Sufficient conditions for the projective freeness of Banach algebras
type: Item
instance of: scholarly article
title: Sufficient conditions for the projective freeness of Banach algebras
zbMATH Open document ID: 1186.46053
author: Alexander Brudnyi
author: Amol Sasane
published in: Journal of Functional Analysis
publication date: 2009-12-18T00:00:00Z
review text: The ring \(R\) is projective free if every finitely generated projective \(R\)-module is free. In this paper, it is shown that, if \(R\) is a semi-simple commutative unital complex Banach algebra for which the Banach algebra \(C(M(R))\) of complex continuous functions on the maximal ideal space \(M(R)\) is a projective free ring, then \(R\) itself is also a projective free ring. Moreover, if \(M(R)\) is connected and each complex vector bundle of finite rank on \(M(R)\) is topologically trivial, then \(C(M(R))\) is a projective free ring. Let \(U\) be an open set in a normal space \(X\). As an application, it is shown that the Hardy algebra \(H^{\infty}(U)\) of bounded holomorphic functions on \(U\) is projective free. Also, the algebras of almost periodic functions and of measures are considered.
reviewed by: Mart Abel
Mathematics Subject Classification ID: 46J20
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 5649985
zbMATH Keywords: projective free ring
zbMATH Keywords: semi-simple commutative unital complex Banach algebra
zbMATH Keywords: maximal ideal space
zbMATH Keywords: holomorphic functions
zbMATH Keywords: Hardy algebra
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.05.003
OpenAlex ID: W2043657727
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DOI: 10.1016/J.JFA.2009.05.003
description: scientific article; zbMATH DE number 5649985