Orthogonally additive and multiplicative polynomials and holomorphic maps between Fourier algebras (Q2794441)

Let \(A(G_1)\) and \(A(G_2)\) be the Fourier algebras of the amenable locally compact groups \(G_1\) and \(G_2\), respectively, and let \(H: B_{A(G_1)}(0,r)\to A(G_2)\) be a holomorphic map for some \(r\in\mathbb{R}^+\) (where \(B_{A(G_1)}(0,r)\) stands for the open unit ball in \(A(G_1)\) with centre \(0\) and radius \(r\)). Suppose that the following conditions hold, (i) the \(n\)th differential \(D^nH(0)\) of \(H\) at \(0\) is a completely bounded \(n\)-linear map for each \(n\in\mathbb{N}\); (ii) \(f,g\in B_{A(G_1)}(0,r), \;fg=0 \;\Rightarrow \;H(f+g)=H(f)+H(g)\), and (iii) \(f,g\in B_{A(G_1)}(0,r), \;fg=0 \;\Rightarrow \;H(f)H(g)=0\). Then the authors show that there exist a sequence \((\omega_n)\) in the Fourier-Stieltjes algebra \(B(G_2)\) of \(G_2\) and a piecewise affine homeomorphism \(\alpha: G_2\to G_1\) such that \(H(f)(t)=\sum_{n=1}^\infty\omega_n(t)f(\alpha(t))^n\) for all \(f\in B_{A(G_1)}(0,r')\) and \(t\in G_2\), for some \(0<r'\leq r\).