# Skew Category Algebras Associated with Partially Defined Dynamical Systems - Mathematics > Rings and Algebras

Abstract: We introduce partially defined dynamical systems defined on a topologicalspace. 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 $A times^{\sigma} G$. We study the connection between topological freeness of$s$ and, on the one hand, ideal properties of $A times^{\sigma} G$ and, onthe other hand, maximal commutativity of $A$ in $A times^{\sigma} G$. Inparticular, we show that if $G$ is a groupoid and for each $e \in \obG$ thegroup of all morphisms $e ightarrow e$ is countable and the topological space$se$ 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 $A times^{\sigma} G$, then $I \cap A eq \{0\}$; iii the ring $A$ is a maximal abelian complex subalgebra of $A times^{\sigma} G$. Thereby, we generalize a result by Svensson, Silvestrovand de Jeu from the additive group of integers to a large class of groupoids.

Author: Patrik Lundström, Johan Öinert

Source: https://arxiv.org/