# Journal of Operator Theory

Volume 83, Issue 2, Spring 2020 pp. 353-389.

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

**Authors**:
Dawn E. Archey (1), N. Christopher Phillips (2)

**Author institution:**(1) Department of Mathematics,
University of Detroit Mercy,
Detroit, MI 48221-3038, U.S.A.

(2) Department of Mathematics,
University of Oregon, Eugene, OR 97403-1222, U.S.A.

**Summary: **We prove that if $A$ is an infinite dimensional simple separable
unital $C^*$-algebra which contains a centrally large subalgebra
with stable rank one, then $A$ has stable rank one.
We use this result to prove that the Giol--Kerr examples
of minimal homeomorphisms 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.

**DOI: **http://dx.doi.org/10.7900/jot.2018oct10.2236

**Keywords: **simple $C^*$-algebra, minimal homeomorphism, crossed product,
stable rank one, large subalgebra, Elliott classification program

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