Stable Recursive Subhomogeneous Algebras

TL;DR

This paper introduces stable recursive subhomogeneous algebras (SRSHAs), extending the concept of RSHAs to infinite-dimensional irreducible representations, and proves that certain inductive limits of SRSHAs have topological stable rank one.

Contribution

The paper defines SRSHAs and demonstrates that simple inductive limits of SRSHAs with no dimension growth have stable rank one.

Findings

SRSHAs have infinite-dimensional irreducible representations.

Inductive limits of SRSHAs with no dimension growth have topological stable rank one.

SRSHAs serve as an analogue to RSHAs in the study of real number actions on compact spaces.

Abstract

In this paper, we introduce stable recursive subhomogeneous algebras (SRSHAs), which is analogous to recursive subhomogeneous algebras (RSHAs) introduced by N. C. Phillips in the studies of free minimal integer actions on compact metric spaces. The difference between the stable version and the none stable version is that the irreducible representations of SRSHAs are infinite dimensional, but the irreducible representations of the RSHAs are finite dimensional. While RSHAs play an important role in the study of free minimal integer actions on compact metric spaces, SRSHAs play an analogous role in the study of free minimal actions by the group of the real numbers on compact metric spaces. In this paper, we show that simple inductive limits of SRSHAs with no dimension growth in which the connecting maps are injective and non-vanishing have topological stable rank one.