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On the Banach *-algebra crossed product associated with a topological dynamical system

de Jeu, Marcel; Svensson, Christian LU and Tomiyama, Jun (2012) In Journal of Functional Analysis 262(11). p.4746-4765
Abstract
Given a topological dynamical system Sigma = (X, sigma), where X is a compact Hausdorff space and a a homeomorphism of X, we introduce the Banach *-algebra crossed product l(1) (E) most naturally associated with Sigma and initiate its study. It has a richer structure than its well investigated C*-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of C(X) in the algebra, which is shown to... (More)
Given a topological dynamical system Sigma = (X, sigma), where X is a compact Hausdorff space and a a homeomorphism of X, we introduce the Banach *-algebra crossed product l(1) (E) most naturally associated with Sigma and initiate its study. It has a richer structure than its well investigated C*-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of C(X) in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal. (C) 2012 Elsevier Inc. All rights reserved. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Involutive Banach algebra, Crossed product, Ideal structure, Topological, dynamical system
in
Journal of Functional Analysis
volume
262
issue
11
pages
4746 - 4765
publisher
Elsevier
external identifiers
  • wos:000303087000005
  • scopus:84859426221
ISSN
0022-1236
DOI
10.1016/j.jfa.2012.03.013
language
English
LU publication?
yes
id
561493d8-ee78-4d12-8915-d8d295a4f64c (old id 2570834)
date added to LUP
2012-06-04 09:43:00
date last changed
2017-03-26 03:52:04
@article{561493d8-ee78-4d12-8915-d8d295a4f64c,
  abstract     = {Given a topological dynamical system Sigma = (X, sigma), where X is a compact Hausdorff space and a a homeomorphism of X, we introduce the Banach *-algebra crossed product l(1) (E) most naturally associated with Sigma and initiate its study. It has a richer structure than its well investigated C*-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of C(X) in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal. (C) 2012 Elsevier Inc. All rights reserved.},
  author       = {de Jeu, Marcel and Svensson, Christian and Tomiyama, Jun},
  issn         = {0022-1236},
  keyword      = {Involutive Banach algebra,Crossed product,Ideal structure,Topological,dynamical system},
  language     = {eng},
  number       = {11},
  pages        = {4746--4765},
  publisher    = {Elsevier},
  series       = {Journal of Functional Analysis},
  title        = {On the Banach *-algebra crossed product associated with a topological dynamical system},
  url          = {http://dx.doi.org/10.1016/j.jfa.2012.03.013},
  volume       = {262},
  year         = {2012},
}