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The Second Duals of Beurling Algebras
H. G. Dales, University of Leeds, England, and A. T.-M. Lau, University of Alberta, Edmonton, Alberta, Canada

Memoirs of the American Mathematical Society
2005; 191 pp; softcover
Volume: 177
ISBN-10: 0-8218-3774-5
ISBN-13: 978-0-8218-3774-0
List Price: US$72
Individual Members: US$43.20
Institutional Members: US$57.60
Order Code: MEMO/177/836
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Let \(A\) be a Banach algebra, with second dual space \(A''\). We propose to study the space \(A''\) as a Banach algebra. There are two Banach algebra products on \(A''\), denoted by \(\,\Box\,\) and \(\,\Diamond\,\). The Banach algebra \(A\) is Arens regular if the two products \(\Box\) and \(\Diamond\) coincide on \(A''\). In fact, \(A''\) has two topological centres denoted by \(\mathfrak{Z}^{(1)}_t(A'')\) and \(\mathfrak{Z}^{(2)}_t(A'')\) with \(A \subset \mathfrak{Z}^{(j)}_t(A'')\subset A''\;\,(j=1,2)\), and \(A\) is Arens regular if and only if \(\mathfrak{Z}^{(1)}_t(A'')=\mathfrak{Z}^{(2)}_t(A'')=A''\). At the other extreme, \(A\) is strongly Arens irregular if \(\mathfrak{Z}^{(1)}_t(A'')=\mathfrak{Z}^{(2)}_t(A'')=A\). We shall give many examples to show that these two topological centres can be different, and can lie strictly between \(A\) and \(A''\).

We shall discuss the algebraic structure of the Banach algebra \((A'',\,\Box\,)\); in particular, we shall seek to determine its radical and when this algebra has a strong Wedderburn decomposition. We are also particularly concerned to discuss the algebraic relationship between the two algebras \((A'',\,\Box\,)\) and \((A'',\,\Diamond\,)\).

Most of our theory and examples will be based on a study of the weighted Beurling algebras \(L^1(G,\omega)\), where \(\omega\) is a weight function on the locally compact group \(G\). The case where \(G\) is discrete and the algebra is \({\ell}^{\,1}(G, \omega )\) is particularly important. We shall also discuss a large variety of other examples. These include a weight \(\omega\) on \(\mathbb{Z}\) such that \(\ell^{\,1}(\mathbb{Z},\omega)\) is neither Arens regular nor strongly Arens irregular, and such that the radical of \((\ell^{\,1}(\mathbb{Z},\omega)'', \,\Box\,)\) is a nilpotent ideal of index exactly \(3\), and a weight \(\omega\) on \(\mathbb{F}_2\) such that two topological centres of the second dual of \(\ell^{\,1}(\mathbb{F}_2, \omega)\) may be different, and that the radicals of the two second duals may have different indices of nilpotence.

Table of Contents

  • Introduction
  • Definitions and preliminary results
  • Repeated limit conditions
  • Examples
  • Introverted subspaces
  • Banach algebras of operators
  • Beurling algebras
  • The second dual of \(\ell^1(G,\omega)\)
  • Algebras on discrete, Abelian groups
  • Beurling algebras on \(\mathbb{F}_2\)
  • Topological centres of duals of introverted subspaces
  • The second dual of \(L^1(G,\omega)\)
  • Derivations into second duals
  • Open questions
  • Bibliography
  • Index
  • Index of symbols
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