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 Γ-locally analytic vectors, Wˡᵃ, is useful because Wˡᵃ can be analyzed via the Lie algebra Lie(Γ), which simplifies the action of Γ. Additionally, Wˡᵃ 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 ↦ Wˡᵃ acts as a descent, and the derived locally analytic vectors Rₗₐⁱ(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 ALT,la in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring AQₚ from the theory of (φ,Γ)-modules in terms of locally analytic vectors.