In p-adic Hodge theory and the p-adic Langlands program, Banach spaces with
$\mathbf {Q}_p$-coefficients and p-adic Lie group actions are central. Studying the subrepresentation of G-locally analytic vectors,
$W^{\mathrm {la}}$, is useful because
$W^{\mathrm {la}}$ can be studied via the Lie algebra
$\mathrm {Lie}(G)$, which simplifies the action of G. Additionally,
$W^{\mathrm {la}}$ 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
$\mathbf {Z}_p$-Tate algebras. This generalization encompasses the classical definition and also specializes to super-Hölder vectors in characteristic p. Using binomial expansions instead of Taylor series, this new definition bridges locally analytic vectors in characteristic
$0$ and characteristic p.
Our main theorem shows that under certain conditions, the map
$W \mapsto W^{\mathrm {la}}$ acts as a descent, and the derived locally analytic vectors
$\mathrm {R}_{\mathrm {la}}^i(W)$ vanish for
$i \geq 1$. This result extends Theorem C of [Por24], 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
$\widetilde {\mathbf {A}}_{\mathrm {LT}}^{\dagger ,\mathrm {la}}$ in the Lubin-Tate setting, and a novel interpretation of the classical Cohen ring
$\mathbf {{A}}_{\mathbf {Q}_p}$ from the theory of
$(\varphi ,\Gamma )$-modules in terms of locally analytic vectors.