Skew Category Algebras Associated with Partially Defined Dynamical Systems

TL;DR

This paper introduces skew category algebras associated with partially defined dynamical systems, exploring their ideal and commutativity properties, and generalizing existing results from groups to groupoids.

Contribution

It defines skew category algebras for partially defined dynamical systems and establishes their properties, extending previous group-based results to groupoids.

Findings

Topological freeness is equivalent to ideal intersection property.

Maximal abelian subalgebra corresponds to topological freeness.

Generalization from groups to groupoids broadens applicability.

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 $A \rtimes^{\sigma} G$. We study the connection between topological freeness of $s$ and, on the one hand, ideal properties of $A \rtimes^{\sigma} G$ and, on the other hand, maximal commutativity of $A$ in $A \rtimes^{\sigma} G$. 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 $A \rtimes^{\sigma} G$, then $I \cap A \neq \{0\}$; (iii) the ring $A$ is a maximal abelian complex subalgebra of $A \rtimes^{\sigma} G$. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.