Hecke C*-Algebras, Schlichting Completions and Morita Equivalence

Abstract

The Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

Hecke C*-algebras and semi-direct products

“…We shall obtain crossedproduct C * -algebras which are Morita-Rieffel equivalent to the completion of the Hecke algebra inside C * (G), similarly to certain results of [5]. At the end of the section we shall briefly indicate how our results can be applied if the Hecke pair is incomplete.…”

“…We adopt the conventions of [5], which contains more references. A Hecke pair (G, H) comprises a group G and a Hecke subgroup H, i.e., one for which every double coset HxH is a finite union of left cosets {y 1 H, .…”

“…For details on this, we refer to [5,Remark 3.11]. In the special case where Q is abelian, we do have R…”

“…As in [5,Section 7], we specialize to the case where N is abelian. Taking Fourier transforms, the action α of Q on B becomes an action…”

“…We begin in Section 1 by recalling our conventions from [5] regarding Hecke algebras. In Section 2 we describe the main properties of our group-theoretic setup (0.1).…”