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

The following open problem stated that, if T: A⟶B is a dense range homomorphism between Banach algebras A and B such that B is semi-simple. Is T automatically continuous? (see [1]).<br /> In [3] given a partial solution of the above problem as follows:<br /> Let A and B be Fréchet 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.<br /> In this paper, we prove the following result :<br /> 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.