Orthogonally additive and multiplicative maps between Figá-Talamanca-Herz algebras

We study the structure of an n-homogeneous polynomial P:Ap(G1)→Ap(G2) between Figá-Talamanca-Herz algebras. Among other results, we show that there exist a proper affine map α:Y⊂G2→G1 defined in an open coset Y of G2 such thatP(f)(y)=f(α(y))n,∀f∈Ap(G1),y∈Y when the locally compact groups G1,G2 are amenable and P is p-completely contractive, orthogonally additive and multiplicative, and preserves a right identity of Ap(G1)⁎⁎.