Algebraic elements of a Banach algebra modulo an ideal
type: Item
instance of: scholarly article
title: Algebraic elements of a Banach algebra modulo an ideal
zbMATH Open document ID: 0577.46048
author: Bruce Barnes
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let A be a Banach algebra. A classical result states that if \(f\in A\) and \(f^ 2-f\in rad(A)\), the Jacobson radical of A, then \(\exists e\in A\) such that \(e^ 2=e\) and e-f\(\in A\). A more recent result proved by C. Olsen states that when \(T\in B(H)\), the algebra of bounded linear operators on a Hilbert space H, and for some polynomial p, p(T)\(\in K(H)\), the ideal of compact operators, then \(\exists S\in B(H)\) such that \(p(S)=0\) and T-S\(\in K(H)\). In the general situation K is an ideal of A, p is a polynomial, and \(f\in A\) has the property p(f)\(\in K\). This paper is concerned with the question: under what conditions on K and f does there exist \(g\in A\) such that \(p(g)=0\) and f-g\(\in K\). Among the ideals K considered are rad(A), various ideals of B(H) in K(H), and the socle of A.
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 3923445
zbMATH Keywords: Jacobson radical
zbMATH Keywords: algebra of bounded linear operators on a Hilbert space
zbMATH Keywords: ideal of compact operators
zbMATH Keywords: ideal
zbMATH Keywords: socle
MaRDI profile type: Publication
full work available at URL: pjm.1985.117.219
OpenAlex ID: W2062368890
DOI: 10.2140/PJM.1985.117.219
Recommended article: Algebraic ideals of semiprime Banach algebras
Recommended article: Q3756797
Recommended article: The ideal structure of some Banach algebras
Recommended article: Biduals of Banach algebras which are ideals in a Banach algebra
Recommended article: Finite-dimensional ideals in Banach algebras
Recommended article: Ideals in constructive Banach algebra theory
Recommended article: Idempotents dans les algèbres de Banach
Recommended article: Ideal amenability of module extension Banach algebras
Recommended article: Elements of rings and Banach algebras with related spectral idempotents
Recommended article: Ideal Spaces of Banach Algebras
description: scientific article; zbMATH DE number 3923445