  In this paper we consider the algebraic crossed product $\mathcal A := C_K(X)
\rtimes_T \mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$,
where $K$ is an arbitrary field and $C_K(X)$ denotes the $K$-algebra of locally
constant $K$-valued functions on $X$. We investigate the possible Sylvester
matrix rank functions that one can construct on $\mathcal A$ by means of full
ergodic $T$-invariant probability measures $\mu$ on $X$. To do so, we present a
general construction of an approximating sequence of $*$-subalgebras $\mathcal
A_n$ which are embeddable into a (possibly infinite) product of matrix algebras
over $K$. This enables us to obtain a specific embedding of the whole
$*$-algebra $\mathcal A$ into $\mathcal M_K$, the well-known von Neumann
continuous factor over $K$, thus obtaining a Sylvester matrix rank function on
$\mathcal A$ by restricting the unique one defined on $\mathcal M_K$. This
process gives a way to obtain a Sylvester matrix rank function on $\mathcal A$,
unique with respect to a certain compatibility property concerning the measure
$\mu$, namely that the rank of a characteristic function of a clopen subset $U
\subseteq X$ must equal the measure of $U$.
