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Crossed Products by Hecke Pairs


About this Title

Rui Palma

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

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Table of Contents


Chapters

  • Introduction
  • Chapter 1. Preliminaries
  • Chapter 2. Orbit space groupoids and Fell bundles
  • Chapter 3. $^*$-Algebraic crossed product by a Hecke pair
  • Chapter 4. Direct limits of sectional algebras
  • Chapter 5. Reduced $C^*$-crossed products
  • Chapter 6. Other completions
  • Chapter 7. Stone-Von Neumann Theorem For Hecke Pairs
  • Chapter 8. Towards Katayama duality

Abstract


We develop a theory of crossed products by actions of Hecke pairs , motivated by applications in non-abelian -duality. Our approach gives back the usual crossed product construction whenever 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 -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).

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