New condition of automatic continuity of dense range homomorphisms on Jordan – Banach Algebras

Abstract EN

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]).

In [3] given a partial solution of the above problem as follows: Let A and B be Frechet 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 Tis 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 r*+ is continuous on B~ and the spectral rA + is continuous at zero, then T is automatically continuous.