Hecke algebras for protonormal groups

We introduce the term "protonormal" to refer to a subgroup H of a group G such that for every x in G the subgroups x^{-1}Hx and H commute as sets. If moreover (G,H) is a Hecke pair we show that the Hecke algebra H(G,H) is generated by the range of a canonical partial representation of G vanishing on H. As a consequence we show that there exists a maximum C*-norm on H(G,H), generalizing previous results by Brenken, Hall, Laca, Larsen, Kaliszewski, Landstad and Quigg. When there exists a normal subgroup N of G, containing H as a normal subgroup, we prove a new formula for the product of the generators and give a very clean description of H(G,H) in terms of generators and relations. We also give a description of H(G,H) as a crossed product relative to a twisted partial action of the group G/N on the group algebra of N/H. Based on our presentation of H(G,H) in terms of generators and relations we propose a generalized construction for Hecke algebras in case (G,H) does not satisfy the Hecke condition.