Norm preserving extensions of bounded holomorphic functions

A relatively polynomially convex subset V of a domain Ω has the extension property if for every polynomial p there is a bounded holomorphic function φ on Ω that agrees with p on V and whose H∞ norm on Ω equals the sup-norm of p on V. We show that if Ω is either strictly convex or strongly linearly convex in ℂ², or the ball in any dimension, then the only sets that have the extension property are retracts. If Ω is strongly linearly convex in any dimension and V has the extension property, we show that V is a totally geodesic submanifold. We show how the extension property is related to spectral sets.