Crossed products by Hecke pairs II: C*-completions

TL;DR

This paper explores the different C*-completions of crossed products by Hecke pairs, establishing their properties, relations to existing theories, and applications like a Stone-von Neumann theorem and duality frameworks.

Contribution

It introduces a comprehensive framework for C*-completions of crossed products by Hecke pairs, connecting with prior constructions and extending their applications.

Findings

Defined reduced and full C*-crossed products for Hecke pairs

Proved the equivalence with Laca, Larsen, and Neshveyev's constructions

Established a Stone-von Neumann theorem for Hecke pairs

Abstract

In this second article on crossed products by "actions" of Hecke pairs we study their different C*-completions, namely we show how reduced and full C*-crossed products can be defined. We also establish that our construction coincides with that of Laca, Larsen and Neshveyev whenever they are both definable. 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 and we lay down the foundations for obtaining a form of Katayama duality with respect to the Echterhoff-Quigg crossed product.