On the ideals of extended quasi-nilpotent Banach algebras (Q1178787)

Let \(A\) be a complex Banach algebra, \(\sigma(a)\) denote the spectrum of \(a\in A\). The author calls \(b\) a nonperturbing element of \(A\) if \(\sigma(a+b)\cap\sigma(a)\neq\emptyset\) for every \(a\in A\). It is easy to see that all elements \(b\) of any (proper) closed, two-sided ideal of \(A\) enjoy this property, but the converse is false. Let \(\bar A:=A\oplus\mathbb{C}\). The author shows that if every element of \(A\) is quasi- nilpotent, then every nonperturbing element of the algebra \(\bar A\) lies in a closed, two-sided ideal of \(\bar A\) and there are two closed, two- sided ideals in the algebra \(\overline{\overline{A}}\) whose union comprises all the nonperturbing elements of \(\overline{\overline{A}}\), although this union is not itself an ideal.