On an Approximate Solution to Rodriguez Conjecture

Rickart Theorem ensures the automatic continuity of a dense range homomorphism from a Banach algebra into a strongly Semisimple Banach algebra. Rodriguez conjecture is an extension of Rickart theorem in order to include the nonassociative algebras as follows: Rodriguez conjecture:Every densely valued homomorphism from a complete normed nonassociative algebra into another one with zero strong radical is continuous. There is an affirmative answer of Rodriguez conjecture in particular case of power-associative algebras. In this work, we give an approximate solution of Rodriguez conjecture: If A and B are complete normed nonassociative algebras and if f is a dense range homomorphism from A into B such that M(A) (the multiplication algebra of A) is full and B is strongly Semisimple, then f is continuous. Finally, we give a Gelfand theorem on automatic continuity as a corollary and as an applied example of our approximate solution of Rodriguez conjecture.