Composition operators between algebras of differentiable functions

Abstract:

Let $E$, $F$ be real Banach spaces, $U \subseteq E$ and $V \subseteq F$ nonvoid open subsets and ${C^k}(U)$ the algebra of real-valued $k$-times continuously Fréchet differentiable functions on $U$, endowed with the compact open topology of order $k$. It is proved that, for $m \geq p$, the nonzero continuous algebra homomorphisms $A:{C^m}(U) \to {C^p}(V)$ are exactly those induced by the mappings $g:V \to U$ satisfying $\phi \circ g \in {C^p}(V)$ for each $\phi \in {E^\ast }$, in the sense that $A(f) = f \circ g$ for every $f \in {C^m}(U)$. Other homomorphisms are described too. It is proved that a mapping $g:V \to {E^{\ast \ast }}$ belongs to ${C^k}(V,({E^{\ast \ast }},{w^\ast }))$ if and only if $\phi \circ g \in {C^k}(V)$ for each $\phi \in {E^\ast }$. It is also shown that if a mapping $g:V \to E$ verifies $\phi \circ g \in {C^k}(V)$ for each $\phi \in {E^\ast }$, then $g \in {C^{k - 1}}(V,E)$.