An optimal linear approximation for a class of nonlinear operators between uniform algebras

Assume that A,B are uniform algebras on compact Hausdorff spaces X and Y, respectively. Let T:A→B be a map (nonlinear in general) satisfying T(A−1)=B−1 and T1=1. We show that, if there exist constants α,β≥1 such that β−1‖f⋅g−1‖≤‖Tf⋅(Tg)−1‖≤α‖f⋅g−1‖ for all f∈A and g∈A−1, then there exists a homeomorphism τ:∂B→∂A between the Šilov boundaries of A and B such that (αβ)−1|f(τ(y))|≤|(Tf)(y)|≤αβ|f(τ(y))| for all f∈A and for all y∈∂B. In particular ∂A and ∂B are homeomorphic. Moreover we give an example which shows that the multiple αβ in the above inequality is best possible.