Scattered elements of Banach algebras

Peng Cao¹

¹ School of Mathematics Beijing Institute of Technology and Key Laboratory of Mathematics, Informatics and Behavioral Semantics Ministry of Education Beijing, China, 100081

Studia Mathematica, Tome 214 (2013) no. 2, pp. 195-200

Résumé

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) $[\mathcal {S}(\mathcal {A}),\mathcal {A}]\subset \mathcal {R}_{\rm sc}(\mathcal {A})$.

Détail

DOI : 10.4064/sm214-2-6

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