Stable rank of some crossed product 𝐶*-algebras

Poon, Yiu Tung

doi:10.1090/S0002-9939-1989-0989097-3

Stable rank of some crossed product $C^ *$-algebras
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Proc. Amer. Math. Soc. 105 (1989), 868-875 Request permission

Abstract:

Let $C\left ( X \right ){ \times _T}Z$ be the crossed product associated to a dynamical system $\left ( {X,T} \right )$. We give a necessary and sufficient condition for $C\left ( X \right ){ \times _T}Z$ to have a dense set of invertible elements. When $X$ is zero-dimensional, we obtain more equivalent conditions which involve the isomorphism between the $K$-groups of $C\left ( X \right ){ \times _T}Z$ and ${C^ * }$-algebras defined by some $T$-invariant closed subsets of $X$. As an application, we show that these conditions are not satisfied by most subshifts and all nontrivial irreducible Markov shifts. When $\left ( {X,T} \right )$ is indecomposable, an equivalent condition is that the intersection of all $T$-invariant nonempty closed subsets of $X$ is nonempty.

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