Functional description of the $C^*$-algebra associated with a group-graded system

We consider a group-graded system as an involutive semigroup equipped with a special family of Banach subspaces (a variant of the Fell bundle) and juxtpoint to each such system a $C^*$-algebra generated by a regular representation in the Hilbert module associated with the system. In the case when the base group is discrete and Abelian, we give its functional description as an algebra of continuous mappings on a dual group with values in the initial system.