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For the $C^*$-crossed product $C^*(\Sigma)$ associated with an arbitrarytopological dynamical system $\Sigma = (X, \sigma)$, we provide a detailedanalysis of the commutant, in $C^* (\Sigma)$, of $C(X)$ and the commutant ofthe image of $C(X)$ under an arbitrary Hilbert space representation$\tilde{\pi}$ of $C^* (\Sigma)$. In particular, we give a concrete descriptionof these commutants, and also determine their spectra. We show that, regardlessof the system $\Sigma$, the commutant of $C(X)$ has non-zero intersection withevery non-zero, not necessarily closed or self-adjoint, ideal of $C^*(\Sigma)$. We also show that the corresponding statement holds true for thecommutant of $\tilde{\pi}(C(X))$ under the assumption that a certain family ofpure states of $\tilde{\pi}(C^* (\Sigma))$ is total. Furthermore we establishthat, if $C(X) \subsetneq C(X)'$, there exist both a $C^*$-subalgebra properlybetween $C(X)$ and $C(X)'$ which has the aforementioned intersection property,and such a $C^*$-subalgebra which does not have this property. We also discussexistence of a projection of norm one from $C^*(\Sigma)$ onto the commutant of$C(X)$.

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