Hecke algebras and semigroup crossed product C∗-algebras

Abstract

For an almost normal subgroup 0 of a discrete group , conditions are given which allow one to dene a universal C-norm on the Hecke algebra H( ; 0). If is a semidirect product of a normal subgroup N containing 0 by a group G satisfying some order relations arising from a naturally dened subsemigroup T , and if the normalizer of N is also normal in , then a presentation of H( ; 0) is given. In this situation the C-completion of H( ; 0 )i s-isomorphic with the semigroup crossed product C-algebra C(N= 0)oT. In their paper introducing a number theoretical model of a quantum statistical system exhibiting a phase transition with symmetry breaking, Bost and Connes introduce the notion of an almost normal subgroup 0 of a discrete group , along with the associated Hecke algebra H( ; 0) and its reduced C-algebra completion C r ( ; 0 )( [BC]). They also provide a presentation of the Hecke algebra in the context of the specic almost normal subgroup they consider in their model. A connection between these relations and some relations occurring in a stable C-algebra associated with certain examples of dynamical systems described in [B] provided the motivation for considering the Hecke algebras further. An overview of the structure of the paper follows. After some preliminaries on almost normal subgroup pairs ( ; 0) we introduce a fundamental semigroup T in the group , which contains the normalizer N 0 of 0. A basic representation of this semigroup as isometries in the convolution Hecke algebra H( ; 0) is described. In the presence of a normal subgroup N of containing 0 and contained in N 0 , a natural semigroup C-dynamical system occurs which possesses a universal property with respect to-representations of the Hecke algebra. In the second section we discuss some properties of group partial pre-order relations arising from a subsemigroup of the group in much the same spirit as Nica in [N]. Applying this to our situation, withT as the subsemigroup of , and introducing a notion of solvable least upper bounds, we obtain some conditions allowing a denition of a universal C-norm on the Hecke algebra. Assuming some more structure for the pair ( ; 0), namely that is an extension of a normal subgroup N containing 0, we obtain that