Skew category algebras associated with partially defined dynamical systems
(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)
Please use this url to cite or link to this publication:
https://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 Publishing
- 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
- 2016-04-01 11:03:30
- date last changed
- 2022-01-26 05:05:06
@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}},
keywords = {{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 Publishing}},
series = {{International Journal of Mathematics}},
title = {{Skew category algebras associated with partially defined dynamical systems}},
url = {{http://dx.doi.org/10.1142/S0129167X12500401}},
doi = {{10.1142/S0129167X12500401}},
volume = {{23}},
year = {{2012}},
}