Dense subsets of Banach \(*\)-algebras (Q1300137)

In a normed \(*\)-algebra \(A\) an element \(x\) is called quasi-normal if it commutes with \(x^*x\), and it is called anti-normal if \(x^m\) does not commute with \(x^{*n}\) for any natural numbers \(m\) and \(n\). Let \(A\) be noncommutative and have an identity \(e\), then \(e\) is shown to belong to the closure of the set of those elements of \(A\) for which no power is quasi-normal. If \(A\) is a Banach \(*\)-algebra with continuous involution, if it does not possess any nonzero nilpotent ideal, and if it is still noncommutative, then the set of anti-normal elements is dense.