Double Derivations, Higher Double Derivations and Automatic Continuity

Abstract

Let be a Banach algebra. Let be linear mappings on . First we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple Banach algebras, in certain case. Afterwards we define a new vocabulary called “-higher double derivation" and present a relation between this subject and derivations and finally give some theorems and corollaries about the automatic continuity of this notion.