CUNTZ–KRIEGER ALGEBRAS ASSOCIATED TO SELF-SIMILAR GROUPOIDS
Isnie Yusnitha · Bulletin of the Australian Mathematical Society · 2024
In this thesis, we study the self-similar action of groupoids on row-finite directed graphs and their associated Cuntz-Krieger algebras.Roughly speaking, if all scales of parts of the object reiterate the whole, then the object is self-similar.For algebraic objects such as groups or groupoids that have this self-similarity property, we simply call them self-similar groups or self-similar groupoids.As an illustration for this self-similarity property, we recall the addition algorithm of integers that we learn from primary school; there is a so-called carrying operation taking place to deal with addition of larger integers.Analogous to this carrying operation, there is a so-called restriction map that encodes the self-similarity property.In the 1980s, Grigorchuk, Gupta and Sidki introduced the notion of self-similar groups to address the question of whether there exist groups with intermediate growth.Groups act self-similarly on the path-spaces of graphs with a single vertex.For self-similar action on more general directed graphs, Laca, Raeburn, Rammage and Whittaker in 2018 introduced the notion of a self-similar groupoid as a system of partial isomorphisms of the path-spaces of a finite directed graph.They considered self-similar groupoids and their associated C * -algebras to study the KMS states on the associated dynamical systems.In contrast to their work using the machinery of Hilbert modules and Cuntz-Pimsner algebras, we work solely through generators and relations and the associated Cuntz-Krieger algebras.We also work for a large class of self-similar groupoids focusing on the ideal structure of their associated C * -algebras.The main objective of this thesis is to acquire the gauge-invariant G-invariant ideal structure of Cuntz-Krieger algebras pertaining to self-similar groupoids.