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

Abstract

We give a simple method for constructing commutative Frechet algebras which admit two inequivalent Frechet 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 Frechet algebra topology of certain Frechet algebras with finite dimensional radicals.

Read used "tensor product by rows" method to show that a certain algebra admits two inequivalent topology and showed that Singer-Wermer conjecture fails in the Frechet case. We study discontinuity of derivation/functional in detail. The 2nd paper gives examples with countably many inequivalent topologies; the 2nd example satisfies this conjecture, and admits countably many equivalent topologies