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

For the C ∗-crossed product C ∗ (Σ) associated with an arbitrary topological dynamical system Σ = (X, σ), we provide a detailed analysis of the commutant, in C ∗ (Σ), of C(X) and the commutant of the image of C(X) under an arbitrary Hilbert space representation ˜π of C ∗ (Σ). In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system Σ, the commutant of C(X) has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of C ∗ (Σ). We also show that the corresponding statement holds true for the commutant of π˜ (C(X)) under the assumption that a certain family of pure states of π˜ (C ∗ (Σ)) is total. Furthermore we establish that, if C(X) C(X) , there exist both a C ∗-subalgebra properly between C(X) and C(X) which has the aforementioned intersection property, and such a C ∗-subalgebra which does not have this property. We also discuss existence of a projection of norm one from C ∗ (Σ) onto the commutant of C(X). © 2009 Elsevier Inc. All rights reserved.