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Sylvester matrix rank functions on crossed products

Pere Ara, Joan Claramunt

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 $μ$ 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 $μ$, namely that the rank of a characteristic function of a clopen subset $U \subseteq X$ must equal the measure of $U$.

Sylvester matrix rank functions on crossed products

Abstract

In this paper we consider the algebraic crossed product induced by a homeomorphism on the Cantor set , where is an arbitrary field and denotes the -algebra of locally constant -valued functions on . We investigate the possible Sylvester matrix rank functions that one can construct on by means of full ergodic -invariant probability measures on . To do so, we present a general construction of an approximating sequence of -subalgebras which are embeddable into a (possibly infinite) product of matrix algebras over . This enables us to obtain a specific embedding of the whole -algebra into , the well-known von Neumann continuous factor over , thus obtaining a Sylvester matrix rank function on by restricting the unique one defined on . This process gives a way to obtain a Sylvester matrix rank function on , unique with respect to a certain compatibility property concerning the measure , namely that the rank of a characteristic function of a clopen subset must equal the measure of .

Paper Structure

This paper contains 13 sections, 22 theorems, 112 equations.

Table of Contents

  1. Introduction and motivation
  2. Background and preliminaries
  3. Von Neumann regular rings and pseudo-rank functions
  4. Sylvester matrix rank functions
  5. -regular rings
  6. The main construction
  7. Quasi-partitions and a -representation of
  8. Example: the lamplighter group algebra
  9. Sylvester matrix rank functions on
  10. Approximation algebras
  11. A rank function on
  12. Relation with measures on
  13. The space

Key Result

Lemma 3.1

Let $\mu$ be an ergodic $T$-invariant probability measure on $X$, and take $E$ to be a Borel subset of $X$ with positive measure. Consider the first return map $r_E : E \rightarrow {\mathbb N} \cup \{ \infty \}$, defined by in case there is $l>0$ such that $T^l(x) \in E$, and $r_E(x) = \infty$ otherwise. For each $k \in {\mathbb N}$, consider $Y_k^l = T^l(r_E^{-1} (k))$, for $0 \leq l \leq k-1$.

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.6
  • Proposition 3.7
  • ...and 39 more