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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Banach spaces with the $ L\sp 1$-Banach-Stone property


Author: Peter Greim
Journal: Trans. Amer. Math. Soc. 287 (1985), 819-828
MSC: Primary 46E40; Secondary 46B20
MathSciNet review: 768743
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Abstract: It has previously been shown that separable Banach spaces $ V$ with trivial $ L$-structure have the $ {L^1}$-Banach-Stone property, i.e. every surjective isometry between two Bochner spaces $ {L^1}({\mu _i},V)$ induces an isomorphism of the two measure algebras. We remove the separability restriction, employing the topology of the measure algebra's Stonean space.

The result is achieved via a complete description of the $ L$-structure of $ {L^1}(\mu ,V)$.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768743-4
Keywords: Banach-Stone theorem, Bochner $ {L^p}$-space, centralizer, hyperstonean, integral module, $ {L^p}$-structure
Article copyright: © Copyright 1985 American Mathematical Society