On a conjecture concerning some automatic continuity theorems

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.