Surjective isometries on $C^1$ spaces of uniform algebra valued maps

Abstract

Let $C^1([0,1], A)$ be the Banach algebra of all continuously differentiable maps from the closed unit interval $[0,1]$ to a uniform algebra $A$ with respect to certain norms. We prove that every surjective, not necessarily linear, isometry on $C^1([0,1], A)$ is represented by homeomorphisms on $[0,1]$ and the maximal ideal space of $A$.