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On the Exel Crossed Product of Topological Covering Maps

Meier Carlsen, Toke and Silvestrov, Sergei LU (2009) In Acta Applicandae Mathematicae 108(3). p.573-583
Abstract
For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C (*)-algebras C(X)a < S (alpha,a"')a"center dot introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)a < S (alpha,a"')a"center dot is a maximal abelian C (*)-subalgebra of C(X)a < S (alpha,a"')a"center dot;... (More)
For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C (*)-algebras C(X)a < S (alpha,a"')a"center dot introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)a < S (alpha,a"')a"center dot is a maximal abelian C (*)-subalgebra of C(X)a < S (alpha,a"')a"center dot; any nontrivial two sided ideal of C(X)a < S (alpha,a"')a"center dot has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)a < S (alpha,a"')a"center dot is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C (*)-algebras of homeomorphism dynamical systems. (Less)
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author
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Crossed product algebra, Topologically free dynamical, system, Ideals, Maximal abelian subalgebra, Covering map
in
Acta Applicandae Mathematicae
volume
108
issue
3
pages
573 - 583
publisher
Springer
external identifiers
  • wos:000271941500008
  • scopus:71449107536
ISSN
0167-8019
DOI
10.1007/s10440-008-9372-6
language
English
LU publication?
yes
id
ecfae905-7bfe-4869-a416-897878dc1890 (old id 1518585)
date added to LUP
2016-04-01 13:01:39
date last changed
2022-01-27 08:57:02
@article{ecfae905-7bfe-4869-a416-897878dc1890,
  abstract     = {{For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C (*)-algebras C(X)a &lt; S (alpha,a"')a"center dot introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)a &lt; S (alpha,a"')a"center dot is a maximal abelian C (*)-subalgebra of C(X)a &lt; S (alpha,a"')a"center dot; any nontrivial two sided ideal of C(X)a &lt; S (alpha,a"')a"center dot has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)a &lt; S (alpha,a"')a"center dot is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C (*)-algebras of homeomorphism dynamical systems.}},
  author       = {{Meier Carlsen, Toke and Silvestrov, Sergei}},
  issn         = {{0167-8019}},
  keywords     = {{Crossed product algebra; Topologically free dynamical; system; Ideals; Maximal abelian subalgebra; Covering map}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{573--583}},
  publisher    = {{Springer}},
  series       = {{Acta Applicandae Mathematicae}},
  title        = {{On the Exel Crossed Product of Topological Covering Maps}},
  url          = {{http://dx.doi.org/10.1007/s10440-008-9372-6}},
  doi          = {{10.1007/s10440-008-9372-6}},
  volume       = {{108}},
  year         = {{2009}},
}