Shift invariant preduals of ℓ1(ℤ)

Daws, M., Haydon, R., Schlumprecht, T. and White, S. (2012) Shift invariant preduals of ℓ1(ℤ). Israel Journal of Mathematics, 192(2), pp. 541-585. (doi: 10.1007/s11856-012-0040-1)

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Abstract

The Banach space ℓ<sub>1</sub>(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 ℓ<sub>1</sub>(ℤ) weak<sup>*</sup>-continuous. This is equivalent to making the natural convolution multiplication on ℓ<sub>1</sub>(ℤ) separately weak*-continuous and so turning ℓ<sub>1</sub>(ℤ) into a dual Banach algebra. We call such preduals <i>shift-invariant</i>. It is known that the only shift-invariant predual arising from the standard duality between C<sub>0</sub>(K) (for countable locally compact K) and ℓ<sub>1</sub>(ℤ) is c<sub>0</sub>(ℤ). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak<sup>*</sup>-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c<sub>0</sub>. 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<sub>0</sub>.

Item Type:Articles
Additional Information:The original publication is available at www.springerlink.com
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:White, Professor Stuart
Authors: Daws, M., Haydon, R., Schlumprecht, T., and White, S.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Israel Journal of Mathematics
Publisher:Hebrew University Magnes Press / Springer
ISSN:0021-2172
ISSN (Online):1565-8511
Copyright Holders:Copyright © 2012 Hebrew University Magnes Press / Springer
First Published:First published in Israel Journal of Mathematics 192(2):541-585
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher
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