New Condition of Automatic Continuity of Dense Range Homomorphisms on Jordan – Banach Algebras

The following open problem stated that, if T: AB is a dense range homomorphism between Banach algebras A and B such that B is semi-simple. Is T automatically continuous? (see [1]). In [3] given a partial solution of the above problem as follows: Let A and B be Frchet algebras such that B is semi simple, the spectral radius rB is continuous on B and the spectral radius rA is continuous at zero. If T : A B is a dense range homomorphism, then T is automatically continuous. In this paper, we prove the following result : If T : A+ B+ is a dense range homomorphism between Jordan Banach algebras A+ and B+ such that B+ is semi simple, the spectral radius rB+ is continuous on B+ and the spectral rA+ is continuous at zero, then T is automatically continuous.