Linear generalized derivations on Banach ${}^{\ast }$-algebras

This paper deals with some identities on Banach ${}^{\ast }$-algebras that are equipped with linear generalized derivations. As an application of one of our results, we describe the structure of the underlying algebras. Precisely, we prove that for a linear generalized derivation $F$ on a Banach ${}^{\ast }$-algebra $A$, either we obtain the existence of a central idempotent element $e\in Q$, for which $F=0$ on $eQ$ and $(1-e)Q$ satisfies $s_4$, or the set of elements $u\in A$ such that the identity $[F(u{)}^n,F(u^{\ast }{)}^{n}F(u{)}^n]\in Z(A)$ holds for no positive integer $n$ turns out to be dense. In addition to this we consider an identity satisfied by a semisimple Banach ${}^{\ast }$-algebra and look for its commutativity. Moreover, some related results are also established.