On the Bishop–Phelps–Bollobás property for positive functionals

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 \(L_p(\mu )\) $L_p(\mu )$ ( \(1\le p < \infty \) $1\le p<\infty$ ), for any positive measure \(\mu \) $\mu$ , C(K) and \(\mathcal {M} (K)\) $\mathcal{M}(K)$ , for any compact Hausdorff topological space K,  satisfy the Bishop–Phelps–Bollobás property for positive functionals. We also provide some more clarifying examples.