Scattered elements of Banach algebras

Peng Cao

doi:10.4064/sm214-2-6

Scattered elements of Banach algebras

Peng Cao^*

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

A scattered element of a Banach algebra A is an element with at most countable spectrum. The set of all scattered elements is denoted by S(A): The scattered radical R_sc(A) is the largest ideal consisting of scattered elements. We characterize in several ways central elements of A modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) S(A)+S(A) ⊂ S(A); (ii) S(A)S(A) ⊂ S(A); (iii) [S(A);A] ⊂ R_sc(A).

Original language	English
Pages (from-to)	195-200
Number of pages	6
Journal	Studia Mathematica
Volume	214
Issue number	2
DOIs	https://doi.org/10.4064/sm214-2-6
Publication status	Published - 2013

Keywords

Jacobson radical
Scattered element
Scattered radical

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abstract = "A scattered element of a Banach algebra A is an element with at most countable spectrum. The set of all scattered elements is denoted by S(A): The scattered radical Rsc(A) is the largest ideal consisting of scattered elements. We characterize in several ways central elements of A modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) S(A)+S(A) ⊂ S(A); (ii) S(A)S(A) ⊂ S(A); (iii) [S(A);A] ⊂ Rsc(A).",

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N2 - A scattered element of a Banach algebra A is an element with at most countable spectrum. The set of all scattered elements is denoted by S(A): The scattered radical Rsc(A) is the largest ideal consisting of scattered elements. We characterize in several ways central elements of A modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) S(A)+S(A) ⊂ S(A); (ii) S(A)S(A) ⊂ S(A); (iii) [S(A);A] ⊂ Rsc(A).

AB - A scattered element of a Banach algebra A is an element with at most countable spectrum. The set of all scattered elements is denoted by S(A): The scattered radical Rsc(A) is the largest ideal consisting of scattered elements. We characterize in several ways central elements of A modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) S(A)+S(A) ⊂ S(A); (ii) S(A)S(A) ⊂ S(A); (iii) [S(A);A] ⊂ Rsc(A).

KW - Jacobson radical

KW - Scattered element

KW - Scattered radical

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DO - 10.4064/sm214-2-6

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JF - Studia Mathematica

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Scattered elements of Banach algebras

Abstract

Keywords

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