RESTRICTIONS OF FOURIER TRANSFORMS ON A

We denote also by A(Γ) and B{Γ) the algebras of Fourier transforms and Fourier Stieltjes transforms on Γ. As ordinary the norms of A(Γ) and B(Γ) are given by L(G)-norm and ikf(G)-norm, where M(G) is the bounded regular Borel measures on G. In this paper we investigate that the restriction map of Fourier algebra Φ: A(Γ) —* A(Λ) is a bounded linear mapping, and ask that does there exists a linear lifting λ: A(Λ) —> A(Γ) such that Φo\ = IdΛ We give the affirmative answer in some situations. Evidently if a lifting λ exists, then Φ is onto mapping. Concerning liftings, restrictions and their relationship, Herz [7] has investigated in some stages of group algebras. (Note that in his discussion, the groups are general locally