Sylvester matrix rank functions on crossed products

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 $U$.