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

Abstract

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