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 Topop and show that it defines what we call a skew category algebra A ⋊σ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊σ G and, on the other hand, maximal commutativity of A in A ⋊σ G. In particular, we show that if G is a groupoid and for each e ∈ ob (G) the group of all morphisms e → 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, i.e. if I is a nonzero ideal of A ⋊σ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊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.