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

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title = "On the commutant of C(X) in C*-crossed products by Z and their representations",
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.",
keywords = "Commutant, Ideals, Crossed product, Dynamical system, subalgebra, Maximal abelian",
author = "Christian Svensson and Jun Tomiyama",
year = "2009",
doi = "10.1016/j.jfa.2009.02.002",
language = "English",
volume = "256",
pages = "2367--2386",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Elsevier",
number = "7",