New Condition of Automatic Continuity of Dense Range Homomorphisms on Jordan – Banach Algebras
JOURNAL OF EDUCATION AND SCIENCE,
2018, Volume 27, Issue 4, Pages 146-150
AbstractThe 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 ).
In  given a partial solution of the above problem as follows:
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.
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.
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