Grading of a semigroup $C^*$-algebra by a local group

In this paper we consider the grading by a local group of the semigroup $C^\ast$-algebra generated by the free product of abelian semigroups. The simplest of these algebras is the Cuntz-Toeplitz algebra $TO_n$. For such algebras, an abstract version of the Fourier series by the local group is constructed. A number of properties of the "harmonics" of this series are given.