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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.

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

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