On the Automatic Continuity on Fréchet algebras

Abstract

\begin{abstract}
A topological algebra $A$ over $\mathbb{C} $ is called functionally continuous if each complex homomorphism on $A$ is continuous. It is well-known that each Banach algebra is functionally continuous, but it is not known whether or not each commutative Fr\'{e}chet algebra has this property; This is a famous question called Michael's problem. Many mathematicians have been trying to answer this old problem in the past 70 years. To give an affirmative answer to this conjecture we use two great theorems of functional analysis, namely the closed graph theorem and the open image theorem: Let $ \varphi $ be complex homomorphism on a commutative Fr\'{e}chet algebra $ \left( A; \left(|| \, ||_{n}\right) _{n\geqslant 0} \right) $. Our first result shows that there exists an integer $q$ such that $\ker \left(||\, ||_{q} \right)\subset \ker \left(\varphi \right) $ (Theorem 2.1). Next, we will provide $A$ with a sequence $ \left( ||.||'_{n}\right) _{n} $ of semi-norms under which $A$ is a Fr\'{e}chet algebra (Lemma2.2 ) and which make the character $ \varphi $ continuous. The Closed graph theorem [2 , B.2, p.335] and the open image theorem [2, B.1, p.335 ] allow us to show that the two topologies are equivalent on $A$ (Lemma2.3) and therefore $\varphi$ is $ \left(|| \, ||_{n}\right) _{n} $-continuous (Theorem2.3).
\end{abstract}