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On the uniqueness question for Hahn-Banach extensions from the space of $ \mathcal{L}^1$ analytic functions


Author: Edgar Reich
Journal: Proc. Amer. Math. Soc. 88 (1983), 305-310
MSC: Primary 46E15; Secondary 30H05
DOI: https://doi.org/10.1090/S0002-9939-1983-0695263-1
MathSciNet review: 695263
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Abstract: Let $ \Omega $ be a region of the complex plane, $ \mathcal{B}\left( \Omega \right)$ the space of analytic $ {\mathcal{L}^1}$ functions over $ \Omega $, and $ \kappa \in {\mathcal{L}^\infty }\left( \Omega \right)$. An evident necessary condition for the linear functional $ {\Lambda _\kappa }\left[ \varphi \right] = \smallint {\smallint _\Omega }\kappa \varphi dxdy(\varphi \in \mathcal{B}(\Omega ))$ to have a unique Hahn-Banach extension from $ \mathcal{B}(\Omega )$ to $ {\mathcal{L}^1}(\Omega )$ is that $ \vert\vert {{\Lambda _{\kappa \vert G}}} \vert\vert = {\vert\vert {\kappa \vert G} \vert\vert _\infty }$ for every restriction $ \kappa \vert G$ of $ \kappa $ to a subregion $ G$ of $ \Omega $. An example is constructed to show that not even a considerably stronger condition is sufficient for uniqueness of the Hahn-Banach extension. Remarks on the problem of whether $ \left\vert {\kappa (z)} \right\vert$ is necessarily constant a.e. if the Hahn-Banach extension is unique indicate that this question is still open, contrary to an assertion in the literature.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0695263-1
Article copyright: © Copyright 1983 American Mathematical Society

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