On the commutant of C(X) in C*crossed products by Z and their representations
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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 nonzero intersection with every nonzero, not necessarily closed or selfadjoint, 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.
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY
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Original language  English 

Pages (fromto)  23672386 
Journal  Journal of Functional Analysis 
Volume  256 
Issue number  7 
Publication status  Published  2009 
Publication category  Research 
Peerreviewed  Yes 