Skew Category Algebras Associated with Partially Defined Dynamical Systems

Abstract

We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor $s$ from a category $G$ to $\Top^{\op}$ and show that it defines what we call a skew category algebra $A \rtimes^σ G$. We study the connection between topological freeness of $s$ and, on the one hand, ideal properties of $A \rtimes^σ G$ and, on the other hand, maximal commutativity of $A$ in $A \rtimes^σ G$. In particular, we show that if $G$ is a groupoid and for each $e \in \ob(G)$ the group of all morphisms $e \rightarrow e$ is countable and the topological space $s(e)$ is Tychonoff and Baire, then the following assertions are equivalent: (i) $s$ is topologically free; (ii) $A$ has the ideal intersection property, that is if $I$ is a nonzero ideal of $A \rtimes^σ G$, then $I \cap A \neq \{0\}$; (iii) the ring $A$ is a maximal abelian complex subalgebra of $A \rtimes^σ G$. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.