Normed algebras with involution
type: Item
instance of: scholarly article
title: Normed algebras with involution
zbMATH Open document ID: 0781.46041
DOI: 10.1007/BF02567080
author: Vladimír Müller
published in: Manuscripta Mathematica
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/155719
review text: Let \(A\) be a (complex) normed algebra with an identity element 1 and an involution \(*\), which may not be continuous. Denote by \(\widetilde {A}\) the completion of \(A\). Then \(\widetilde {A}\) is not a Banach \(*\)-algebra in general. For each \(x\) in \(A\), let \(r(x)\) be the spectral radius and \(p(x)= r(x^* x)^{1/2}\) the Pták function. Then \(A\) is said to be an \(H\)-algebra if \(r(x+y)\leq p(x)+p(y)\) for all \(x\), \(y\) in \(A\). The main result of this paper is: Let \(A\) be a normed \(*\)-algebra and \(H(A)\) the set of all selfadjoint elements of \(A\). Then the following conditions are equivalent: (i) \(A\) is an \(H\)-algebra. (ii) \(r(x)\leq p(x)\) and \(x+y\geq 0\) whenever \(x,y\in A\), \(x\geq 0\), \(y\geq 0\). (iii) \(r(x)\leq p(x)\) and \(p(x+y)\leq p(x)+p(y)\). (iv) There exists a Hilbert space \(H\) and a \(*\)-representation \(\pi: A\to B(H)\) such that \(r(\pi(x))=r(x)\) for every \(x\in A\). (v) \(1+a+b+ic\) is invertible in \(\widetilde A\) for every \(a,b,c\in H(A)\), \(a\geq 0\), \(b\geq 0\). (iv) \(1+a+b+ic\) is not a left (right) topological divisior of zero in \(A\) for all \(a,b,c\in H(A)\), \(a\geq 0\), \(b\geq 0\). This generalizes the known results for Hermitian Banach algebras.
Mathematics Subject Classification ID: 46K05
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 91458
zbMATH Keywords: Banach \(*\)-algebra
zbMATH Keywords: spectral radius
zbMATH Keywords: Pták function
zbMATH Keywords: \(H\)-algebra
zbMATH Keywords: \(*\)- representation
reviewed by: Wong, Pak-ken
MaRDI profile type: Publication
cites work: Q3215824
cites work: Q3730456
cites work: A Square Root Lemma for Banach (* )-Algebras
cites work: Q5836035
cites work: Unitary operators in \(C^ *\)-algebras
cites work: Q5606150
cites work: Symmetry of Banach Algebras
cites work: Locally multiplicatively-convex topological algebras
cites work: Characterizations of 𝐶*-algebras
cites work: Banach algebras with involution
cites work: A note on unitary operators in \(C^ *\)-algebras
cites work: Symmetry of complex involutory Banach algebras. II
cites work: Eine metrische Kennzeichnung der selbstadjungierten Operatoren
OpenAlex ID: W2017154521
description: scientific article; zbMATH DE number 91458