On the Bishop–Phelps–Bollobás property for positive functionals

Abstract

We introduce the so-called Bishop–Phelps–Bollobás property for positive functionals, a particular case of the Bishop–Phelps–Bollobás property for positive operators. First we show a version of the Bishop–Phelps–Bollobás theorem where all the elements and functionals are positive. We also characterize the Bishop–Phelps–Bollobás property for positive functionals and positive elements in a Banach lattice. We prove that any finite-dimensional Banach lattice has the Bishop–Phelps–Bollobás property for positive functionals. A sufficient condition and also a necessary condition to have the Bishop–Phelps–Bollobás property for positive functionals are also provided. As a consequence of this result, we obtain that the spaces $L_p(\mu )$ ($1\le p < \infty$), for any positive measure $\mu$, C(K) and $\mathcal {M} (K)$, for any compact Hausdorff topological space K,  satisfy the Bishop–Phelps–Bollobás property for positive functionals. We also provide some more clarifying examples.