Tripotents in algebras: ideals and commutators (Q2095856)

An element \(A\) in algebra \(\mathcal{A}\) is said to be \(n\)-potent (\(n\in \mathbb{N}\)) if \(A^{n}=A\). For the particular cases \(n=2\) and \(n=3,\) we find the standard definitions of idempotents and tripotents, respectively. The main goal of this paper is to study \(n\)-potent elements in unital algebras. Obtaining in this way the compactness conditions for the product \(AB\) of a Hilbert space tripotents \(A\) and \(B\). Also, the authors discovered conditions assured that \(A+B\) is an idempotent.