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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Classification of crossed-product $ C\sp *$-algebras associated with characters on free groups

Author: Hong Sheng Yin
Journal: Trans. Amer. Math. Soc. 320 (1990), 105-143
MSC: Primary 46L55; Secondary 46L80
MathSciNet review: 962286
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Abstract: We study the classification problem of crossed-product $ {C^ * }$-algebras of the form $ C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf{Z}}$, where $ G$ is a discrete group, $ \chi$ is a one-dimensional character of $ G$, and $ {\alpha_\chi}$ is the unique $ *$-automorphism of $ C_r^ * (G)$ such that if $ U$ is the left regular representation of $ G$, then $ {\alpha_{\chi}(U_{g})=\chi(g)U_{g}}$, $ g \in G$. When $ {G = F_{n}}$, the free group on $ n$ generators, we have a complete classification of these crossed products up to $ *$-isomorphism which generalizes the classification of irrational and rational rotation $ {C^ * }$-algebras. We show that these crossed products are determined by two $ K$-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural $ \operatorname{Aut} ({F_n})$-action, and that these two orbit invariants completely classify the characters up to automorphisms of $ {F_n}$. The classification of crossed products follows from these results.

We also consider the same problem for $ G$ some other groups.

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Article copyright: © Copyright 1990 American Mathematical Society