A generalization of the Arens extension for Banach algebras
type: Item
instance of: scholarly article
title: A generalization of the Arens extension for Banach algebras
zbMATH Open document ID: 1321.46028
author: Olena Taras
author: A. V. Zagorodnyuk
published in: Indagationes Mathematicae. New Series
publication date: 2015-03-23T00:00:00Z
review text: Let \(A\) be a complex Banach algebra and \(H_b(A)\) the Fréchet algebra of all holomorphic functions on \(A\) bounded on bounded sets. The authors introduce a multiplicative convolution \(\star\) in the strong dual space \(H_b(A)^*\) in the following way: Every \(x\in A\) determines a linear operator on \(A\) according to \(m_x(y)=xy\). Then the composition operator \(Q_x\) on \(H_b(A)\) with symbol \(m_x\) leads to its dual operator \(Q_x^*\) on \(H_b(A)^*\). With its help, the multiplicative convolution of \(f\in H_b(A)\) and \(\psi\in H_b(A)^*\) is defined by \((\psi\star f)(x)=Q_x^*(\psi)(f)\). And finally, the multiplicative convolution of \(\varphi,\psi\in H_b(A)^*\) is defined by \((\varphi\star \psi)(f)=\varphi(\psi\star f)\). It is proved that \((\varphi\star \psi)\in H_b(A)^*\) and that \(\star\) is an extension of the Arens product in \(A^{**}\) if one considers \(A^{**}\) as a subset of \(H_b(A)^*\) by identifying every \(x\in A^{**}\) with the evaluation functional \(\tilde{\delta}_x(f)= \tilde{f}(x)\), where \( \tilde{f}\) is the Aron-Berner extension of \(f\).
reviewed by: Pablo Galindo
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 6418454
zbMATH Keywords: Arens-Berner extension
zbMATH Keywords: Arens product
zbMATH Keywords: Banach algebras
zbMATH Keywords: holomorphic functions of bounded type
MaRDI profile type: Publication
full work available at URL: j.indag.2014.10.003
OpenAlex ID: W2039081183
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DOI: 10.1016/J.INDAG.2014.10.003
description: scientific article; zbMATH DE number 6418454