# Sylvester matrix rank functions on crossed products

## TL;DR

This paper constructs Sylvester matrix rank functions on crossed product algebras from Cantor minimal systems, linking algebraic properties to ergodic measures and embedding into von Neumann factors.

## Contribution

It introduces a method to derive Sylvester matrix rank functions on crossed products using ergodic measures and embeddings into von Neumann continuous factors.

## Key findings

- Constructed a sequence of subalgebras approximating the crossed product.
- Embedded the algebra into a von Neumann continuous factor.
- Established a rank function compatible with ergodic measures.

## Abstract

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 $U$.