Commutativity theorems for Banach algebras (Q804901)

Commutativity assertions for rings, mostly due to Herstein, are sharpened in the case when the ring is a Banach algebra A, and results as the following ones are obtained: Supposed that A has no nontrivial nilpotent ideal and there exists a nonvoid open set G in A such that for each x,y\(\in G\) there are (individual) m,n\(\in {\mathbb{N}}\) for which \(x^ my^ n-y^ nx^ m\) lies in the center of A, then A is commutative. Further an element a of A lies in the center of A, if there exists a nonvoid open set G in A with the property that for each \(x\in G\) there exists \(n\in {\mathbb{N}}\), \(n\geq 2\), such that a commutes with \(x^ n-x\). Now let A be a Banach\({}^*\)-algebra with continuous involution. When A has no nontrivial nilpotent ideal it must be commutative or else the set of \(x\in A\) such that \(x^ r\) is nonnormal for each \(r\in {\mathbb{N}}\) is dense in A. If there exists no universal \(r\in {\mathbb{N}}\) such that \(x^ r\) lies in the center of A for each \(x\in A\), then the set of \(x\in A\) such that \((xx^*)^ n\neq (x^*y)^ n\) for each \(n\in {\mathbb{N}}\) is dense in A.