Finite dimensionality in socle of Banach algebras
type: Item
instance of: scholarly article
title: Finite dimensionality in socle of Banach algebras
zbMATH Open document ID: 0581.46043
DOI: 10.1155/S0161171284000570
author: S.-E. Takahasi
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46000
review text: It is shown that if the socle soc(A) of a semisimple Banach algebra A is norm-closed, then soc(A) is already finite dimensional. This is an extension of Al-Moajil's result which states that if A is a semi-simple Banach algebra such that \(lan(comp(A))=\{0\}\) and soc(A) is norm-closed, then A is finite dimensional. Here comp(A) and lan(comp(A)) denote respectively the compactrum of A and the left annihilator of comp(A). The proof makes use of the above Al-Moajil's result and the fact that if A is a semisimple Banach algebra with socle soc(A) and min(A) is the set of all minimal idempotents of A, then \(lan(comp(A))=lan(soc(A))=lan(\min (A))=ran(\min (A))=ran(comp(A))\), where ran(X) denotes the right annihilator of X.
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 3929719
zbMATH Keywords: minimal left ideal
zbMATH Keywords: compactrum
zbMATH Keywords: left annihilator
zbMATH Keywords: semisimple Banach algebra with socle
zbMATH Keywords: minimal idempotents
zbMATH Keywords: right annihilator
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3929719