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Stable rank of some crossed product $ C\sp *$-algebras


Author: Yiu Tung Poon
Journal: Proc. Amer. Math. Soc. 105 (1989), 868-875
MSC: Primary 46L80; Secondary 46L55, 54H15
DOI: https://doi.org/10.1090/S0002-9939-1989-0989097-3
MathSciNet review: 989097
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0989097-3
Keywords: Stable rank of $ {C^ * }$-algebras, density of invertible elements, isomorphism of $ K$-groups
Article copyright: © Copyright 1989 American Mathematical Society

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