Operator algebras associated with graphs and categories of paths: a Survey

TL;DR

This survey reviews the construction of operator algebras from directed graphs, higher rank graphs, and categories of paths, highlighting properties and introducing a new construction method.

Contribution

It provides a comprehensive overview of operator algebras from graphs and categories, and proposes a novel construction applying free semigroupoid methods to categories of paths.

Findings

Examples of algebras from specific graphs

Discussion of properties like semisimplicity and reflexivity

Introduction of a new construction method

Abstract

Many interesting examples of operator algebras, both self-adjoint and non-self-adjoint, can be constructed from directed graphs. In this survey, we overview the construction of $C^*$-algebras from directed graphs and from two generalizations of graphs: higher rank graphs and categories of paths. We also look at free semigroupoid algebras generated from graphs and higher rank graphs, with an emphasis on the left regular free semigroupoid algebra. We give examples of specific graphs and the algebras they generate, and we discuss properties such as semisimplicity and reflexivity. Finally, we propose a new construction: applying the left regular free semigroupoid construction to categories of paths.