Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras

Abstract

In 1971 R. L. Carpenter proved that every derivation on a semisimple commutative Frechet algebra with identity is continuous. The concept of almost derivations on Frechet algebras is introduced in this article. Also, R. L. Carpenter result motivates us to ask an open question: Is every almost derivation on semisimple commutative Frechet algebras continuous?. Moreover, a partial answer to this open question is derived in the sense that every almost derivation T on semisimple commutative Frechet Q-algebras A, with an additional condition on A, is continuous. Furthermore, an example is provided to illustrate our main result.