Composition operators between algebras of differentiable functions

Abstract

Let E, F be real Banach spaces, U subset-or-equal-to E and V subset-equal-to F non-void open subsets and C(k)(U) the algebra of real-valued k-times continuously Frechet differentiable functions on U, endowed with the compact open topology of order k. It is proved that, for m greater-than-or-equal-to p, the nonzero continuous algebra homomorphisms A: C(m)(U) --> C(p)(V) are exactly those induced by the mappings g: V --> U satisfying phi . g is-an-element-of C(p)(V) for each phi is-an-element-of E*, in the sense that A(f) = fog for every f is-an-element-of C(m)(U). Other homomorphisms are described too. It is proved that a mapping g: V --> E** belongs to C(k)(V, (E**, w*)) if and only if phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*. It is also shown that if a mapping g: V --> E verifies phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*, then g is-an-element-of C(k-1)(V, E).