Simple Skew Category Algebras Associated with Minimal Partially Defined Dynamical Systems

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.