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

Abstract

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 $\tilde{\pi}$ of $C^* (\Sigma)$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $\Sigma$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^* (\Sigma)$. We also show that the corresponding statement holds true for the commutant of $\tilde{\pi}(C(X))$ under the assumption that a certain family of pure states of $\tilde{\pi}(C^* (\Sigma))$ 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 $C^*(\Sigma)$ onto the commutant of $C(X)$.