On the commutant of C(X) in C*-crossed products by Z and their representations
(2009) In Journal of Functional Analysis 256(7). p.2367-2386- Abstract
- For the C*-crossed product C*(Sigma) associated with an arbitrary topological dynamical system Sigma = (X, sigma), we provide a detailed analysis of the commutant, in C*(Sigma), of C(X) and the commutant of the image of C(X) under an arbitrary Hilbert space representation (pi) over tilde of C*(E), In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system E, the commutant of C(X) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C*(Z). We also show that the corresponding statement holds true for the commutant of (pi) over tilde (C(X)) tinder the assumption that a certain family of pure states of (pi) over tilde... (More)
- For the C*-crossed product C*(Sigma) associated with an arbitrary topological dynamical system Sigma = (X, sigma), we provide a detailed analysis of the commutant, in C*(Sigma), of C(X) and the commutant of the image of C(X) under an arbitrary Hilbert space representation (pi) over tilde of C*(E), In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system E, the commutant of C(X) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C*(Z). We also show that the corresponding statement holds true for the commutant of (pi) over tilde (C(X)) tinder the assumption that a certain family of pure states of (pi) over tilde (C*(Z)) is total. Furthermore we establish that, if C(X) subset of C(X)', there exist both a C*-Kibalgebra properly between C(X) and C(X)' which has the aforementioned intersection property, and such a C*-subalgebra which does not have this properly. We also discuss existence of* a projection of norm one from C*(Sigma) onto the commutant of C(X). (c) 2009 Elsevier Inc. All rights reserved. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1404780
- author
- Svensson, Christian LU and Tomiyama, Jun
- organization
- publishing date
- 2009
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Commutant, Ideals, Crossed product, Dynamical system, subalgebra, Maximal abelian
- in
- Journal of Functional Analysis
- volume
- 256
- issue
- 7
- pages
- 2367 - 2386
- publisher
- Elsevier
- external identifiers
-
- wos:000264078100012
- scopus:60649095407
- ISSN
- 0022-1236
- DOI
- 10.1016/j.jfa.2009.02.002
- language
- English
- LU publication?
- yes
- id
- 8b345789-a930-44a6-be0d-946faf04e1a6 (old id 1404780)
- date added to LUP
- 2016-04-01 14:22:12
- date last changed
- 2022-02-19 18:39:45
@article{8b345789-a930-44a6-be0d-946faf04e1a6,
abstract = {{For the C*-crossed product C*(Sigma) associated with an arbitrary topological dynamical system Sigma = (X, sigma), we provide a detailed analysis of the commutant, in C*(Sigma), of C(X) and the commutant of the image of C(X) under an arbitrary Hilbert space representation (pi) over tilde of C*(E), In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system E, the commutant of C(X) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C*(Z). We also show that the corresponding statement holds true for the commutant of (pi) over tilde (C(X)) tinder the assumption that a certain family of pure states of (pi) over tilde (C*(Z)) is total. Furthermore we establish that, if C(X) subset of C(X)', there exist both a C*-Kibalgebra properly between C(X) and C(X)' which has the aforementioned intersection property, and such a C*-subalgebra which does not have this properly. We also discuss existence of* a projection of norm one from C*(Sigma) onto the commutant of C(X). (c) 2009 Elsevier Inc. All rights reserved.}},
author = {{Svensson, Christian and Tomiyama, Jun}},
issn = {{0022-1236}},
keywords = {{Commutant; Ideals; Crossed product; Dynamical system; subalgebra; Maximal abelian}},
language = {{eng}},
number = {{7}},
pages = {{2367--2386}},
publisher = {{Elsevier}},
series = {{Journal of Functional Analysis}},
title = {{On the commutant of C(X) in C*-crossed products by Z and their representations}},
url = {{http://dx.doi.org/10.1016/j.jfa.2009.02.002}},
doi = {{10.1016/j.jfa.2009.02.002}},
volume = {{256}},
year = {{2009}},
}