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Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics

Gonçalves, Daniel; Öinert, Johan LU and Royer, Danilo (2014) In Journal of Algebra 420. p.201-216
Abstract
Let A be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of A, all of which have local units. We show that A is maximal commutative in the partial skew group ring A*G if and only if A has the ideal intersection property in A*G. From this we derive a criterion for simplicity of A*G in terms of maximal commutativity and G-simplicity of A. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Partial skew group ring, Leavitt path algebra, Partial topological dynamics, Simplicity
in
Journal of Algebra
volume
420
pages
201 - 216
publisher
Elsevier
external identifiers
  • wos:000343020900011
  • scopus:84908556969
ISSN
0021-8693
DOI
10.1016/j.jalgebra.2014.07.027
language
English
LU publication?
yes
id
56a2aaac-d83a-4736-a1ad-efdafeeaf95f (old id 4631450)
date added to LUP
2014-09-11 14:10:53
date last changed
2017-07-23 03:02:13
@article{56a2aaac-d83a-4736-a1ad-efdafeeaf95f,
  abstract     = {Let A be a commutative and associative ring (not necessarily unital), G a group and α a partial action of G on ideals of A, all of which have local units. We show that A is maximal commutative in the partial skew group ring A*G if and only if A has the ideal intersection property in A*G. From this we derive a criterion for simplicity of A*G in terms of maximal commutativity and G-simplicity of A. We also provide two applications of our main results. First, we give a new proof of the simplicity criterion for Leavitt path algebras, as well as a new proof of the Cuntz–Krieger uniqueness theorem. Secondly, we study topological dynamics arising from partial actions on clopen subsets of a compact set.},
  author       = {Gonçalves, Daniel and Öinert, Johan and Royer, Danilo},
  issn         = {0021-8693},
  keyword      = {Partial skew group ring,Leavitt path algebra,Partial topological dynamics,Simplicity},
  language     = {eng},
  pages        = {201--216},
  publisher    = {Elsevier},
  series       = {Journal of Algebra},
  title        = {Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics},
  url          = {http://dx.doi.org/10.1016/j.jalgebra.2014.07.027},
  volume       = {420},
  year         = {2014},
}