2010/06/30 by Patrik Lundström, Johan Öinert · 30 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Dynamical systems theory #Mathematics #Pure mathematics #Skew #math.RA #msc:16S99 #msc:16W50
paper · pdf · doi:10.1142/s0129167x12500401
published in International Journal of Mathematics 23(04), 1250040 (World Scientific) · 16 pages. This article is an improvement of, and hereby a replacement for, version 1 (arXiv:1006.4776v1) entitled "Category Dynamical Systems and Skew Category Algebras"
arxiv created 2011/06/06 · openalex publication_date 2011/10/19 · arxiv updated 2013/01/08 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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 ⋊ 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.