K-theoretic aspects of Graph C*-Algebras

Gothe, Atul (2022) K-theoretic aspects of Graph C*-Algebras. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Atul Gothe (17MS008)) 17MS008_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (712kB)

Abstract

In this project, weaim to provide an introduction to the structural and K-theoretic properties of graph C*-algebras. Firstly, we begin with a survey of general K-theory of C*-algebras in Chapter 1. This includes the definitions and a few relevant properties of the K-groups. After this, we move on to the relations between the K-groups and state an important exact sequence in the final section of Chapter 1. The primary references for details and proofs of the statements for this chapter are [1] [2] [3]. Chapter 2 begins with the definition of row-finite graphs and the C*-algebras generated by them (Cuntz-Kriger C*-algebras). After this, we investigate the structural properties of the graph C*-algebras in terms of the combinatorial properties of the underlying graphs. Finally, we conclude the chapter by introducing the method to treat arbitrary graphs instead of row-finite graphs. References are [4] [5]. In the last chapter, we study the essential and helpful method to study the K-theory of graph C*-algebra. We finish this chapter with the computation of a few examples, particularly those arising as the compact matrix quantum groups. Finally, the appendix gives the necessary background for the compact matrix quantum groups.

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