Hecke C*-algebras and semi-direct products

Abstract

We analyse Hecke pairs (G,H) and the associated Hecke algebra when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊_βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)^R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K², where K = ℚ or K = ℤ[p⁻¹]. In particular we look at the ax + b group of a quadratic extension of K.