  In this paper we describe the commutant of an arbitrary subalgebra $A$ of the
algebra of functions on a set $X$ in a crossed product of $A$ with the
integers, where the latter act on $A$ by a composition automorphism defined via
a bijection of $X$. The resulting conditions which are necessary and sufficient
for $A$ to be maximal abelian in the crossed product are subsequently applied
to situations where these conditions can be shown to be equivalent to a
condition in topological dynamics. As a further step, using the Gelfand
transform we obtain for a commutative completely regular semi-simple Banach
algebra a topological dynamical condition on its character space which is
equivalent to the algebra being maximal abelian in a crossed product with the
integers.
