On Commutativity of Banach $^*$-Algebras with Derivation

Abstract

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 Banach $^*$-algebras.