  If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a
compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed
product Banach $\sp{*}$-algebra $\ell^1(\Sigma)$ is naturally associated with
these data. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group
algebra of the integers. The commutant $C(X)'_1$ of $C(X)$ in $\ell^1(\Sigma)$
is known to be a maximal abelian subalgebra which has non-zero intersection
with each non-zero closed ideal, and the same holds for the commutant $C(X)'_*$
of $C(X)$ in $C^*(\Sigma)$, the enveloping $C^*$-algebra of $\ell^1(\Sigma)$.
This intersection property has proven to be a valuable tool in investigating
these algebras. Motivated by this pivotal role, we study $C(X)'_1$ and
$C(X)'_*$ in detail in the present paper. The maximal ideal space of $C(X)'_1$
is described explicitly, and is seen to coincide with its pure state space and
to be a topological quotient of $X\times\mathbb{T}$. We show that $C(X)'_1$ is
hermitian and semisimple, and that its enveloping $C^*$-algebra is $C(X)'_*$.
Furthermore, we establish necessary and sufficient conditions for projections
onto $C(X)'_1$ and $C(X)'_*$ to exist, and give explicit formulas for such
projections, which we show to be unique. In the appendix, topological results
for the periodic points of a homeomorphism of a locally compact Hausdorff space
are given.
