On multiplicatively closed subsets of normed algebras
type: Item
instance of: scholarly article
title: On multiplicatively closed subsets of normed algebras
zbMATH Open document ID: 1200.46042
author: A. Moreno
author: Ángel Rodríguez Palacios
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: It is well-known that, if \(S\) is a bounded and multiplicatively closed subset of an associative normed algebra (\(A,| \cdot |\)), then there exists an equivalent algebra norm \(\|\cdot \|\) on \(A\) such that \(\|s\|\leqslant 1\) for every \(s\in S\). It is not very hard to find examples of nonassociative normed algebras \(A\) where such a result fails. In this paper, the authors prove that a similar result remains true in the nonassociative case whenever the subset \(S\) is assumed to be contained in the nucleus of \(A\). It is also shown that, in the particular case that the subset \(S\) reduces to a nonzero nuclear idempotent \(p\), the equivalent algebra norm \(|| \cdot ||\) above can be chosen so that \(p\) becomes a strongly exposed point of the closed unit ball of (\(A,|| \cdot ||\)). Normed (not necessarily associative) algebras \(A\) satisfying the ``norm-one boundedness property'' (i.e., for every bounded and multiplicatively closed subset \(S\) of \(A\), there exists an equivalent algebra norm \(|| \cdot ||\) on \(A\) such that \(|| s || \leqslant 1\) for every \(s\in S\)) are especially studied. It is shown that absolute-valued algebras, \(J\)B-algebras, and nilpotent normed algebras fulfil the NBP. In the case when an anti-commutative complete normed algebraic algebra \(A\) satisfies the NBP, there exists \(n \in\mathbb N\) such that \(L^n_a = 0\) for every \(a\in A\), where \(L_a\) denotes the operator of left multiplication by \(a\). It follows from a celebrated theorem of E.\,I.\thinspace Zel'manov on the so-called Engel Lie algebras that a complete normed algebraic Lie algebra satisfies the NBP if and only if it is nilpotent.
reviewed by: Mart Abel
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 5696921
zbMATH Keywords: normed algebras
zbMATH Keywords: multiplicatively closed set
zbMATH Keywords: spectral radius
zbMATH Keywords: nearly associated algebra
zbMATH Keywords: norm-one boundedness property
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.09.034
OpenAlex ID: W2079465222
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DOI: 10.1016/J.JALGEBRA.2009.09.034
description: scientific article; zbMATH DE number 5696921