الملخص

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 Fréchet 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, then is 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.