Hecke algebras and semigroup crossed product C*-algebras

For an almost normal subgroup Gamma(o) of a discrete group Gamma, conditions are given which allow one-to define a universal C*-norm on the Hecke algebra H(Gamma,Gamma(o)). If Gamma is a semidirect product of a normal subgroup N containing Gamma(o) by a group G satisfying some order relations arising from a naturally defined subsemigroup T, and if the normalizer of N is also normal in Gamma, then a presentation of H(Gamma,Gamma(o)) is given. In this situation the C*-completion of H(Gamma,Gamma(o)) is *-isomorphic with the semigroup crossed product C*-algebra C*(N/Gamma(o)) x T.