Skew Category Algebras Associated with Partially Defined Dynamical Systems

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 ${\text{Top}}^{\text{op}}$ and show that it defines what we call a skew category algebra $A \rtimes^{\sigma} G$\mathrm{.}Westudytheconnectionbetweentopologicalfreenessof$s$and,ontheonehand,idealpropertiesof$A \rtimes^{\sigma} G$ and, on the other hand, maximal commutativity of $A$ in $A{⋊}^{\sigma }G$. In particular, we show that if $G$ is a groupoid and for each $e\in \text{ob}(G)$ the group of all morphisms $e\to e$ 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 $A{⋊}^{\sigma }G$, then $I \cap A \neq {0}$;(iii)thering$A$isamaximalabeliancomplexsubalgebraof$A \rtimes^{\sigma} G$. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.