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 with involution 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 $\unicode[STIX]{x1D707}$ 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 $\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset $U\subseteq X$ must equal the measure of $U$.

For a division ring $D$, denote by ${{\mathcal{M}}_{D}}$ the $D$-ring obtained as the completion of the direct limit $\underset{\to n}{\mathop \lim }\,{{M}_{{{2}^{n}}}}(D)$ with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial $D$-ring $B$ and any non-discrete extremal pseudo-rank function $N$ on $B$, there is an isomorphism of $D$-rings $\overline{B}\,\cong \,{{\mathcal{M}}_{D}}$, where $\overline{B}$ stands for the completion of $B$ with respect to the pseudo-metric induced by $N$. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for $*$-algebras over fields $\text{F}$ with positive definite involution, where the algebra ${{\mathcal{M}}_{\text{F}}}$ is endowed with its natural involution coming from the $*$-transpose involution on each of the factors ${{M}_{{{2}^{n}}}}\,(F)$.

This work is concerned with the question of when a von Neumann regular ring is expressible as a directed union of Artinian subrings.