Locally analytic vectors and decompletion in mixed characteristic

In p-adic Hodge theory and the p-adic Langlands program, Banach spaces with Qₚ-coefficients and p-adic Lie group actions are central. Studying the subrepresentation of -locally analytic vectors, W^la, is useful because W^la can be analyzed via the Lie algebra Lie (), which simplifies the action of. Additionally, W^la often behaves as a decompletion of W, making it closer to an algebraic or geometric object. This article introduces a notion of locally analytic vectors for W in a mixed characteristic setting, specifically for Zₚ-Tate algebras. This generalization encompasses the classical definition and also specializes to super-H\"older vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic 0 and p. Our main theorem shows that under certain conditions, the map W W^la acts as a descent, and the derived locally analytic vectors R₋₀ⁱ (W) vanish for i 1. This result extends Theorem C of Po24, providing new tools for propagating information about locally analytic vectors from characteristic 0 to characteristic p. We provide three applications: a new proof of Berger-Rozensztajn's main result using characteristic 0 methods, the introduction of an integral multivariable ring A₋ₓ^, la in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring Aₐ䂹 from the theory of (, ) -modules in terms of locally analytic vectors.