An introduction to separated graphs and their type semigroups

TL;DR

This paper introduces $C^*$-algebras linked to directed and separated graphs, explores their dynamical properties, and provides formulas for computing associated type semigroups.

Contribution

It generalizes the concept of $C^*$-algebras to separated graphs and derives formulas for their type semigroups based on recent research.

Findings

Formula for the type semigroup of self-similar group actions on graphs

Formula for the type semigroup of finite bipartite separated graphs

Review of structural results on type semigroups in dynamical systems

Abstract

We introduce $C^*$-algebras associated with directed graphs, along with two generalizations of this concept, namely Exel-Pardo $C^*$-algebras associated with a self-similar action of a group on a directed graph, and the $C^*$-algebras associated with separated graphs. These constructions have in common that they have a dynamical behavior, being the groupoid $C^*$-algebras associated to certain topological groupoids, which are built from the combinatorial structure. An important invariant one may associate to these dynamical systems is the so-called type semigroup. We will find a formula to compute the type semigroup for a general self-similar action of a group on a row-finite graph $E$ without sources, following a recent paper by Kwa\'sniewski, Meyer and Prasad, and for any finite bipartite separated graph, following a paper by Exel and the author. In addition, we will review various results concerning the structure of the type semigroup for different dynamical systems.