# The Second Duals of Beurling Algebras

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*H. G. Dales; A. T.-M. Lau*

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

# Table of Contents

## The Second Duals of Beurling Algebras

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. Definitions and Preliminary Results 714 free
- Chapter 3. Repeated Limit Conditions 2532
- Chapter 4. Examples 3542
- Chapter 5. Introverted Subspaces 4552
- Chapter 6. Banach Algebras of Operators 5360
- Chapter 7. Beurling Algebras 6572
- Chapter 8. The Second Dual of l[sup(1)](G,w) 95102
- Chapter 9. Algebras on Discrete, Abelian Groups 111118
- Chapter 10. Beurling Algebras on F[sub(2)] 131138
- Chapter 11. Topological Centres of Duals of Introverted Subspaces 141148
- Chapter 12. The Second Dual of L[sup(1)](G,w) 153160
- Chapter 13. Derivations into Second Duals 167174
- Chapter 14. Open Questions 175182
- Bibliography 177184
- Index 185192
- Index of Symbols 189196 free