SAMS Subject Classification Number: 11

In this talk \(A\) will represent a non commutative and unital Banach algebra. We let \(\sigma(a)\) represent the usual spectrum of \(a \in A\). It is well known that for \(a, b \in A\) we have \[\lambda \in \sigma(ab) \setminus \{0\} \iff \lambda \in \sigma(ba) \setminus \{0\}.\qquad{(1)}\] We are interested in subsets of \(A\) that have the Jacobson Property, i.e. \(X \subset A\) such that for \(a, b \in A\): \[1 - ab \in X \iff 1 - ba \in X.\qquad{(2)}\]