  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
$C(X)$.
