Published online by Cambridge University Press: 06 June 2019
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
$U$ .