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

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.

arXiv comment32 pages; 2 figures; AMSLaTeX. Changes from first version: Improved exposition. Corrected misprints. Added mention of work using centrally large subalgebras constructed using methods which are not generalizations of Putnam's construction. Added references