Grading of $C^*$-algebras generated by free products of abelian semigroups

The talk is devoted to the study of the grading of the semigroup C*-algebra generated by the regular representation of free products of abelian semigroups. It is claimed that the grading of such algebras can be given by a local subgroup of a free group. Thus, one can obtain an abstract version of the Fourier theorem on the representation of an element of a semigroup $C^*$-algebra as a formal series whose coefficients are indexed by a local group.