A real valued homomorphism on algebras of differentiable functions

Abstract

In this paper we prove that, for every homomorphism A A on C k ( E ) {C^k}\left ( E \right ) , there exists x ∈ E x \in E such that A ( f ) = f ( x ) A\left ( f \right ) = f\left ( x \right ) for f ∈ C k ( E ) f \in {C^k}\left ( E \right ) . Here C k ( E ) ( k = 1 , 2 , … , ∞ ) {C^k}\left ( E \right )\;\left ( {k = 1,2, \ldots , \infty } \right ) denotes the algebra of all k k -times differentiable real functions on a real and separable Banach space E E .