Scattered elements of Banach algebras
Volume 214 / 2013
Abstract
A scattered element of a Banach algebra $\mathcal {A}$ is an element with at most countable spectrum. The set of all scattered elements is denoted by $\mathcal {S}(\mathcal {A}).$ The scattered radical $\mathcal {R}_{\rm sc}(\mathcal {A})$ is the largest ideal consisting of scattered elements. We characterize in several ways central elements of $\mathcal {A}$ modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) $\mathcal {S}(\mathcal {A})+\mathcal {S}(\mathcal {A})\subset \mathcal {S}(\mathcal {A});$ (ii) $\mathcal {S}(\mathcal {A})\mathcal {S}(\mathcal {A})\subset \mathcal {S}(\mathcal {A});$ (iii) $[\mathcal {S}(\mathcal {A}),\mathcal {A}]\subset \mathcal {R}_{\rm sc}(\mathcal {A})$.
