On the Banach *-algebra crossed product associated with a topological dynamical system
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/2570834
- author
- de Jeu, Marcel ; Svensson, Christian LU and Tomiyama, Jun
- organization
- publishing date
- 2012
- 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
- 2016-04-01 13:33:45
- date last changed
- 2022-04-14 01:47:52
@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}}, keywords = {{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}}, doi = {{10.1016/j.jfa.2012.03.013}}, volume = {{262}}, year = {{2012}}, }