Second Duals of Beurling Algebras (eBook)

H. G Dales (Autor)

eBook Download: PDF

191 Seiten
American Mathematical Society (Verlag)
978-1-4704-0437-6 (ISBN)

Infos

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.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• ISBN-10: 1-4704-0437-0 / 1470404370
• ISBN-13: 978-1-4704-0437-6 / 9781470404376