Generalized derivations and commutativity of prime Banach algebras (Q515731)

The first result on the relation between derivations and commutativity of rings is due to \textit{E. C. Posner} [Proc. Amer. Math. Soc. 8, 1093--1100 (1958; Zbl 0082.03003)] who showed that if \(d\) is a derivation of a prime ring \(R\) such that \(ad(a)-d(a)a\) is in the center of \(R\) for all \(a\in R\), then either \(d\) is zero or \(R\) is commutative. In this paper, the authors study several conditions on generalized derivations of a unital prime Banach algebra that ensure its commutativity. Let \(A\) be a unital prime Banach algebra with a continuous linear generalized derivation \(g\) associated with a nonzero continuous linear derivation \(d\). They prove that \(A\) is commutative if either {\parindent=0.7cm\begin{itemize}\item[1.] \(g((xy)^n)-d(x^n)d(y^n)\in Z(A)\), or \item[2.] \(g((xy)^n)-d(y^n)d(x^n)\in Z(A)\), \end{itemize}} for sufficiently many \(x, y\in A\) and an integer \(n = n(x, y) > 1\). They also give some conditions on continuous left multipliers on \(A\) that ensure the commutativity of \(A\).