Jordan derivations on prime rings and their applications in Banach algebras. I (Q2851108)

An additive mapping \(D\) from a ring \(R\) to \(R\) is called a derivation if \(D(xy)= D(x)y+xD(y)\) holds for all \(x, y\in R\). It is called a Jordan derivation if \(D(x^2)=D(x)x+xD(x)\) holds for all \(x\in R\). \textit{B. E. Johnson} and \textit{A. M. Sinclair} [Am. J. Math. 90, 1067--1073 (1968; Zbl 0179.18103)] proved that any linear derivation on a semisimple Banach algebra is continuous. A result of \textit{I. M. Singer} and \textit{J. Wermer} [Math. Ann. 129, 260--264 (1955; Zbl 0067.35101)] states that every continuous linear derivation on a commutative Banach algebra maps the algebra into its radical. From these two results, we can conclude that there are no nonzero linear derivations on a commutative semisimple Banach algebra. \textit{M. P. Thomas} [Ann. Math., II. Ser. 128, 435--460 (1988; Zbl 0681.47016)] proved that any linear derivation on a commutative Banach algebra maps the algebra into its radical. A noncommutative version of Singer and Wermer's Conjecture states that every continuous linear derivation on a noncommutative Banach algebra maps the algebra into its radical.NEWLINENEWLINEThe purpose of the paper under review is to prove that the noncommutative version of the Singer-Wermer Conjecture holds true under certain conditions. Let \(A\) be a noncommutative Banach algebra. Suppose that there exists a continuous linear Jordan derivation \(D: A\to A\) such that \(D(x)^3[D(x), x]\in \text{rad}(A)\) for all \(x\in A\). In this case, the author shows that \(D(A)\subseteq \text{rad}(A)\).