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Let $G$ be a $p$-adic reductive group and $\mathfrak{g}$ its Lie algebra. Weconstruct a functor from the extension closure of the Bernstein-Gelfand-Gelfandcategory $\mathcal{O}$ associated to $\mathfrak{g}$ into the category oflocally analytic representations of $G$, thereby expanding on an earlierconstruction of Orlik-Strauch. A key role in this new construction is played by$p$-adic logarithms on tori. This functor is shown to be exact with image inthe subcategory of admissible representations in the sense of Schneider andTeitelbaum. En route, we establish some basic results in the theory of modulesover distribution algebras and related subalgebras, such as a tensor-homadjunction formula. We also relate our constructions to certain representationsconstructed by Breuil and Schraen in the context of the $p$-adic Langlandsprogram.

From category $\mathcal{O}^\infty$ to locally analytic representations