On a conjecture concerning some automatic continuity theorems

TL;DR

This paper addresses a conjecture in automatic continuity theorems for uniform topological algebras, providing a characterization of when inverse homomorphisms are continuous based on radical properties.

Contribution

It offers a new characterization of the continuity of inverse homomorphisms in uniform topological algebras, resolving a conjecture in the field.

Findings

Characterization of inverse homomorphism continuity via radical properties.

Provides an answer to a conjecture on automatic continuity theorems.

Links radical intersections with the image of homomorphisms.

Abstract

Let A and B be commutative locally convex algebras with unit. A is assumed to be a uniform topological algebra. Let h be an injective homomorphism from A to B. Under additional assumptions, we characterize the continuity of the homomorphism h^(-1) / Im(h) by the fact that the radical (or strong radical) of the closure of Im(h) has only zero as a common point with Im(h). This gives an answer to a conjecture concerning some automatic continuity theorems on uniform topological algebras.