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

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On the commutant of C(X) in C*-crossed products by Z and their representations. / Svensson, Christian; Tomiyama, Jun.

In: Journal of Functional Analysis, Vol. 256, No. 7, 2009, p. 2367-2386.

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Svensson, Christian ; Tomiyama, Jun. / On the commutant of C(X) in C*-crossed products by Z and their representations. In: Journal of Functional Analysis. 2009 ; Vol. 256, No. 7. pp. 2367-2386.

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TY - JOUR

T1 - On the commutant of C(X) in C*-crossed products by Z and their representations

AU - Svensson, Christian

AU - Tomiyama, Jun

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Commutant

KW - Ideals

KW - Crossed product

KW - Dynamical system

KW - subalgebra

KW - Maximal abelian

U2 - 10.1016/j.jfa.2009.02.002

DO - 10.1016/j.jfa.2009.02.002

M3 - Article

VL - 256

SP - 2367

EP - 2386

JO - Journal of Functional Analysis

T2 - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 7

ER -