we will discuss higher rank graphs (or k-graphs) as an appropriate higher dimensional analogue of a directed graph which will be studied in other lectures.

5. Roozbeh Hazrat (4-hour course).
Title : The graded structure of Leavitt path algebras
Abstract : In ring theory, in many instances there is a possibility of partitioning the structure of the ring and then rearranging the partitions. These rings are called graded rings. This adds an extra layer of structure (and complexity) to the theory. We will discuss the natural grading of Leavitt path algebras, and consider the category of graded modules. We study the information comes out of this grading. We recall the concept of the graded Grothendieck group and summarise the results on the classification of Leavitt path algebras via graded K-theory.

6. Astrid an Huef (3-hour course) [Denoted by An Huef-I in the schedule]
Title : Equilibrium states of the C*-algebras of finite directed graphs
Abstract : An action of the real numbers on a space or a C*-algebra is used to describe the time evolution in mathematical formulations of physical systems. In systems arising in statistical and quantum mechanics, there should be many "equilibrium states’’ ; their mathematical formulation is called the KMS condition. Here I will consider the action of the real numbers on the graph algebras induced from the "gauge action’’ of the circle. To illustrate the basic ideas, I will compute the KMS states of the Cuntz algebra, which is the graph algebra of the graph with one vertex and n loops at the vertex (this work goes back to Olesen and Pedersen in 1978). Then I will show that the Toeplitz-Cuntz algebra, which is an extension of the Cuntz algebra, has many more KMS states. Finally, I will discuss the theory for the graph algebras of finite directed graphs.

7. Astrid an Huef (2-hour course) [Denoted by An Huef-II in the schedule]
Title : Analogues of Leavitt path algebras associated to higher-rank graphs
Abstract : The Leavitt path algebras are purely algebraic analogues of the C*-algebras of directed graphs. The higher-rank graph C*- algebras, introduced by Kumjian and Pask in 2000, have provided many new and interesting examples of tractable C*-algebras. Again, there is a striking connection between properties of the higher-rank graph and the algebra. In these 2 lectures, I will talk about an analogue of the Leavitt path algebras for higher-rank graphs which we call the Kumjian-Pask algebras. I will discuss how the Kumjian-Pask algebra is defined, its universal property, and the uniqueness theorems which say when a representation of this algebra is faithful.

8. Kulumani Rangaswamy (5-hour course)
Title : Leavitt path algebras with special ring theoretic properties
Abstract : Leavitt path algebras are endowed with many idempotents. In this course, we shall investigate graph-theoretic conditions under which a Leavitt path algebra belongs to some of the well-known classes of rings possessing large supply of idempotents such as von Neumann regular rings, pi-regular rings, self-injective rings, Zorn rings, and other generalized regular rings. A useful tool will be two types of direct limit constructions.

9. Pere Ara (5-hour course)
Title : Leavitt path algebras and graph C*-algebras associated to separated graphs
Abstract : A separated graph is a pair (E,C), where E is a directed graph, C is the union indexed by the vertices of the graph v in E^0 of the sets C_v, and C_v is a partition of r^-1(v) (into pairwise disjoint nonempty subsets) for every vertex v. We will introduce the theory of Leavitt path algebras of separated graphs, which has been recently developed by Goodearl and the presenter. These algebras allow to incorporate the Leavitt algebras of any type (m,n) into the theory of graph algebras. We will also outline some of the basic facts on the construction by Exel and the presenter, attaching to a finite bipartite separated graph (E,C) a partial dynamical system (\Omega(E,C), \mathbb F,\alpha) and the corresponding crossed product algebra. The theory will be illustrated with several representative examples.

10. Lia Vas (5-hour course).
Title : The role of involution in graph algebras
Abstract : Both Leavitt path and graph C algebras are equipped with involution. After a brief introduction to involutive rings, we study the impact of the presence of involution on some algebraic properties of these two classes of algebras. Whenever possible, we shall point out the similarities and differences between Leavitt path and graph C*-algebras. We shall also present a class of open conjectures related to the presence of the involution in these algebras.

11. Muge Kanuni (4-hour course).
Title : Leavitt path algebras and invariant basis property
Abstract : A ring R is said to have the Invariant Basis Number property or more simply IBN in case no two free left R-modules of different rank are isomorphic. W. G. Leavitt constructed some non-IBN algebras — what we now call Leavitt algebras — in the 1960’s. In 2005, Abrams-Aranda Pino, and Ara-Moreno-Pardo introduced the Leavitt path algebra as a quotient of a path algebra constructed on a given quiver. The Leavitt path algebra of the quiver with one vertex and n loops turns out to be the Leavitt algebra R of type (1,m), that is a non-IBN algebra where R is isomorphic to m-copies of R as a left module and not isomorphic to n-copies of R for any 1< n < m. On the other hand, there is an abundance of examples of Leavitt path algebras which have IBN.
We will talk about a certain class of Leavitt path algebras, namely Cohn path algebras, which all satisfy the IBN property. Moreover, give an algorithm to decide whether a Leavitt path algebra has IBN or not.

Research Talks

Ayten Koç (1- hour research talk)
Title : Finite Dimensional Quotients of Leavitt Path Algebras of Separated Digraphs
Abstract : We investigate necessary and sufficient conditions for the Leavitt path algebra of a separated di(rected)graph to be finite dimensional. We give a necessary and sufficient criterion to determine whether the Leavitt path algebra of a separated digraph has a finite dimensional quotient. We also provide an algorithm to determine this in the case of a Leavitt path algebra of a finite digraph.