Simple Skew Category Algebras Associated with Minimal Partially Defined Dynamical Systems

TL;DR

This paper investigates the simplicity of skew category algebras derived from minimal partially defined dynamical systems, establishing conditions under which these algebras are simple and generalizing previous results.

Contribution

It extends the understanding of simplicity conditions for skew category algebras associated with minimal dynamical systems, generalizing known results from skew group algebras.

Findings

Simplicity implies inverse connectedness, minimality, and faithfulness of the system.

For locally abelian groupoids, simplicity is equivalent to inverse connectedness, minimality, and faithfulness.

Generalizes results from skew group algebras to a broader class of skew category algebras.

Abstract

In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $\Top^{\op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e \in \ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by \"{O}inert for skew group algebras to a large class of skew category algebras.