Restricted algebras on inverse semigroups III, Fourier algebra

The Fourier and Fourier-Stieltjes algebras $A(G)$ and $B(G)$ of a locally compact group $G$ are introduced and studied in 60's by Piere Eymard in his PhD thesis. If $G$ is a locally compact abelian group, then $A(G)\simeq L^1(\hat{G})$, and $B(G)\simeq M(\hat{G})$, via the Fourier and Fourier-Stieltjes transforms, where $\hat{G}$ is the Pontryagin dual of $G$. Recently these algebras are defined on a (topological or measured) groupoid and have shown to share many common features with the group case. This is the last in a series of papers in which we have investigated a "restricted" form of these algebras on a unital inverse semigroup $S$.