An inverse semigroup approach to self-similar k-graph $C^{\ast }$-algebras and simplicity

Abstract

We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda )$ and its $C^{\ast }$-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 $C^{\ast }$-algebra via inverse semigroup methods.

38 pages