Skew category algebras associated with partially defined dynamical systems
(2012) In International Journal of Mathematics 23(4). p.11250040 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 ⋊ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊ G and, on the other hand, maximal commutativity of A in A ⋊ 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 ⋊ G, then I ∩ A ≠ {0}; (iii) the... (More)
 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 ⋊ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊ G and, on the other hand, maximal commutativity of A in A ⋊ 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 ⋊ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊ G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/3409315
 author
 Lundström, Patrik and Öinert, Johan ^{LU}
 organization
 publishing date
 2012
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 partially defined dynamical systems, category dynamical systems, Skew category algebras, topological freeness, maximal commutative subrings, ideals
 in
 International Journal of Mathematics
 volume
 23
 issue
 4
 pages
 1  1250040
 publisher
 World Scientific
 external identifiers

 scopus:84860197662
 ISSN
 0129167X
 DOI
 10.1142/S0129167X12500401
 language
 English
 LU publication?
 no
 id
 db3006a203c347ae8b7fefa63b60bbbf (old id 3409315)
 alternative location
 http://arxiv.org/abs/1006.4776
 date added to LUP
 20130322 21:25:23
 date last changed
 20170101 04:07:49
@article{db3006a203c347ae8b7fefa63b60bbbf, 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 ⋊ G. We study the connection between topological freeness of s and, on the one hand, ideal properties of A ⋊ G and, on the other hand, maximal commutativity of A in A ⋊ 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 ⋊ G, then I ∩ A ≠ {0}; (iii) the ring A is a maximal abelian complex subalgebra of A ⋊ G. Thereby, we generalize a result by Svensson, Silvestrov and de Jeu from the additive group of integers to a large class of groupoids.}, author = {Lundström, Patrik and Öinert, Johan}, issn = {0129167X}, keyword = {partially defined dynamical systems,category dynamical systems,Skew category algebras,topological freeness,maximal commutative subrings,ideals}, language = {eng}, number = {4}, pages = {11250040}, publisher = {World Scientific}, series = {International Journal of Mathematics}, title = {Skew category algebras associated with partially defined dynamical systems}, url = {http://dx.doi.org/10.1142/S0129167X12500401}, volume = {23}, year = {2012}, }