Automatic continuity of almost Jordan derivations on special Jordan Banach algebras

The following is the question form of Kaplansky conjecture of ۱۹۵۸. Is every derivation on semisimple Banach algebra continuous? Kaplansky conjecture was proved by Johnson and Sinclair in ۱۹۶۸. The concept of almost Jordan derivations on Jordan Banach algebras is introduced in this article. Also, Kaplansky conjecture is extended to Jordan Banach algebras as an open question: Is every almost Jordan derivations on semisimple Jordan Banach algebras continuous?. Moreover, a partial answer to this open question is derived in the sense that every almost Jordan derivation T on semisimple special Jordan Banach algebras \Omega^{+}, with an additional condition on \Omega^{+}, is continuous.

Keywords:

Automatic continuity , Almost derivation , Banach algebras