A structure theorem for discontinuous derivations of Banach algebras of differential functions

Abstract:

Let $D:{C^n}\left [ {0,1} \right ] \to \mathcal {M}$ be a derivation from the Banach algebra of $n$ times continuously differentiable functions on $\left [ {0,1} \right ]$ into a Banach ${C^n}\left [ {0,1} \right ]$-module $\mathcal {M}$. If $D$ is continuous then it is completely determined by $D\left ( z \right )$ where $z\left ( t \right ) = t,0 \leq t \leq 1$. For the case when $D$ is discontinuous we show that $D\left ( f \right )$ is determined by $D\left ( z \right )$ for all $f$ in an ideal $\mathcal {T}{\left ( D \right )^2}$ of ${C^n}\left [ {0,1} \right ]$ where its closure $\overline {\mathcal {T}{{\left ( D \right )}^2}}$ is of finite codimension.