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Crossed Products by Hecke Pairs
About this Title
Rui Palma, Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, NO-0316 Oslo, Norway
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 252, Number 1204
ISBNs: 978-1-4704-2809-9 (print); 978-1-4704-4377-1 (online)
DOI: https://doi.org/10.1090/memo/1204
Published electronically: January 29, 2018
Keywords: Crossed product,
Hecke pair,
Hecke algebra,
$C^*$-dynamical system,
Fell bundle,
covariant representation
MSC: Primary 46L55; Secondary 20C08
Table of Contents
Chapters
- Introduction
- 1. Preliminaries
- 2. Orbit space groupoids and Fell bundles
- 3. $^*$-Algebraic crossed product by a Hecke pair
- 4. Direct limits of sectional algebras
- 5. Reduced $C^*$-crossed products
- 6. Other completions
- 7. Stone-Von Neumann Theorem For Hecke Pairs
- 8. Towards Katayama duality
Abstract
We develop a theory of crossed products by actions of Hecke pairs $(G, \Gamma )$, motivated by applications in non-abelian $C^*$-duality. Our approach gives back the usual crossed product construction whenever $G / \Gamma$ is a group and retains many of the aspects of crossed products by groups. We start by laying the $^*$-algebraic foundations of these crossed products by Hecke pairs and exploring their representation theory, and then proceed to study their different $C^*$-completions. We establish that our construction coincides with that of Laca, Larsen and Neshveyev (2007) whenever they are both definable and, as an application of our theory, we prove a Stone-von Neumann theorem for Hecke pairs which encompasses the work of an Huef, Kaliszewski and Raeburn (2008).- Pere Ara and Martin Mathieu, Local multipliers of $C^*$-algebras, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2003. MR 1940428
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