The ideal structure of some analytic crossed products

Shpigel, Miron

doi:10.1090/S0002-9947-99-02221-7

The ideal structure of some analytic crossed products
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Trans. Amer. Math. Soc. 351 (1999), 2515-2538 Request permission

Abstract:

We study the ideal structure of a class of some analytic crossed products. For an $r$-discrete, principal, minimal groupoid $G$, we consider the analytic crossed product $C^*(G,\sigma )\times _\alpha \mathbb {Z}_+$, where $\alpha$ is given by a cocycle $c$. We show that the maximal ideal space $\mathcal {M}$ of $C^*(G,\sigma )\times _\alpha \mathbb {Z}_+$ depends on the asymptotic range of $c$, $R_\infty (c)$; that is, $\mathcal {M}$ is homeomorphic to $\overline { \mathbb {D}}\mid R_\infty (c)$ for $R_\infty (c)$ finite, and $\mathcal {M}$ consists of the unique maximal ideal for $R_\infty (c)= \mathbb {T}$. We also prove that $C^*(G,\sigma )\times _\alpha \mathbb {Z}_+$ is semisimple in both cases, and that $R_\infty (c)$ is invariant under isometric isomorphism.

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