Permanence of stable rank one for centrally large subalgebras and crossed products by minimal homeomorphisms

TL;DR

This paper introduces the concept of centrally large subalgebras in simple unital C*-algebras and proves that stable rank one is preserved in certain crossed product algebras, impacting classification theory.

Contribution

It establishes that stable rank one is retained in a broad class of crossed products by minimal homeomorphisms, extending the understanding of structural properties in C*-algebras.

Findings

C*-algebras with centrally large subalgebras of stable rank one also have stable rank one

Large subalgebras of crossed product type are centrally large

Crossed products by minimal homeomorphisms have stable rank one regardless of dimension

Abstract

We define centrally large subalgebras of simple unital C*-algebras, strengthening the definition of large subalgebras in previous work. We prove that if A is any infinite dimensional simple separable unital C*-algebra which contains a centrally large subalgebra with stable rank one, then A has stable rank one. We also prove that large subalgebras of crossed product type are automatically centrally large. We use these results to prove that if X is a compact metric space which has a surjective continuous map to the Cantor set, and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable rank one, regardless of the dimension of X or the mean dimension of h. In particular, the Giol-Kerr examples give crossed products with stable rank one but which are not stable under tensoring with the Jiang-Su algebra and are therefore not classifiable in terms of the Elliott invariant.