Automatic Continuity of Dense Range Homomorphisms into Multiplicatively Semisimple Complete Normed Algebras

The following open problem state that: If is a dense range homomorphism from Banach algebra into Banach algebra such that is semisimple. Is automatically continuous? (see[1]) In [5] given a partial solution of the above problem as follows: Let and be a Frchet algebras such that is semisimple, the spectral radius is continuous on and the spectral radius is continuous at zero. If is a dense range homomorphism, thenis automatically continuous. In this paper, we prove the following result: If is a dense range homomorphism from a complete normed nonassociative algebrainto a complete normed nonassociative algebra such that is semisimple and multiplication algebra of is also semisimple, the spectral radius is continuous on and the spectral radius is continuous at zero, then is automatically continuous.