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Skew category algebras associated with partially defined dynamical systems

Lundström, Patrik and Öinert, Johan LU (2012) In International Journal of Mathematics 23(4). p.1-1250040
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)
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author
organization
publishing date
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
0129-167X
DOI
10.1142/S0129167X12500401
language
English
LU publication?
no
id
db3006a2-03c3-47ae-8b7f-efa63b60bbbf (old id 3409315)
alternative location
http://arxiv.org/abs/1006.4776
date added to LUP
2013-03-22 21:25:23
date last changed
2017-01-01 04:07:49
@article{db3006a2-03c3-47ae-8b7f-efa63b60bbbf,
  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         = {0129-167X},
  keyword      = {partially defined dynamical systems,category dynamical systems,Skew category algebras,topological freeness,maximal commutative subrings,ideals},
  language     = {eng},
  number       = {4},
  pages        = {1--1250040},
  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},
}