Endomorphisms of Banach algebras of infinitely differentiable functions on compact plane sets

This note is a sequel to [7] where we investigated the endomorphisms of a certain class of Banach algebras of infinitely differentiable functions on the unit interval. Start with a perfect, compact plane setX. We say that a complex-valued function f defined on X is complex-differentiable at a point a ∈ X if the limit f (a) = lim z→a, z∈X f(z)− f(a) z − a exists. We call f ′(a) the complex derivative of f at a. Using this concept of derivative, we define the terms complex–differentiable on X, continuously complex–differentiable on X, and infinitely complex–differentiable on X in the obvious way. We denote the n-th complex derivative of f at a by f (n)(a), and we denote the set of infinitely differentiable functions on X by D∞(X).