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 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 under the assumption that a certain family of pure states of 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).