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

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Commutant

Ideals

Crossed product

Dynamical system

subalgebra

Maximal abelian

Publikations- och innehållstyp

art (ämneskategori)

ref (ämneskategori)