Bishop-Phelps-Bollob\'as property for positive functionals

We introduce the so-called Bishop-Phelps-Bollob\'as property for positive functionals, a particular case of the Bishop-Phelps-Bollob\'as property for positive operators. First we show a version of the Bishop-Phelps-Bollob\'as theorem for positive elements and positive functionals in the dual of any Banach lattice. We also characterize the strong Bishop-Phelps-Bollob\'as property for positive functionals in a Banach lattice. We prove that any finite-dimensional Banach lattice has the the Bishop-Phelps-Bollob\'as property for positive functionals. A sufficient and a necessary condition to have the Bishop-Phelps-Bollob\'as 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 and Hausdorff topological space $K,$ satisfy the Bishop-Phelps-Bollob\'as property for positive functionals. We also provide some more clarifying examples.