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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
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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

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.

• 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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