Topologization of Hecke pairs and Hecke C*-algebras

Let (G, S) be a Hecke pair, i.e., G is a group and S an almost normal subgroup, meaning that every double coset SgS is the union of finitely many left cosets of S. We show that there exists a homomorphism phi from G to a totally disconnected, locally compact group (G) over tilde such that (S) over tilde := <(phi(S))over bar> is a compact, open subgroup of (G) over tilde, and such that the Hecke algebras H(G, S) and H((G) over tilde(S) over tilde) are isomorphic. This "topologization" construction is then used to solve a problem in the theory of Hecke C*-algebras.