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| Paper IPM / M / 17874 |
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| Abstract: | |
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We introduce the so-called Bishop-Phelps- Bollob�ás property for positive functionals, a particular case of the Bishop-Phelps- Bollob�ás property for positive operators. First we show a version of the Bishop-Phelps- Bollob�ás theorem â??where all the elements and functionals are positive. We also characterize the â??Bishop-Phelps- Bollob�ás property for positive functionals and positive elements in a Banach lattice. We prove that any finite-dimensional Banach lattice â??has the Bishop-Phelps- Bollob�ás â?? property for positive functionals. A sufficient condition and also a necessary condition to have the Bishop-Phelps- Bollob�ás property for positive functionals are also provided. As a consequence of this result, we obtain that the spaces L_(p )(�ü) (1â?äp<â??), for any positive measure �ü ,C(K) and M(K), for any â??compact Hausdorff topological space K, satisfy the Bishop-Phelps- Bollob�ás property for positive functionals. We also provide some more clarifying examples.
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