A subgroup of a group is said to be almost normal, or a Hecke subgroup, if every double coset contains finitely many right cosets.
To each almost normal inclusion, one associates a Hecke algebra and a Hecke C*-algebra, which play the roles of the group algebra and groupC*-algebra of the (nonexisting) quotient group. I will discuss this construction and some examples, arising from number theory, that can be
viewed as semigroup crossed products.