The Jacobson property in rings and Banach algebras

Abstract

In a Banach algebra A it is well known that the usual spectrum has the following property: $$\begin{aligned} \sigma (ab) \setminus \{0\} = \sigma (ba) \setminus \{0\} \end{aligned}$$ σ ( a b ) \ { 0 } = σ ( b a ) \ { 0 } for elements $$a, b \in A$$ a , b ∈ A . In this note we are interested in subsets of A that have the Jacobson Property, i.e. $$X \subset A$$ X ⊂ A such that for $$a, b \in A$$ a , b ∈ A : $$\begin{aligned} 1 - ab \in X \implies 1 - ba \in X. \end{aligned}$$ 1 - a b ∈ X ⇒ 1 - b a ∈ X . We are interested in sets with this property in the more general setting of a ring. We also look at the consequences of ideals having this property. We show that there are rings for which the Jacobson radical has this property.