On Commutativity of Banach (^*)-Algebras with Derivation (original) (raw)

The aim of this paper is to apply purely ring theoretic results to discuss the commutativity of a Banach algebra and Banach \(^*\)-algebra via derivations. We prove that if \(\mathfrak {A}\) is a semiprime Banach algebra and \(\mathscr {G}\) a nonempty open subsets of \(\mathfrak {A}\) which admits a nonzero continuous linear derivation \(d:\mathfrak {A}\rightarrow \mathfrak {A}\) such that \(d([x^m-x,y])\in Z(\mathfrak {A})\) for each x in \(\mathscr {G}\) and an integer \(m=m(x)>1\), then \(\mathfrak {A}\) is commutative. Further, we discuss the commutativity of Banach \(^*\)-algebra. In particular, it is shown that either a semiprime Banach \(^*\)-algebra \(\mathfrak {A}\) with continuous involution and derivation is commutative or the set of \(x\in \mathfrak {A}\) for which \([d(x^k),d((x^k)^*)]\in Z(\mathfrak {A})\) for no positive integer \(k\ge 1\), is dense in \(\mathfrak {A}\). Finally, few more parallel results have been established about the commutativity of Banach and...