Daws, M, Haydon, R, Schlumprecht, T et al. (1 more author) (2013) Shift invariant preduals of $\ell_1(\Z)$. Israel Journal of Mathematics, 192 (2). 541 - 585. ISSN 0021-2172
Abstract
The Banach space $\ell_1(\Z)$ admits many non-isomorphic preduals, for example, $C(K)$ for any compact countable space $K$, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on $\ell_1(\Z)$ weak$^*$-continuous. This is equivalent to making the natural convolution multiplication on $\ell_1(\Z)$ separately weak*-continuous and so turning $\ell_1(\Z)$ into a dual Banach algebra. We call such preduals \emph{shift-invariant}. It is known that the only shift-invariant predual arising from the standard duality between $C_0(K)$ (for countable locally compact $K$) and $\ell_1(\Z)$ is $c_0(\Z)$. We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak$^*$-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to $c_0$. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of $\Z$. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to $c_0$.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2013, Springer Verlag. This is an author produced version of a paper published in Israel Journal of Mathematics. Uploaded in accordance with the publisher's self-archiving policy. The final publication is available at link.springer.com. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 10 Dec 2013 10:32 |
Last Modified: | 01 Dec 2014 01:38 |
Published Version: | http://dx.doi.org/10.1007/s11856-012-0040-1 |
Status: | Published |
Publisher: | Springer Verlag |
Identification Number: | 10.1007/s11856-012-0040-1 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:77165 |