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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The ideal structure
of some analytic crossed products


Author: Miron Shpigel
Journal: Trans. Amer. Math. Soc. 351 (1999), 2515-2538
MSC (1991): Primary 47D25; Secondary 46H10, 46L05
Published electronically: February 15, 1999
MathSciNet review: 1475694
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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 $\cal M$ 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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Additional Information

Miron Shpigel
Affiliation: Department of Mathematics, Technion — Israel Institute of Technology, 3200 Haifa, Israel
Email: meshpigel@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02221-7
PII: S 0002-9947(99)02221-7
Received by editor(s): December 2, 1996
Published electronically: February 15, 1999
Article copyright: © Copyright 1999 American Mathematical Society