On discontinuity of derivations, inducing non-unique complete metric topologies

Abstract

We give a simple method for constructing commutative Fréchet algebras which admit two inequivalent Fréchet algebra topologies. The result is applied to show that the action of any non-algebraic analytic function may fail to be uniquely defined among other useful applications. We give an affirmative answer to a question of Loy from 1974. We also obtain the uniqueness of the Fréchet algebra topology of certain Fréchet algebras with finite dimensional radicals.