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}$$inducedbyahomeomorphism$T$ontheCantorset$X$, 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 $\ast$-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 $\ast$-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 $U \subseteq X$mustequalthemeasureof$U$.