Extensions of semigroups by the dihedral groups and semigroup C*-algebras

We consider the semidirect product Z(sic)phi Z(x) of the additive group Z of all integers and the multiplicative semigroup Z(x) of integers without zero relative to a semigroup homomorphism phi from Z(x) to the endomorphism semigroup of Z. It is shown that this semidirect product is a normal extension of the semigroup Z x N by the dihedral group, where N is the multiplicative semigroup of all natural numbers. Further, we study the structure of C*-algebras associated with this extension. In particular, we prove that the reduced semigroup C*-algebra of the semigroup Z(sic)phi Z(x) is topologically graded over the dihedral group. As a consequence, there exists a structure of a free Banach module over the reduced semigroup C*-algebra of Z x N in the underlying Banach space of the reduced semigroup C*-algebra of Z(sic)phi Z(x).