On the commutant of $C(X)$ in $C^*$-crossed products by $\mathbb{Z}$ and their representations

Abstract

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 $\tildeπ$ 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 $\tildeπ(C(X))$ under the assumption that a certain family of pure states of $\tildeπ(C^* (Σ))$ is total. Furthermore we establish that, if $C(X) \subsetneq C(X)'$, 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 $C(X)$.