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On the duals of Lebesgue-Bochner $ L\sp p$ spaces


Author: Bahattin Cengiz
Journal: Proc. Amer. Math. Soc. 114 (1992), 923-926
MSC: Primary 46E40
DOI: https://doi.org/10.1090/S0002-9939-1992-1027088-7
MathSciNet review: 1027088
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Abstract: Let $ (X,\mathcal{A},\mu )$ be an arbitrary positive measure space. We prove that there exist an extremally disconnected (locally) compact Hausdorff space $ Y$ and a perfect (regular) Borel measure $ \nu $ on $ Y$ such that $ {L^p}(\mu ,{\rm E}) \simeq {L^p}(\nu ,E)$ for all $ 1 \leq p < \infty $ and any Banach space $ E$. If $ {E^*}$ is separable, then $ {L^p}(\mu ,{\rm E})* \simeq {L^q}(\mu ,{{\rm E}^*})$ for all $ 1 < p < \infty ,\;\frac{1}{p} + \frac{1}{q} = 1$ , and $ {L^1}(\mu ,{\rm E})* \simeq {L^\infty }(\nu ,{{\rm E}^*}) \simeq C(\beta Y,{\rm E}_*^*)$, where $ E_*^*$ denotes $ {E^*}$ endowed with the weak* topology. In particular $ {L^1}{(\mu )^*} \simeq {L^\infty }(\nu )$.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1027088-7
Article copyright: © Copyright 1992 American Mathematical Society

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