An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity

TL;DR

This paper extends the concept of self-similar $k$-graph $C^*$-algebras using inverse semigroup techniques, providing a new framework for analyzing their structure and simplicity.

Contribution

It generalizes the notion of self-similarity in $k$-graph $C^*$-algebras and introduces an inverse semigroup model for their analysis.

Findings

Developed an inverse semigroup model for $ ext{O}_{G, extLambda}$

Analyzed the tight groupoid and $C^*$-algebra structure

Provided insights into simplicity conditions

Abstract

We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda)$ and its $C^*$-algebra $\mathcal{O}_{G,\Lambda}$ to any finitely aligned $k$-graph $\Lambda$. We then introduce an inverse semigroup model for $\mathcal{O}_{G,\Lambda}$ and analyze its tight groupoid and $C^*$-algebra via inverse semigroup methods.