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On the commutant of C(X) in C*-crossed products by Z and their representations

Svensson, Christian LU and Tomiyama, Jun (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)
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author
organization
publishing date
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
2009-06-11 14:56:43
date last changed
2017-11-19 03:55:59
@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},
  keyword      = {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},
  volume       = {256},
  year         = {2009},
}