# Dagger groups and $p$-adic distribution algebras

Aranya Lahiri, Claus Sorensen, Matthias Strauch

###### Abstract.

Let $(G,\omega)$ be a $p$-saturated group and $K/\mathbb{Q}_{p}$ a finite
extension. In this paper we introduce the space of $K$-valued overconvergent
functions $\mathcal{C}^{\dagger}(G,K)$. In the process we promote the rigid
analytic group attached to $(G,\omega)$ in [LS23] to a dagger group. A main
result of this article is that under certain assumptions (satisfied for
example when $G$ is a uniform pro-$p$ group) the distribution algebra
$D^{\dagger}(G,K)$, i.e. the strong dual of $\mathcal{C}^{\dagger}(G,K)$, is a
Fréchet-Stein algebra in the sense of [ST03].

In the last section we introduce overconvergent representations and show that
there is an anti-equivalence of categories between overconvergent
$G$-representations of compact type and continuous $D^{\dagger}(G,K)$-modules
on nuclear Fréchet spaces. This is analogous to the anti-equivalence between
locally analytic representations and modules over the locally analytic
distribution algebra as proved in [ST02].

##  1\. Introduction

The theory of locally analytic representations plays a central role in the
$p$-adic Langlands program. For now, let $G$ be any locally analytic group
(over $\mathbb{Q}_{p}$ for simplicity). In [ST02, Cor. 3.3] Schneider and
Teitelbaum establish a duality between locally analytic $G$-representations on
$K$-vector spaces of compact type and certain modules over the locally
analytic distribution algebra $D(G,K)$. (Here $K/\mathbb{Q}_{p}$ is a fixed
finite extension which serves as our coefficient field throughout this paper.)
In the sequel [ST03] they introduce the notion of a Fréchet-Stein algebra, and
one of their main results is that $D(G,K)$ has this property when $G$ is
compact. This gives rise to an abelian category of coadmissible
$D(G,K)$-modules, and by duality an abelian category of admissible locally
analytic representations. Moreover, in [ST03, Sect. 8] they develop a
dimension theory for modules over a Fréchet-Stein algebra. The Fréchet-Stein
structure on $D(G,K)$ has many other applications. We mention the
irreducibility criterion for locally analytic principal series representations
in [OS10], the proof of which makes critical use of the Fréchet-Stein
structure.

In this paper we will narrow our focus and only consider $p$-valuable groups.
Instead of $D(G,K)$ we will introduce and study a distribution algebra
$D^{\dagger}(G,K)$ which is dual to the space of overconvergent functions on
$G$. One of our goals (Theorem 1.2 below) is to show $D^{\dagger}(G,K)$ is
Fréchet-Stein when $G$ is uniform.

To state our results precisely, let $(G,\omega)$ be a $p$-valued group. Thus
$G$ is a pro-$p$-group endowed with a function
$\omega:G\backslash\\{1\\}\longrightarrow(0,\infty)$ satisfying the axioms
(a)–(d) in [Sch11, p. 169]. Moreover, the topology of $G$ is defined by
$\omega$. Note that $G$ is necessarily torsion-free since
$\omega(g^{p})=\omega(g)+1$ for all $g\in G$.

Without loss of generality we may and will assume $\omega$ is
$\mathbb{Q}$-valued, see [Sch11, Cor. 33.3]. Furthermore, we will exclusively
consider saturated groups $(G,\omega)$. This means
$\omega(g)>\frac{p}{p-1}\Rightarrow g\in G^{p}$. Also, $(G,\omega)$ is assumed
to be of finite rank $d$, and we may therefore choose an ordered basis
$(g_{1},\ldots,g_{d})$ by [Sch11, Prop. 26.6]. This puts a global chart on $G$
via the homeomorphism

$\displaystyle\psi:\mathbb{Z}_{p}^{d}$ $\displaystyle\longrightarrow G$
$\displaystyle(x_{1},\ldots,x_{d})$ $\displaystyle\longmapsto
g_{1}^{x_{1}}\cdots g_{d}^{x_{d}}.$

Recall that $\omega(g_{1}^{x_{1}}\cdots
g_{d}^{x_{d}})=\min_{i=1,\ldots,d}\\{\omega(g_{i})+v(x_{i})\\}$ by virtue of
$(g_{1},\ldots,g_{d})$ being an ordered basis. (Here $v$ is the standard
valuation on $\mathbb{Z}_{p}$ with $v(p)=1$.) Via $\psi$ we view functions on
$G$ as functions on $\mathbb{Z}_{p}^{d}$, and vice versa.

The construction in [LS23] associates a rigid analytic group
$\mathbb{G}^{\text{rig}}$ over $\mathbb{Q}_{p}$ with any saturated group
$(G,\omega)$ as above. Its underlying space is the polydisc
$\text{Sp}(\mathbb{Q}_{p}\langle Z_{1},\ldots,Z_{d}\rangle)$, and there is a
natural isomorphism of abstract groups
$\mathbb{G}^{\text{rig}}(\mathbb{Q}_{p})\overset{\sim}{\longrightarrow}G$. In
this paper we first refine this construction and promote
$\mathbb{G}^{\text{rig}}$ to a dagger group $\mathbb{G}$ (by which we mean a
group object in the category of dagger spaces, cf. [GK00]).

In the process we introduce the space $\mathcal{C}^{\dagger}(G,K)$ of
overconvergent functions $f:G\longrightarrow K$ taking values in a finite
extension $K/\mathbb{Q}_{p}$. Via $\psi$ such $f$ correspond to functions on
$\mathbb{Z}_{p}^{d}$ which extend to rigid functions on slightly larger
polydiscs. See Definition 5.4 below for a more precise formulation. The space
$\mathcal{C}^{\dagger}(G,K)$ is endowed with a natural locally convex topology
known as the fringe (or direct limit) topology.

The central object in this article is the continuous dual $D^{\dagger}(G,K)$
equipped with the strong topology. This becomes an algebra under convolution,
essentially due to the existence of the dagger group $\mathbb{G}$. In this
paper we establish some foundational structural properties of these
$\dagger$-distribution algebras.

Our first general result is:

######  Theorem 1.1.

For any saturated $p$-valued group $(G,\omega)$ as above, $D^{\dagger}(G,K)$
is a nuclear Fréchet algebra. (That is, it can be expressed as an inverse
limit of Banach $K$-algebras with compact transition maps.)

The Banach algebras in this Theorem are rigid distribution algebras
$D^{\text{rig}}(\mathbb{G}_{N},K)$ (see [Em17, Cor. 5..1.8] for their general
definition) of a sequence of affinoid rigid analytic groups
$(\mathbb{G}_{N})_{N\geq 1}$ which we call the strict neighborhood groups of
the dagger group $\mathbb{G}$. The underlying space of $\mathbb{G}_{N}$ is the
polydisc $\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$ where
$\tau_{N,i}=\frac{1}{N+1}(\omega(g_{i})-\frac{1}{p-1})$. One of the key
technical aspects of our work is to endow these polydiscs with a group
structure compatible with the one on $\mathbb{G}$.

Unfortunately, rigid distribution algebras such as
$D^{\text{rig}}(\mathbb{G}_{N},K)$ are not Noetherian in general. This was
observed by Berthelot, and the argument is exposed in [Clo18, App. A.2] for
example. As a result, it is not at all clear whether $D^{\dagger}(G,K)$ is
Fréchet-Stein – in the sense of [ST03, Sect. 3]. Although we believe this to
be true in general, we can currently only prove it for certain equi-$p$-valued
groups (the main example being uniform groups). This is our second main
result:

######  Theorem 1.2.

Suppose $(G,\omega)$ is equi-$p$-valued, with
$\omega(g_{1})=\cdots=\omega(g_{d})=\omega_{0}$. Assume
$2\omega_{0}>\frac{p}{p-1}$. Then there is an isomorphism of topological
$K$-algebras,

(1.1)
$D^{\dagger}(G,K)\overset{\sim}{\longrightarrow}{\varprojlim}_{s<\theta^{1/\omega_{0}}}D_{s}(G,K),$

where $\theta=p^{-1/(p-1)}$ denotes the radius of convergence of the $p$-adic
exponential function, and $D_{s}(G,K)$ are the algebras from [ST03, p. 162].
In particular $D^{\dagger}(G,K)$ is a Fréchet-Stein algebra.

The inequality $2\omega_{0}>\frac{p}{p-1}$ is exactly the condition (HYP) in
[ST03, p. 163]. As we explain in Example 8.7 this is satisfied for a uniform
pro-$p$-group $G$ endowed with the canonical $p$-valuation
$\omega^{\text{can}}$ which gives the lower $p$-series. In this example
$\omega_{0}=\begin{cases}1&\text{if $p>2$}\\\ 2&\text{if $p=2$}\end{cases}$.

We emphasize that, in the situation of Theorem 1.2, the Fréchet-Stein
structure on $D^{\dagger}(G,K)$ follows immediately by appealing to [ST03,
Thm. 4.5, Thm. 4.9]. Our contribution is the isomorphism (1.1).

In Section 9 we introduce the notion of an overconvergent $G$-representation
(of compact type) and establish a duality à la [ST02] with continuous
$D^{\dagger}(G,K)$-modules on nuclear Fréchet spaces.

As a future project we intend to view the base change map in [Clo17, Thm. 4.1]
through the lens of $\dagger$-distribution algebras. Classically, unramified
base change is given by a morphism
$b:\mathcal{H}_{\mathbb{Q}_{p^{r}}}\longrightarrow\mathcal{H}_{\mathbb{Q}_{p}}$
between spherical Hecke algebras (for $\text{GL}_{n}$ say, to fix ideas).
Clozel has defined an analogous base change map
$b^{\text{rig}}:D^{\text{rig}}(I_{r},K)\longrightarrow D^{\text{rig}}(I,K)$
between the rigid distribution algebras for the pro-$p$ Iwahori subgroups $I$
and $I_{r}$ in $\text{GL}_{2}(\mathbb{Q}_{p})$ and
$\text{GL}_{2}(\mathbb{Q}_{p^{r}})$ respectively – assuming $p$ is large
enough. (Our notation differs from Clozel’s – he denotes our $b^{\text{rig}}$
by $b:\mathcal{D}_{r}\rightarrow\mathcal{D}$.) A natural question is whether
there is a $\dagger$-version $b^{\dagger}:D^{\dagger}(I_{r},K)\longrightarrow
D^{\dagger}(I,K)$. As noted above, we expect the algebra $D^{\dagger}(I,K)$ to
be Fréchet-Stein (and similarly for $I_{r}$) but we do not currently have a
proof of this.

##  2\. Notation

We let $\mathbb{N}_{0}=\\{0,1,2,\ldots\\}$. If $\mathcal{K}$ is a field, a
power series in $\mathcal{K}[\\![X_{1},\ldots,X_{n}]\\!]$ will be written as
$\sum_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}X^{\alpha}$. Here
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ is a tuple of non-negative integers
and $X^{\alpha}=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$. We adopt the
notation $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$.

When $\mathcal{K}$ is a non-archimedean local field, $T_{n}$ denotes the Tate
algebra of all power series
$\sum_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}X^{\alpha}$ satisfying the
condition $c_{\alpha}\longrightarrow 0$ as $|\alpha|\longrightarrow\infty$.
The Gauss norm of such a power series is

$\|{\sum}_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}X^{\alpha}\|={\sup}_{\alpha\in\mathbb{N}_{0}^{n}}|c_{\alpha}|,$

and $T_{n}$ thus becomes a Banach algebra over $\mathcal{K}$.

##  3\. The category of dagger spaces

Let $\mathcal{K}$ be a non-archimedean local field with absolute value
$|\cdot|$. It extends uniquely to an absolute value on the algebraic closure
$\overline{\mathcal{K}}$, and we let
$\Gamma=|\overline{\mathcal{K}}^{\times}|$ denote the valuation subgroup of
$\mathbb{R}_{>0}$. (Eventually we will take $\mathcal{K}=\mathbb{Q}_{p}$, in
which case $\Gamma=p^{\mathbb{Q}}$.) In this section we review the category of
dagger spaces over $\mathcal{K}$ introduced in [GK00, Df. 2.12].

###  3.1. Dagger algebras

We start with the Washnitzer algebra $W_{n}=\mathcal{K}\langle
X_{1},\ldots,X_{n}\rangle^{\dagger}$ in the variables $X_{1},\ldots,X_{n}$.
This is the subalgebra of the Tate algebra $T_{n}$ consisting of all power
series
$\sum_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}X^{\alpha}\in\mathcal{K}[\\![X_{1},\ldots,X_{n}]\\!]$
such that $|c_{\alpha}|r^{|\alpha|}\longrightarrow 0$ as
$|\alpha|\longrightarrow\infty$, for some $r>1$. Since $\Gamma$ is dense in
$\mathbb{R}_{>0}$ we may and will assume $r\in\Gamma$.

######  Definition 3.1.

For $r\in\Gamma_{>1}$ we let

$T_{n,r}=\bigg{\\{}{\sum}_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}X^{\alpha}\in\mathcal{K}[\\![X_{1},\ldots,X_{n}]\\!]:{\lim}_{|\alpha|\longrightarrow\infty}|c_{\alpha}|r^{|\alpha|}=0\bigg{\\}}.$

By [BGR84, Sect. 6.1.5, Thm. 4] this is a $\mathcal{K}$-affinoid algebra with
norm

$\|{\sum}_{\alpha\in\mathbb{N}_{0}^{n}}c_{\alpha}X^{\alpha}\|_{r}={\sup}_{\alpha\in\mathbb{N}_{0}^{n}}|c_{\alpha}|r^{|\alpha|}.$

There are two natural ways to put a topology on $W_{n}$. There is the topology
induced by the Gauss norm on the ambient $T_{n}$, but to us the following
finer topology will be more important in the sequel. Obviously
$W_{n}=\bigcup_{r\in\Gamma_{>1}}T_{n,r}$, and we endow $W_{n}$ with the finest
locally convex topology for which all the inclusions $T_{n,r}\hookrightarrow
W_{n}$ are continuous. See [Sch02, p. 22] for more details on locally convex
final topologies. In [GK00, Sect. 4.2] the latter is referred to as the direct
limit topology on $W_{n}$. In [Ked06, Df. 2.3.7] it is called the fringe
topology.

A dagger algebra is a $\mathcal{K}$-algebra $A$ which is isomorphic to
$W_{n}/I$ for some ideal $I\subset W_{n}$, for some $n$. The quotient semi-
norm on $W_{n}/I$ turns out to be a norm [GK00, Prop. 1.6]. However, we prefer
to endow $A$ with the direct limit topology as in [GK00, Sect. 4.2]:
$A\simeq\varinjlim_{r\rightarrow 1}T_{n,r}/IT_{n,r}$, where we take the limit
over sufficiently small $r\in\Gamma_{>1}$ (smaller than an $r_{0}>1$ for which
$T_{n,r_{0}}\supset I$).

######  Definition 3.2.

For $A$ as above, we let $\hat{A}$ denote its completion with respect to the
natural equivalence class of norms on $A\simeq W_{n}/I$. (In [GK00] the
completion is denoted by $A^{\prime}$. To avoid confusion with duals later on,
we adopt the notation $\hat{A}$.)

The completion $\hat{A}$ is an affinoid $\mathcal{K}$-algebra, with an
associated rigid space $\text{Sp}(\hat{A})$.

###  3.2. Dagger spaces

$\text{Sp}(A)$ is the set of maximal ideals of a dagger algebra $A$. By [GK00,
Thm. 1.7] there is a bijection
$\text{Sp}(\hat{A})\overset{\sim}{\longrightarrow}\text{Sp}(A)$ given by
pulling back along the completion $A\rightarrow\hat{A}$. Via this bijection we
transfer the usual $G$-topology on $\text{Sp}(\hat{A})$ to a $G$-topology on
$\text{Sp}(A)$. As in rigid geometry, the structure sheaf
$\mathcal{O}_{\text{Sp}(A)}$ is determined by its values on affinoid
subdomains $U\subset\text{Sp}(A)$. We will not need the exact definitions in
our paper, and we refer the reader to [GK00, Sect. 2] for all details. We
emphasize that $\mathcal{O}_{\text{Sp}(A)}(U)$ is again a dagger algebra for
such $U$.

A dagger space over $\mathcal{K}$ is then a locally $G$-ringed space
$(X,\mathcal{O}_{X})$ which admits an admissible covering $X=\bigcup_{i}U_{i}$
such that all $(U_{i},\mathcal{O}_{X}|_{U_{i}})$ are affinoid
$\mathcal{K}$-dagger spaces (that is, isomorphic to $\text{Sp}(A_{i})$ for
some dagger algebras $A_{i}$). Morphisms between dagger spaces are morphisms
of locally $G$-ringed spaces. This gives a category $\text{Dag}_{\mathcal{K}}$
which admits fiber products by [GK00, 1.16, 2.13]. Moreover, by [GK00, Thm.
2.19] there is a faithful functor
$\widehat{(\cdot)}:\text{Dag}_{\mathcal{K}}\longrightarrow\text{Rig}_{\mathcal{K}}$
to the category of rigid spaces over $\mathcal{K}$ which takes an affinoid
dagger space $\text{Sp}(A)$ to the rigid space $\text{Sp}(\hat{A})$.

###  3.3. Distributions

Let $A$ be a dagger algebra over $\mathcal{K}$, expressed as a locally convex
inductive limit of $\mathcal{K}$-affinoid algebras $A_{r}=T_{n,r}/IT_{n,r}$,
indexed by small enough $r\in\Gamma_{>1}$ as above. If $r_{1}\leq r_{2}$, the
transition map $A_{r_{2}}\rightarrow A_{r_{1}}$ is compact by [Em17, Prop.
2.1.16] for example. (Note that $\text{Sp}(A_{r})$ is a closed subvariety of a
rigid polydisc.) See also [Sch02, p. 98]. In other words, $A$ is a locally
convex space of compact type, cf. [Em17, Df. 1.1.16, part (ii)]. By passing to
quotients of the $A_{r}$, one can write $A$ as an inductive limit of affinoid
algebras with compact and injective transition maps, as explained in [Em17, p.
15] for instance.

######  Definition 3.3.

The space of distributions on $\text{Sp}(A)$ is the strong dual
$A_{b}^{\prime}=\mathcal{L}(A,\mathcal{K})_{b}$. (We stress that $A$ carries
the direct limit topology here.)

Thus $A_{b}^{\prime}$ is a nuclear Fréchet space. If the transition maps
$A_{r_{2}}\rightarrow A_{r_{1}}$ are injective (this happens when $A=W_{n}$
for example) [Sch02, Prop. 16.10] gives a topological isomorphism

$A_{b}^{\prime}\overset{\sim}{\longrightarrow}\varprojlim(A_{r})_{b}^{\prime}.$

Moreover, each $(A_{r})_{b}^{\prime}$ is just the dual
$\mathcal{L}(A_{r},\mathcal{K})$ with the operator norm topology, cf. [Sch02,
Rem. 6.7].

##  4\. Dagger groups

###  4.1. Group objects

$\text{Dag}_{\mathcal{K}}$ has a terminal object $\text{Sp}(\mathcal{K})$, and
finite products such as $X\times_{\text{Sp}(\mathcal{K})}Y$ exist. It
therefore makes sense to talk about group objects in
$\text{Dag}_{\mathcal{K}}$.

######  Definition 4.1.

A dagger group over $\mathcal{K}$ is a group object in the category
$\text{Dag}_{\mathcal{K}}$.

Thus a dagger group $G$ comes with three morphisms in
$\text{Dag}_{\mathcal{K}}$,

$G\times_{\text{Sp}(\mathcal{K})}G\overset{m}{\longrightarrow}G,\hskip
6.0pt\hskip 6.0pt\hskip
6.0pt\text{Sp}(\mathcal{K})\overset{e}{\longrightarrow}G,\hskip 6.0pt\hskip
6.0pt\hskip 6.0ptG\overset{i}{\longrightarrow}G,$

satisfying the usual conditions ($m$ is associative, $e$ is a two-sided unit,
and $i$ is a two-sided inverse). Equivalently
$X\mapsto\text{Hom}_{\text{Dag}_{\mathcal{K}}}(X,G)$ gives a contravariant
functor from $\text{Dag}_{\mathcal{K}}$ to the category of groups.

All dagger groups considered later in this paper will be affinoid. Thus
$G=\text{Sp}(A)$ for a dagger algebra $A$ equipped with morphisms of
$\mathcal{K}$-algebras

(4.1)
$A\overset{m^{*}}{\longrightarrow}A\otimes_{\mathcal{K}}^{\dagger}A,\hskip
6.0pt\hskip 6.0pt\hskip
6.0ptA\overset{e^{*}}{\longrightarrow}\mathcal{K},\hskip 6.0pt\hskip
6.0pt\hskip 6.0ptA\overset{i^{*}}{\longrightarrow}A,$

such that the standard diagrams commute. See [Spr09, p. 22] for instance. We
recall the definition of the tensor product $\otimes_{\mathcal{K}}^{\dagger}$
from [GK00, 1.16]: If $A_{1}$ and $A_{2}$ are dagger algebras, first write
them as quotients of Washnitzer algebras $W_{n_{i}}\twoheadrightarrow A_{i}$,
for $i=1,2$. Then $A_{1}\otimes_{\mathcal{K}}^{\dagger}A_{2}$ is defined to be
the image of $W_{n_{1}+n_{2}}$ under the natural map
$T_{n_{1}+n_{2}}\rightarrow\hat{A}_{1}\hat{\otimes}_{\mathcal{K}}\hat{A}_{2}$.
For example,
$W_{n_{1}}\otimes_{\mathcal{K}}^{\dagger}W_{n_{2}}=W_{n_{1}+n_{2}}$.

######  Remark 4.2.

As a locally convex space, $A_{1}\otimes_{\mathcal{K}}^{\dagger}A_{2}$ is the
same as the tensor product $A_{1}\widehat{\otimes}_{\mathcal{K},\pi}A_{2}$
considered in [Sch02, p. 107] and [Em17, Prop. 1.1.32] for example. (The
latter is the Hausdorff completion of $A_{1}\otimes_{\mathcal{K}}A_{2}$ with
the projective tensor product topology, cf. [Sch02, Ch. 17, part B]. Under our
assumptions $A_{1}\otimes_{\mathcal{K},\pi}A_{2}$ coincides with the inductive
tensor product $A_{1}\otimes_{\mathcal{K},\iota}A_{2}$, cf. [Sch02, Ch. 17,
part A], and we may drop the subscripts $\pi$ and $\iota$.) To see this, let
$A_{i}=W_{n_{i}}$ for simplicity. Then, by [Em17, Prop. 1.1.32],

$W_{n_{1}}\widehat{\otimes}_{\mathcal{K},\pi}W_{n_{2}}=\varinjlim_{r_{1},r_{2}}T_{n_{1},r_{1}}\widehat{\otimes}_{\mathcal{K}}T_{n_{2},r_{2}}=\varinjlim_{r}T_{n_{1},r}\widehat{\otimes}_{\mathcal{K}}T_{n_{2},r}=\varinjlim_{r}T_{n_{1}+n_{2},r}=W_{n_{1}+n_{2}}.$

######  Remark 4.3.

We note that the morphisms in (4.1) are automatically continuous for the norm
topology by [GK00, Prop. 1.6]. See also [Ked06, Df. 2.2.2]. In fact, any
$\mathcal{K}$-algebra morphism between dagger algebras is automatically
continuous – not just for the norm topology, but also with respect to the
direct limit topology by [GK00, Lem. 1.8] and [Ked06, Cor. 2.3.3, Df. 2.3.7].

###  4.2. Distribution algebras

We consider the duals of the maps in (4.1). First we observe that by Remark
4.2, combined with [Sch02, Prop. 20.13] and [Em17, Prop. 1.1.32], we have

$(A\otimes_{\mathcal{K}}^{\dagger}A)_{b}^{\prime}=(A\widehat{\otimes}_{\mathcal{K},\pi}A)_{b}^{\prime}=A_{b}^{\prime}\widehat{\otimes}_{\mathcal{K},\pi}A_{b}^{\prime}.$

Thus the dual of $m^{*}$ gives a jointly continuous and associative
$\mathcal{K}$-bilinear multiplication map on the space of distributions

$\ast:A_{b}^{\prime}\times A_{b}^{\prime}\longrightarrow A_{b}^{\prime}.$

(Associativity follows from dualizing the first diagram in [Spr09, p. 22].)
Furthermore, the dual of $e^{*}$ gives a two-sided unit
$\mathcal{K}\longrightarrow A_{b}^{\prime}$ for $\ast$ (by dualizing the third
diagram in [Spr09, p. 22]).

We conclude that any dagger group of the form $G=\text{Sp}(A)$ gives rise to a
distribution algebra, whose underlying space $A_{b}^{\prime}$ is nuclear
Fréchet. One of the themes of this article is to show $A_{b}^{\prime}$ is a
nuclear Fréchet algebra, see [Em17, Df. 1.2.12], for $G$ arising naturally
from certain $p$-valued groups. Essentially what this means is we want to
promote $\text{Sp}(A_{r})$ to a rigid analytic group for a cofinal set of
$r\in\Gamma_{>1}$. In general, for a rank $d$ saturated group, we are forced
to replace $r$ by a sequence of polyradii $(r_{1},\ldots,r_{d})$ for this to
work. See 6.2 below.

##  5\. Examples of dagger groups

###  5.1. A refinement of [Sch11, Prop. 29.2]

Let $(G,\omega)$ be a $p$-valued group with ordered basis
$(g_{1},\ldots,g_{d})$. For any (finite) sequence of elements
$(h_{1},\ldots,h_{s})$ from $G$, and any tuple
$(y_{1},\ldots,y_{s})\in\mathbb{Z}_{p}^{s}$, we form the product
$h_{1}^{y_{1}}\cdots h_{s}^{y_{s}}$ and express it as $g_{1}^{x_{1}}\cdots
g_{d}^{x_{d}}$ for a uniquely determined tuple
$(x_{1},\ldots,x_{d})\in\mathbb{Z}_{p}^{d}$. [Sch11, Prop. 29.2] tells us that
$x_{i}=F_{i}(y_{1},\ldots,y_{s})$ for certain power series
$F_{i}\in\mathbb{Q}_{p}\langle Y_{1},\ldots,Y_{s}\rangle$.

A close inspection of the proof immediately gives the following strengthening:

######  Proposition 5.1.

With notation as above, $F_{i}\in\mathbb{Q}_{p}\langle
Y_{1},\ldots,Y_{s}\rangle^{\dagger}$ for all $i=1,\ldots,d$.

###### Proof.

For ease of comparison we will adopt the notation from [Sch11, Prop. 29.2] and
its proof, except that the rank of $(G,\omega)$ is $d$ and not $r$.

The power series $F_{i}\in\mathbb{Q}_{p}[\\![Y_{1},\ldots,Y_{s}]\\!]$ are
given explicitly in [Sch11, p. 204]. They have the form

(5.1)
$F_{i}=\sum_{\beta\in\mathbb{N}_{0}^{s}}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}F_{\beta},$

where $F_{\beta}=\prod_{j=1}^{s}Y_{j}(Y_{j}-1)\cdots(Y_{j}-\beta_{j}+1)$ is a
degree $|\beta|$ polynomial in $\mathbb{Z}[Y_{1},\ldots,Y_{s}]$, and the
coefficients $c_{\alpha,\beta}\in\mathbb{Z}_{p}$ are defined by (43) in
[Sch11, p. 203]. For $\alpha$ we are taking the tuple
$\underline{i}=(\ldots,0,1,0,\ldots)$ with a $1$ in the $i^{\text{th}}$ slot,
and zeroes elsewhere. The key estimate in the proof of [Sch11, Prop. 29.2] is
the following inequality from [Sch11, p. 204]:

$v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}\geq-\omega(g_{i})+\sum_{j=1}^{s}\beta_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)}.$

Pick any $\rho\in\mathbb{Q}$ in the interval $0<\rho<M-\frac{1}{p-1}$, where
$M=\min_{j}\omega(h_{j})$. Then the inequalities

$v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}-\rho|\beta|\geq-\omega(g_{i})+\sum_{j=1}^{s}\beta_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}-\rho\bigg{)}\geq-\omega(g_{i})+\big{(}M-\frac{1}{p-1}-\rho\big{)}|\beta|$

show that
$v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}-\rho|\beta|\longrightarrow\infty$
as $|\beta|\longrightarrow\infty$. This shows that the terms of (5.1) converge
to zero relative to the norm $\|\cdot\|_{p^{\rho}}$, and therefore all the
$F_{i}$ belong to the subalgebra $T_{s,p^{\rho}}\subset\mathbb{Q}_{p}\langle
Y_{1},\ldots,Y_{s}\rangle^{\dagger}$. ∎

######  Remark 5.2.

If $(G,\omega)$ is saturated (see [Sch11, p. 187] for the definition) the
power series $F_{i}$ have coefficients in $\mathbb{Z}_{p}$. This is [Sch11,
Rem. 29.3].

###  5.2. A refinement of [LS23, Prop. 4.1]

Start with a saturated $p$-valued group $(G,\omega)$ and choose an ordered
basis $(g_{1},\ldots,g_{d})$. In [LS23] two of us (A. L. and C. S.) defined an
affinoid rigid analytic group $\mathbb{G}^{\text{rig}}$ over $\mathbb{Q}_{p}$
associated with this data. The defining properties are:

  * (a)

$\mathbb{G}^{\text{rig}}=\text{Sp}(\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle)$ as a rigid analytic space over $\mathbb{Q}_{p}$;

  * (b)

The map $\varphi\mapsto g_{1}^{\varphi(Z_{1})}\cdots g_{d}^{\varphi(Z_{d})}$
gives an isomorphism of abstract groups

$\mathbb{G}^{\text{rig}}(\mathbb{Q}_{p})=\text{Hom}_{\mathbb{Q}_{p}-\text{alg}}(\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle,\mathbb{Q}_{p})\overset{\sim}{\longrightarrow}G.$

Here we will promote $\mathbb{G}^{\text{rig}}$ to a dagger group $\mathbb{G}$.
Thus $\widehat{\mathbb{G}}=\mathbb{G}^{\text{rig}}$ where $\widehat{(\cdot)}$
is the completion functor from Section 3.2.

######  Proposition 5.3.

Let $(G,\omega)$ be a saturated group with ordered basis
$(g_{1},\ldots,g_{d})$. Then there is a unique dagger group $\mathbb{G}$ over
$\mathbb{Q}_{p}$ such that

  * (a)

$\mathbb{G}=\text{Sp}(\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger})$ as a dagger space over $\mathbb{Q}_{p}$;

  * (b)

The map $\varphi\mapsto g_{1}^{\varphi(Z_{1})}\cdots g_{d}^{\varphi(Z_{d})}$
gives an isomorphism of abstract groups

$\mathbb{G}(\mathbb{Q}_{p})=\text{Hom}_{\mathbb{Q}_{p}-\text{alg}}(\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger},\mathbb{Q}_{p})\overset{\sim}{\longrightarrow}G.$

Moreover, if $(g_{1}^{\prime},\ldots,g_{d}^{\prime})$ is another ordered basis
for $(G,\omega)$ with associated dagger group $\mathbb{G}^{\prime}$, then
there is a unique isomorphism of dagger groups
$\mathbb{G}\overset{\sim}{\longrightarrow}\mathbb{G}^{\prime}$ such that the
diagram below commutes:

${\mathbb{G}(\mathbb{Q}_{p})}$${\mathbb{G}^{\prime}(\mathbb{Q}_{p})}$${G}$$\scriptstyle{\sim}$

(The horizontal arrow is the isomorphism induced on $\mathbb{Q}_{p}$-points.
The slanted arrows are the ones from property (b).)

###### Proof.

In light of Proposition 5.1, the proofs of [LS23, Prop. 4.1, Prop. 4.3] apply
almost verbatim in this context. For example, 5.1 yields $d$ power series
$F_{1},\ldots,F_{d}\in\mathbb{Q}_{p}\langle
X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}\rangle^{\dagger}$ such that

$(g_{1}^{x_{1}}\cdots g_{d}^{x_{d}})(g_{1}^{y_{1}}\cdots
g_{d}^{y_{d}})=g_{1}^{F_{1}(x,y)}\cdots g_{d}^{F_{d}(x,y)}$

for all tuples $x=(x_{1},\ldots,x_{d})$ and $y=(y_{1},\ldots,y_{d})$ in
$\mathbb{Z}_{p}^{d}$. Furthermore, the $F_{i}$ have coefficients in
$\mathbb{Z}_{p}$ by Remark 5.2. This enables us to define a
$\mathbb{Q}_{p}$-algebra homomorphism between Washnitzer algebras,

$\displaystyle m^{*}:\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}$
$\displaystyle\longrightarrow\mathbb{Q}_{p}\langle
X_{1},\ldots,X_{d}\rangle^{\dagger}\otimes_{\mathbb{Q}_{p}}^{\dagger}\mathbb{Q}_{p}\langle
Y_{1},\ldots,Y_{d}\rangle^{\dagger}=\mathbb{Q}_{p}\langle
X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}\rangle^{\dagger}$ $\displaystyle Z_{i}$
$\displaystyle\longmapsto F_{i}.$

(Note that dagger algebras are weakly complete, cf. [GK00, Sect. 1.9]. Since
the $F_{i}$ are power-bounded relative to the Gauss norm the homomorphism
$Z_{i}\mapsto F_{i}$ extends from $\mathbb{Q}_{p}[Z_{1},\ldots,Z_{d}]$ to
$\mathbb{Q}_{p}\langle Z_{1},\ldots,Z_{d}\rangle^{\dagger}$.) Similar
arguments give a map $i^{*}:\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}\longrightarrow\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}$ reflecting inversion in $G$. Finally, let
$e^{*}:\mathbb{Q}_{p}\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}\longrightarrow\mathbb{Q}_{p}$ be the
obvious augmentation map (evaluation at the origin). Since
$(F_{1},\ldots,F_{d})$ is a formal group law, altogether this gives
$\text{Sp}(\mathbb{Q}_{p}\langle Z_{1},\ldots,Z_{d}\rangle^{\dagger})$ the
structure of a dagger group; see (4.1) above.

The statements about uniqueness are proved word-for-word as in [LS23] by
appealing to Proposition 5.1 rather than [Sch11, Prop. 29.2]. ∎

###  5.3. Overconvergent functions on a saturated group

We keep our saturated $p$-valued group $(G,\omega)$. For the most part we will
suppress the $p$-valuation $\omega$ from the notation. Fix a finite field
extension $K/\mathbb{Q}_{p}$ and consider the space $\mathcal{C}(G,K)$ of all
continuous functions $f:G\rightarrow K$. In this section we will define the
subspace $\mathcal{C}^{\dagger}(G,K)$ of overconvergent functions. This is
parallel to the definition of the space of rigid functions
$\mathcal{C}^{\text{rig}}(G,K)$ in [LS23, Df. 4.5].

Pick an ordered basis $(g_{1},\ldots,g_{d})$ for $(G,\omega)$. Then each power
series

$\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}Z^{\alpha}\in K\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}$

gives a function
$f(g)=\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}x_{1}^{\alpha_{1}}\cdots
x_{d}^{\alpha_{d}}$ where $g=g_{1}^{x_{1}}\cdots g_{d}^{x_{d}}$. The resulting
map $\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}Z^{\alpha}\mapsto f$ is
injective by [Sch11, Cor. 5.8], and we let $\mathcal{C}^{\dagger}(G,K)$ be the
image. Mimicking the discussion leading up to [LS23, Df. 4.5] one easily
checks that $\mathcal{C}^{\dagger}(G,K)$ is independent of the choice of
basis.

This justifies the following definition.

######  Definition 5.4.

A function $f:G\rightarrow K$ is overconvergent if
$f(g)=\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}x_{1}^{\alpha_{1}}\cdots
x_{d}^{\alpha_{d}}$ for all $g=g_{1}^{x_{1}}\cdots g_{d}^{x_{d}}$, where the
coefficients $c_{\alpha}\in K$ satisfy
$|c_{\alpha}|r^{|\alpha|}\longrightarrow 0$ as $|\alpha|\longrightarrow\infty$
for some $r>1$. (This notion is independent of the choice of basis
$(g_{1},\ldots,g_{d})$.)

The space $\mathcal{C}^{\dagger}(G,K)$ of overconvergent functions carries a
topology by identifying it with $K\langle Z_{1},\ldots,Z_{d}\rangle^{\dagger}$
(with the direct limit topology). Again, this topology is independent of the
choice of basis since any algebra morphism between dagger algebras is
automatically continuous ([GK00, Lem. 1.8]).

######  Remark 5.5.

For a fixed $r\in\Gamma_{>1}$ the subalgebra $T_{d,r}\subset K\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}$ corresponds to a subalgebra of
$\mathcal{C}^{\dagger}(G,K)$ which does depend on the choice of basis in
general. The same with the norm $\|\cdot\|_{r}$.

###  5.4. The $\dagger$-distribution algebra of a saturated group

Observe that $G\times G$ has a natural $p$-valuation given by
$\omega_{2}(g,h)=\min\\{\omega(g),\omega(h)\\}$. An ordered basis for
$(G\times G,\omega_{2})$ is given by

$((g_{1},e),\ldots,(g_{d},e),(e,g_{1}),\ldots,(e,g_{d}))$

where $e\in G$ is the identity. (See [Sch11, p. 210].) In particular $(G\times
G,\omega_{2})$ is saturated since $(G,\omega)$ is saturated. To see this use
[Sch11, Prop. 26.11]. Therefore the space $\mathcal{C}^{\dagger}(G\times G,K)$
is defined, and there is a natural isomorphism of dagger algebras

$\displaystyle\Psi:\mathcal{C}^{\dagger}(G,K)\otimes_{K}^{\dagger}\mathcal{C}^{\dagger}(G,K)$
$\displaystyle\overset{\sim}{\longrightarrow}\mathcal{C}^{\dagger}(G\times
G,K)$ $\displaystyle f_{1}\otimes f_{2}$
$\displaystyle\longmapsto\big{[}(g,h)\mapsto f_{1}(g)f_{2}(h)\big{]}.$

This is immediate after choosing bases as above.

The dagger group structure on $\text{Sp}(\mathcal{C}^{\dagger}(G,K))$ can now
be described, without resorting to bases, by the three morphisms

  * •

$m^{*}:\mathcal{C}^{\dagger}(G,K)\longrightarrow\mathcal{C}^{\dagger}(G,K)\otimes_{K}^{\dagger}\mathcal{C}^{\dagger}(G,K)$
such that $(\Psi\circ m^{*})(f)(g,h)=f(gh)$;

  * •

$e^{*}:\mathcal{C}^{\dagger}(G,K)\longrightarrow K$ such that $e^{*}(f)=f(e)$;

  * •

$i^{*}:\mathcal{C}^{\dagger}(G,K)\longrightarrow\mathcal{C}^{\dagger}(G,K)$
such that $i^{*}(f)(g)=f(g^{-1})$.

As in Section 4.2 this gives an algebra structure on the strong dual.

######  Definition 5.6.

The $\dagger$-distribution $K$-algebra of a saturated group $(G,\omega)$ is
the strong dual

$D^{\dagger}(G,K)=\mathcal{C}^{\dagger}(G,K)_{b}^{\prime}=\mathcal{L}(\mathcal{C}^{\dagger}(G,K),K)_{b}.$

Multiplication in $D^{\dagger}(G,K)$ is the convolution product

$\displaystyle\ast:D^{\dagger}(G,K)\times D^{\dagger}(G,K)$
$\displaystyle\longrightarrow D^{\dagger}(G,K)$
$\displaystyle(\delta_{1},\delta_{2})$
$\displaystyle\longmapsto\big{[}f\mapsto(\delta_{1}\times\delta_{2})(m^{*}(f))\big{]},$

where $(\delta_{1}\times\delta_{2})(f_{1}\otimes
f_{2})=\delta_{1}(f_{1})\delta_{2}(f_{2})$. As noted in 4.2, $\ast$ is jointly
continuous. The two-sided unit element for $\ast$ is the Dirac delta
distribution $e^{*}=\delta_{e}\in D^{\dagger}(G,K)$.

One of the goals of this article is to establish finer structural properties
of these algebras.

##  6\. Strict neighborhood groups

###  6.1. Proposition 5.1 revisited

In this section we improve Proposition 5.1 as follows.

######  Proposition 6.1.

Keep the notation from Proposition 5.1, and assume $(G,\omega)$ is saturated.
For every $j=1,\ldots,s$, let $\rho_{j}$ be a rational number in the range
$0<\rho_{j}<\omega(h_{j})-\frac{1}{p-1}$. Then $F_{i}$ converges on

$\mathbb{B}(p^{\rho_{1}},\ldots,p^{\rho_{s}})=\\{(y_{1},\ldots,y_{s})\in\mathbb{A}_{\text{rig}}^{s},\forall
j:|y_{j}|\leq p^{\rho_{j}}\\}.$

Moreover, on this polydisc $F_{i}$ has supremum at most
$p^{\omega(g_{i})-\frac{1}{p-1}}$.

###### Proof.

In the proof of 5.1 we introduced the polynomials
$F_{\beta}=\prod_{j=1}^{s}Y_{j}(Y_{j}-1)\cdots(Y_{j}-\beta_{j}+1)$, which we
here expand as $F_{\beta}=\sum_{\alpha\in\mathbb{N}_{0}^{s},\forall
j:\alpha_{j}\leq\beta_{j}}a_{\beta,\alpha}Y^{\alpha}$. Clearly
$a_{\beta,\alpha}\in\mathbb{Z}$, and the leading coefficient is
$a_{\beta,\beta}=1$. Correspondingly, the power series $F_{i}$ introduced in
(5.1) has the expansion
$F_{i}=\sum_{\alpha\in\mathbb{N}_{0}^{s}}d_{i,\alpha}Y^{\alpha}$, where

$d_{i,\alpha}={\sum}_{\beta\in\mathbb{N}_{0}^{s},\forall
j:\alpha_{j}\leq\beta_{j}}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}a_{\beta,\alpha}.$

We know $d_{i,\alpha}\in\mathbb{Z}_{p}$ since $(G,\omega)$ is saturated. The
fact that $F_{i}$ converges on $\mathbb{B}(p^{\rho_{1}},\ldots,p^{\rho_{s}})$
follows by tweaking the estimates in the proof of 5.1 slightly. Indeed, the
inequality

$v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}-\sum_{j=1}^{s}\rho_{j}\beta_{j}\geq-\omega(g_{i})+\sum_{j=1}^{s}\beta_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}-\rho_{j}\bigg{)}$

shows the left-hand side goes to infinity as $|\beta|\rightarrow\infty$.

Our claim about the supremum follows immediately from the following bound on
$v(d_{i,\alpha})$.

(6.1)
$v(d_{i,\alpha})\geq-\bigg{(}\omega(g_{i})-\frac{1}{p-1}\bigg{)}+\sum_{j=1}^{s}\alpha_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)}.$

To verify the bound (6.1) we start with the very first inequality in [Sch11,
p. 204], which states that

$v(c_{\underline{i},\beta})\geq-\omega(g_{i})+\sum_{j=1}^{s}\beta_{j}\omega(h_{j}).$

Using the elementary fact that $v(n!)=\frac{n-s_{p}(n)}{p-1}$ for all
$n\in\mathbb{N}_{0}$, where $s_{p}(n)$ is the sum of the digits of $n$ is base
$p$, we obtain the inequality

$\displaystyle
v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}$
$\displaystyle\geq-\omega(g_{i})+\sum_{j=1}^{s}\beta_{j}\omega(h_{j})-\sum_{j=1}^{s}v(\beta_{j}!)$
$\displaystyle=-\omega(g_{i})+\sum_{j=1}^{s}\beta_{j}\omega(h_{j})-\sum_{j=1}^{s}\frac{\beta_{j}-s_{p}(\beta_{j})}{p-1}$
$\displaystyle=-\bigg{(}\omega(g_{i})-\frac{1}{p-1}\bigg{)}+\sum_{j=1}^{s}\beta_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)}-\frac{1}{p-1}+\sum_{j=1}^{s}\frac{s_{p}(\beta_{j})}{p-1}$
$\displaystyle=-\bigg{(}\omega(g_{i})-\frac{1}{p-1}\bigg{)}+\sum_{j=1}^{s}\beta_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)}+S(\beta),$

where we have introduced
$S(\beta)=-\frac{1}{p-1}+\sum_{j=1}^{s}\frac{s_{p}(\beta_{j})}{p-1}$ in the
last step. We need only check (6.1) for nonzero tuples $\alpha$ (it holds
trivially for $\alpha=(0,\ldots,0)$ since the right-hand side is negative and
$d_{i,\alpha}\in\mathbb{Z}_{p}$). So suppose $\alpha$ is nonzero. Then all
$\beta$ contributing to $d_{i,\alpha}$ are nonzero as well, as
$\alpha_{j}\leq\beta_{j}$ for all $j$. Now, clearly $S(\beta)\geq 0$ for all
nonzero $\beta$. Continuing the calculation, we infer that for such $\beta$ we
have

$\displaystyle
v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}$
$\displaystyle\geq-\bigg{(}\omega(g_{i})-\frac{1}{p-1}\bigg{)}+\sum_{j=1}^{s}\beta_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)}$
$\displaystyle\geq-\bigg{(}\omega(g_{i})-\frac{1}{p-1}\bigg{)}+\sum_{j=1}^{s}\alpha_{j}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)}.$

Since $v(a_{\beta,\alpha})\geq 0$ this proves (6.1) and hence Proposition 6.1.
Indeed, by definition of $d_{i,\alpha}$,

$v(d_{i,\alpha})\geq{\min}_{\beta\in\mathbb{N}_{0}^{s},\forall
j:\alpha_{j}\leq\beta_{j}}\bigg{[}v\bigg{(}\frac{c_{\underline{i},\beta}}{\beta_{1}!\cdots\beta_{s}!}\bigg{)}+v(a_{\beta,\alpha})\bigg{]}.$

∎

Proposition 6.1 can be further improved in the following way.

######  Definition 6.2.

For $i=1,\ldots,d$ and $j=1,\ldots,s$ we introduce sequences in
$\mathbb{R}_{>0}$ by

$\rho_{N,j}=\frac{1}{N+1}\bigg{(}\omega(h_{j})-\frac{1}{p-1}\bigg{)},\hskip
6.0pt\hskip 6.0pt\hskip
6.0pt\tau_{N,i}=\frac{1}{N+1}\bigg{(}\omega(g_{i})-\frac{1}{p-1}\bigg{)},$

where $N$ runs over all positive integers.

We will typically assume $\omega$ is $\mathbb{Q}$-valued, in which case
$(\rho_{N,j})_{N\geq 1}$ and $(\tau_{N,i})_{N\geq 1}$ are sequences in
$\mathbb{Q}_{>0}$.

######  Remark 6.3.

Note that $\omega(h_{j})-\frac{1}{p-1}-\rho_{N,j}=N\rho_{N,j}>0$ and
$\omega(g_{i})-\frac{1}{p-1}-\tau_{N,i}=N\tau_{N,i}>0$.

This leads to one of our key observations:

######  Theorem 6.4.

Keep the notation from Propositions 5.1 and 6.1. Assume $(G,\omega)$ is
saturated and $\omega$ is $\mathbb{Q}$-valued. Let $N$ be a positive integer.
Then $F_{i}$ converges on

$\mathbb{B}(p^{\rho_{N,1}},\ldots,p^{\rho_{N,s}})=\\{(y_{1},\ldots,y_{s})\in\mathbb{A}_{\text{rig}}^{s},\forall
j:|y_{j}|\leq p^{\rho_{N,j}}\\}.$

Moreover, on this polydisc $F_{i}$ has supremum at most $p^{\tau_{N,i}}$.

###### Proof.

We fix an $N\geq 1$ and omit it from the notation in this proof. Thus
$\rho_{j}=\rho_{N,j}$ and $\tau_{i}=\tau_{N,i}$, with $i$ and $j$ varying.

We are assuming $(G,\omega)$ is saturated. In other words
$\omega(g_{i})-\frac{1}{p-1}\leq 1$ by [Sch11, Prop. 26.11]. Consequently
$N\tau_{i}=\omega(g_{i})-\frac{1}{p-1}-\tau_{i}<1$. That is,
$\tau_{i}<\frac{1}{N}$.

The convergence of $F_{i}$ on the polydisc is part of Proposition 6.1 (which
applies to $\rho_{j}$ by Remark 6.3). We need to check
$|F_{i}(y_{1},\ldots,y_{s})|\leq p^{\tau_{i}}$ for all $(y_{1},\ldots,y_{s})$
in the polydisc. Adopting the notation from the proof of Proposition 6.1, it
suffices to check the coefficients $d_{i,\alpha}$ of $F_{i}$ satisfy

(6.2) $v(d_{i,\alpha})-\sum_{j=1}^{s}\rho_{j}\alpha_{j}\geq-\tau_{i}$

for all $i$ and $\alpha\in\mathbb{N}_{0}^{s}$. Dividing (6.1) by $N+1$
immediately gives the bound

(6.3)
$\frac{1}{N+1}v(d_{i,\alpha})\geq-\tau_{i}+\sum_{j=1}^{s}\rho_{j}\alpha_{j}.$

If $v(d_{i,\alpha})=0$ we are done, so we may assume $v(d_{i,\alpha})\geq 1$.
Using (6.3) and $\tau_{i}<\frac{1}{N}$ we infer that

$\displaystyle v(d_{i,\alpha})-\sum_{j=1}^{s}\rho_{j}\alpha_{j}$
$\displaystyle\geq-\tau_{i}-N\tau_{i}+N\sum_{j=1}^{s}\rho_{j}\alpha_{j}$
$\displaystyle>-\tau_{i}-1+N\sum_{j=1}^{s}\rho_{j}\alpha_{j}.$

If $-1+N\sum_{j=1}^{s}\rho_{j}\alpha_{j}\geq 0$ are are done; (6.2) follows.
Otherwise $-1+N\sum_{j=1}^{s}\rho_{j}\alpha_{j}<0$, in which case we proceed
as follows:

$v(d_{i,\alpha})\geq
1>N\sum_{j=1}^{s}\rho_{j}\alpha_{j}>-\tau_{i}+N\sum_{j=1}^{s}\rho_{j}\alpha_{j}\geq-\tau_{i}+\sum_{j=1}^{s}\rho_{j}\alpha_{j},$

which is (6.2). ∎

The upshot of Theorem 6.4 is we have two decreasing sequences of rational
numbers $\rho_{N,j}\rightarrow 0^{+}$ and $\tau_{N,i}\rightarrow 0^{+}$, for
any two fixed $i$ and $j$, such that the power series $F_{1},\ldots,F_{d}$
induce morphisms between polydiscs,

$\displaystyle\mathbb{B}(p^{\rho_{N,1}},\ldots,p^{\rho_{N,s}})$
$\displaystyle\longrightarrow\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
$\displaystyle(y_{1},\ldots,y_{s})$
$\displaystyle\longmapsto(F_{1}(y_{1},\ldots,y_{s}),\ldots,F_{d}(y_{1},\ldots,y_{s})).$

We will use this to construct ambient rigid analytic groups of $\mathbb{G}$.

###  6.2. The existence of strict neighborhood groups

The next application is our main motivation behind Theorem 6.4.

######  Corollary 6.5.

Let $(G,\omega)$ be a saturated group with ordered basis
$(g_{1},\ldots,g_{d})$. Assume111This is a mild assumption. By [Sch11, Cor.
33.3] any $p$-valuable group admits a $\mathbb{Q}$-valued $p$-valuation
$\omega$. $\omega$ takes values in $\mathbb{Q}$, and let
$\tau_{N,i}=\frac{1}{N+1}\big{(}\omega(g_{i})-\frac{1}{p-1}\big{)}$ as above.
Then the polydisc $\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$ has a
unique rigid analytic group structure which is compatible with the group
structure on the dagger group $\mathbb{G}$.

###### Proof.

We apply Theorem 6.4 in the setting of the proof of Proposition 5.3. Thus
$F_{1},\ldots,F_{d}$ are the power series in $\mathbb{Q}_{p}\langle
X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}\rangle^{\dagger}$ such that

$(g_{1}^{x_{1}}\cdots g_{d}^{x_{d}})(g_{1}^{y_{1}}\cdots
g_{d}^{y_{d}})=g_{1}^{F_{1}(x,y)}\cdots g_{d}^{F_{d}(x,y)}$

for all tuples $x=(x_{1},\ldots,x_{d})$ and $y=(y_{1},\ldots,y_{d})$ in
$\mathbb{Z}_{p}^{d}$. In this situation the sequence of elements
$h_{1},\ldots,h_{s}$ in Proposition 5.1 is
$g_{1},\ldots,g_{d},g_{1},\ldots,g_{d}$. Moreover,
$\tau_{N,i}=\frac{1}{N+1}\big{(}\omega(g_{i})-\frac{1}{p-1}\big{)}$ and

$\rho_{N,1}=\rho_{N,d+1}=\tau_{N,1},\hskip 6.0pt\hskip
6.0pt\rho_{N,2}=\rho_{N,d+2}=\tau_{N,2},\hskip 6.0pt\hskip 6.0pt\ldots\hskip
6.0pt\hskip 6.0pt,\rho_{N,d}=\rho_{N,2d}=\tau_{N,d}.$

Theorem 6.4 therefore gives a map

$\displaystyle
m:\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})\times_{\text{Sp}(\mathbb{Q}_{p})}\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
$\displaystyle\longrightarrow\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
$\displaystyle(x,y)$ $\displaystyle\longmapsto(F_{1}(x,y),\ldots,F_{d}(x,y)).$

We are building a group object in $\text{Rig}_{\mathbb{Q}_{p}}$. The unit
$e:\text{Sp}(\mathbb{Q}_{p})\rightarrow\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
is given by the origin.

The inversion map is the result of another application of Theorem 6.4. This
time we write

(6.4) $(g_{1}^{x_{1}}\cdots
g_{d}^{x_{d}})^{-1}=(g_{d}^{-1})^{x_{d}}\cdots(g_{1}^{-1})^{x_{1}}=g_{1}^{I_{1}(x)}\cdots
g_{d}^{I_{d}(x)}$

with $I_{1},\ldots,I_{d}\in\mathbb{Q}_{p}\langle
X_{1},\ldots,X_{d}\rangle^{\dagger}$. We switch coordinates and introduce
variables $Y_{1}=X_{d},\ldots,Y_{d}=X_{1}$. Correspondingly, we let
$J_{i}(Y_{1},\ldots,Y_{d})=I_{i}(Y_{d},\ldots,Y_{1})$. In this notation (6.4)
becomes

$(g_{d}^{-1})^{y_{1}}\cdots(g_{1}^{-1})^{y_{d}}=g_{1}^{J_{1}(y)}\cdots
g_{d}^{J_{d}(y)}.$

Now the elements $h_{1},\ldots,h_{s}$ from Proposition 5.1 are
$g_{d}^{-1},\ldots,g_{1}^{-1}$. Since $\omega$ is invariant under inversion,
see [Sch11, p. 169], the associated sequence $(\rho_{N,j})$ in the current
setup therefore amounts to

$\frac{1}{N+1}\big{(}\omega(g_{d}^{-1})-\frac{1}{p-1}\big{)}=\tau_{N,d},\hskip
6.0pt\hskip 6.0pt\ldots\hskip 6.0pt\hskip
6.0pt,\frac{1}{N+1}\big{(}\omega(g_{1}^{-1})-\frac{1}{p-1}\big{)}=\tau_{N,1}.$

By Theorem 6.4 we know that

$|y_{1}|\leq p^{\tau_{N,d}},\ldots,|y_{d}|\leq
p^{\tau_{N,1}}\Longrightarrow\forall i:|J_{i}(y_{1},\ldots,y_{d})|\leq
p^{\tau_{N,i}}.$

Switching back to the original coordinates, and expressing this implication in
terms of $x_{1},\ldots,x_{d}$, gives a morphism between polydiscs

$\displaystyle i:\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
$\displaystyle\longrightarrow\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
$\displaystyle x$ $\displaystyle\longmapsto(I_{1}(x),\ldots,I_{d}(x)).$

Since $(F_{1},\ldots,F_{d})$ is a formal group law, the above morphisms
$(m,e,i)$ endow $\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$ with the
structure of a rigid analytic group over $\mathbb{Q}_{p}$. ∎

######  Definition 6.6.

We refer to $\mathbb{G}_{N}:=\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$
as the $N^{\text{th}}$ strict neighborhood group of $\mathbb{G}$.

##  7\. Fréchet algebras

According to [Em17, Df. 1.2.12] a nuclear Fréchet algebra over $\mathcal{K}$
is a topological $\mathcal{K}$-algebra $S$ for which there exists a projective
system of $\mathcal{K}$-Banach algebras $S_{1}\leftarrow
S_{2}\leftarrow\cdots$, with compact transition maps, and an isomorphism of
topological $\mathcal{K}$-algebras
$S\overset{\sim}{\longrightarrow}\varprojlim_{n}S_{n}$.

######  Theorem 7.1.

Let $(G,\omega)$ be a saturated group where $\omega$ is $\mathbb{Q}$-valued.
Then $D^{\dagger}(G,K)$ is a nuclear Fréchet algebra over $K$.

In preparation for the proof we introduce some notation.

######  Definition 7.2.

$B_{N}=\\{\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}Z^{\alpha}\in
K[\\![Z_{1},\ldots,Z_{d}]\\!]:\lim_{|\alpha|\rightarrow\infty}|c_{\alpha}|p^{\sum_{i=1}^{d}\tau_{N,i}\alpha_{i}}=0\\}$.

By [BGR84, Thm. 6.1.5/4, p. 225] this is an affinoid algebra over $K$ with
norm

$\|\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}Z^{\alpha}\|_{N}={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|c_{\alpha}|p^{\sum_{i=1}^{d}\tau_{N,i}\alpha_{i}}.$

The rigid analytic group structure on
$\mathbb{G}_{N}\times_{\text{Sp}(\mathbb{Q}_{p})}\text{Sp}(K)=\text{Sp}(B_{N})$
is given by the three morphisms

$B_{N}\overset{m^{*}}{\longrightarrow}B_{N}\widehat{\otimes}_{K}B_{N},\hskip
6.0pt\hskip 6.0pt\hskip
6.0ptB_{N}\overset{e^{*}}{\longrightarrow}\mathbb{Q}_{p},\hskip 6.0pt\hskip
6.0pt\hskip 6.0ptB_{N}\overset{i^{*}}{\longrightarrow}B_{N},$

which are restrictions of their counterparts on $K\langle
Z_{1},\ldots,Z_{d}\rangle^{\dagger}$ from the proof of Proposition 5.3.

###### Proof.

(Theorem 7.1.) First note that $B_{N}\subset T_{d,p^{\mu}}$ for
$\mu=\min_{i}\tau_{N,i}$, and $T_{d,r}\subset B_{N}$ for $N>\\!\\!>_{r}0$.
Both inclusions are continuous (indeed contractions). Therefore
$\varinjlim_{N}B_{N}=W_{d}=K\langle Z_{1},\ldots,Z_{d}\rangle^{\dagger}$ as
locally convex vector spaces, where we equip $W_{d}=\varinjlim_{r}T_{d,r}$
with the usual direct limit topology. The transition maps
$B_{1}\hookrightarrow B_{2}\hookrightarrow\cdots$ are (injective and) compact
by [Em17, Prop. 2.1.16]. By [Sch02, Prop. 16.10] we get a topological
isomorphism after passing to strong duals,

(7.1)
$(W_{d})_{b}^{\prime}\overset{\sim}{\longrightarrow}\varprojlim_{N}(B_{N})_{b}^{\prime}.$

The dual transition maps
$(B_{1})_{b}^{\prime}\leftarrow(B_{2})_{b}^{\prime}\leftarrow\cdots$ remain
compact by [Sch02, Lem. 16.4], and each $(B_{N})_{b}^{\prime}$ is a Banach
algebra by Lemma 7.3 below. What is left is to observe (7.1) is an isomorphism
of algebras. This follows because the identification
$\varinjlim_{N}B_{N}=W_{d}$ preserves comultiplication:

${B_{N}}$${B_{N}\widehat{\otimes}_{K}B_{N}}$${W_{d}}$${W_{d}\otimes_{K}^{\dagger}W_{d}}$$\scriptstyle{m^{*}}$$\scriptstyle{m^{*}}$

commutes for all $N\geq 1$. ∎

The next result, used above, is presumably well-known. We include a proof, for
lack of a reference.

######  Lemma 7.3.

Let $B$ be an affinoid $\mathcal{K}$-algebra, and suppose $\text{Sp}(B)$ has a
rigid analytic group structure. Then the distribution algebra $B_{b}^{\prime}$
(which is $\mathcal{L}(B,\mathcal{K})$ with the operator norm topology by
[Sch02, Rem. 6.7]) is a $\mathcal{K}$-Banach algebra: There is a constant
$c>0$ such that $\|\delta_{1}\ast\delta_{2}\|\leq
c\|\delta_{1}\|\cdot\|\delta_{2}\|$ for all $\delta_{1},\delta_{2}\in
B_{b}^{\prime}$.

###### Proof.

Let $B\overset{m^{*}}{\longrightarrow}B\widehat{\otimes}_{\mathcal{K}}B$ be
the morphism defining multiplication on $\text{Sp}(B)$. Then, by definition of
the convolution $\ast$ we have

$(\delta_{1}\ast\delta_{2})(f)=(\delta_{1}\times\delta_{2})(m^{*}(f))$

for $f\in B$, where $(\delta_{1}\times\delta_{2})(f_{1}\otimes
f_{2})=\delta_{1}(f_{1})\delta_{2}(f_{2})$. Expanding $\varphi\in
B\otimes_{\mathcal{K}}B$ as $\varphi=\sum_{i}f_{1,i}\otimes f_{2,i}$ we find
that

$|(\delta_{1}\times\delta_{2})(\varphi)|\leq\max_{i}\\{|\delta_{1}(f_{1,i})|\cdot|\delta_{2}(f_{2,i})|\\}\leq\|\delta_{1}\|\cdot\|\delta_{2}\|\cdot\max_{i}\\{\|f_{1,i}\|\cdot\|f_{2,i}\|\\}$

Taking the infimum over all expansions $\varphi=\sum_{i}f_{1,i}\otimes
f_{2,i}$ this shows
$\|\delta_{1}\times\delta_{2}\|\leq\|\delta_{1}\|\cdot\|\delta_{2}\|$. (See
the definition of the tensor product norm on $B\otimes_{\mathcal{K}}B$ in
[BGR84, Sect. 2.1.7] or [Sch02, Sect. 17, Part B].) Consequently,

$|(\delta_{1}\ast\delta_{2})(f)|\leq\|\delta_{1}\times\delta_{2}\|\cdot\|m^{*}\|\cdot\|f\|\leq\|m^{*}\|\cdot\|\delta_{1}\|\cdot\|\delta_{2}\|\cdot\|f\|.$

In other words, we may take $c=\|m^{*}\|$. ∎

######  Remark 7.4.

If the affinoid algebra $B$ in Lemma 7.3 is a Tate algebra, then the operator
norm on $B_{b}^{\prime}$ is submultiplicative ($c=1$). Indeed, every
$\mathcal{K}$-algebra morphism between Tate algebras is a contraction by
[BGR84, Thm. 5.1.3/4, p. 194], and there are canonical isometric isomorphisms
$T_{m}\widehat{\otimes}_{\mathcal{K}}T_{n}\overset{\sim}{\longrightarrow}T_{m+n}$
by [BGR84, Cor. 6.1.1/8, p. 224]. Hence the comultiplication $m^{*}$ from the
proof of 7.3 automatically has operator norm at most one.

##  8\. The relation to [ST03]

###  8.1. An exercise in $p$-adic analysis

For the reader’s convenience, we include a proof of the following basic result
from $p$-adic analysis. This is based on the proofs of [KCon, Thm. 4.4] and
[S84, Thm. 54.4]. For $\alpha\in\mathbb{N}_{0}^{d}$ and
$x=(x_{1},\ldots,x_{d})\in\mathbb{Z}_{p}^{d}$ we let
$\binom{x}{\alpha}=\binom{x_{1}}{\alpha_{1}}\cdots\binom{x_{d}}{\alpha_{d}}$.
Also, we let $\alpha!=\alpha_{1}!\cdots\alpha_{d}!$.

######  Lemma 8.1.

Let $f\in\mathcal{C}(\mathbb{Z}_{p}^{d},K)$ be a continuous function with
Mahler expansion
$f(x)=\sum_{\alpha\in\mathbb{N}_{0}^{d}}m_{\alpha}\binom{x}{\alpha}$, with
coefficients $m_{\alpha}\in K$. Let $r_{1},\ldots,r_{d}\in p^{\mathbb{Q}}$ be
numbers $>1$. Then $f$ extends (uniquely) to a rigid analytic function on the
polydisc

$\mathbb{B}(r_{1},\ldots,r_{d})=\\{(x_{1},\ldots,x_{d})\in\mathbb{A}_{\text{rig}}^{d},\forall
i:|x_{i}|\leq r_{i}\\}$

if and only if
$\lim_{|\alpha|\rightarrow\infty}\frac{|m_{\alpha}|}{|\alpha!|}r^{|\alpha|}=0$.
(Here we adopt the notation $r^{|\alpha|}=r_{1}^{\alpha_{1}}\cdots
r_{d}^{\alpha_{d}}$.)

Furthermore, if $f(x)=\sum_{\beta\in\mathbb{N}_{0}^{d}}c_{\beta}x^{\beta}$ is
the Taylor expansion of such an $f$, then

$\|f\|_{r}:={\sup}_{\beta\in\mathbb{N}_{0}^{d}}|c_{\beta}|r^{|\beta|}={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}\frac{|m_{\alpha}|}{|\alpha!|}r^{|\alpha|}.$

###### Proof.

We will give the proof for $d=1$, and ask the reader to make the necessary
modifications in the higher-dimensional case. Thus
$f\in\mathcal{C}(\mathbb{Z}_{p},K)$ and $r\in p^{\mathbb{Q}}$ is $>1$.
Moreover, $\alpha\in\mathbb{N}_{0}$.

First assume $\frac{|m_{\alpha}|}{|\alpha!|}r^{\alpha}\rightarrow 0$ as
$\alpha\rightarrow\infty$. Expand the falling powers
$x^{\underline{\alpha}}:=x(x-1)\cdots(x-\alpha+1)$ as $\mathbb{Z}$-linear
combinations of monomials
$x^{\underline{\alpha}}=\sum_{\beta\leq\alpha}a_{\alpha,\beta}x^{\beta}$,
where $a_{\alpha,\alpha}=1$. Then for $x\in\mathbb{Z}_{p}$ we have

$f(x)=\sum_{\alpha}\frac{m_{\alpha}}{\alpha!}x^{\underline{\alpha}}=\sum_{\alpha}\sum_{\beta\leq\alpha}\frac{m_{\alpha}}{\alpha!}a_{\alpha,\beta}x^{\beta}.$

The double sum is absolutely convergent for $|x|\leq r$ since its terms go to
zero as $\alpha+\beta\rightarrow\infty$:

$|\frac{m_{\alpha}}{\alpha!}a_{\alpha,\beta}x^{\beta}|\leq\frac{|m_{\alpha}|}{|\alpha!|}r^{\beta}\leq\frac{|m_{\alpha}|}{|\alpha!|}r^{\alpha}.$

We may therefore interchange the order of summation, and hence
$f(x)=\sum_{\beta}\big{(}\sum_{\alpha\geq\beta}\frac{m_{\alpha}}{\alpha!}a_{\alpha,\beta}\big{)}x^{\beta}$
provided $|x|\leq r$. This extends $f$ to a rigid analytic function on
$\mathbb{B}(r)$.

On the other hand, now suppose $f(x)=\sum_{\beta}c_{\beta}x^{\beta}$ for
$|x|\leq r$ and certain $c_{\beta}\in K$. We now express the monomials
$x^{\beta}$ as $\mathbb{Z}$-linear combinations of falling powers,
$x^{\beta}=\sum_{\alpha\leq\beta}s_{\beta,\alpha}x^{\underline{\alpha}}$. (The
coefficients $s_{\beta,\alpha}$ are known as Stirling numbers of the second
kind.) With this notation we have

$f(x)=\sum_{\beta}\sum_{\alpha\leq\beta}c_{\beta}s_{\beta,\alpha}x^{\underline{\alpha}}=\sum_{\beta}\sum_{\alpha\leq\beta}c_{\beta}s_{\beta,\alpha}\alpha!\binom{x}{\alpha}.$

Again, the double sum is absolutely convergent, at least for
$x\in\mathbb{Z}_{p}$, since
$|c_{\beta}s_{\beta,\alpha}\alpha!\binom{x}{\alpha}|\leq|c_{\beta}|$ for such
$x$ and certainly $c_{\beta}\rightarrow 0$ for $\beta\rightarrow\infty$. (Even
$|c_{\beta}|r^{\beta}$ is a null sequence.). Swapping the order of summation
we find that $f$ has Mahler coefficients

$m_{\alpha}=\sum_{\beta\geq\alpha}c_{\beta}s_{\beta,\alpha}\alpha!.$

To finish the proof of the only if part, we simply check that

$\frac{|m_{\alpha}|}{|\alpha!|}r^{\alpha}\leq\max_{\beta\geq\alpha}\\{|c_{\beta}|r^{\alpha}\\}\leq\max_{\beta\geq\alpha}\\{|c_{\beta}|r^{\beta}\\}.$

We are now done since $|c_{\beta}|r^{\beta}$ is a null sequence by hypothesis.
We conclude $\frac{|m_{\alpha}|}{|\alpha!|}r^{\alpha}$ is a null sequence.

The last claim about $\|f\|_{r}$ follows straightforwardly from the formulas
$m_{\alpha}=\sum_{\beta\geq\alpha}c_{\beta}s_{\beta,\alpha}\alpha!$ and
$c_{\beta}=\sum_{\alpha\geq\beta}\frac{m_{\alpha}}{\alpha!}a_{\alpha,\beta}$
derived above. For example, for all $\beta\in\mathbb{N}_{0}^{d}$ we have

$|c_{\beta}|r^{|\beta|}\leq\max_{\alpha\geq\beta}\frac{|m_{\alpha}|}{|\alpha!|}r^{|\beta|}\leq\max_{\alpha\geq\beta}\frac{|m_{\alpha}|}{|\alpha!|}r^{|\alpha|}\leq{\max}_{\alpha\in\mathbb{N}_{0}^{d}}\frac{|m_{\alpha}|}{|\alpha!|}r^{|\alpha|}.$

Taking the supremum over $\beta$ shows $\|f\|_{r}$ is $\leq$ the right-hand
side. A similar argument for $m_{\alpha}$ gives the other inequality. ∎

######  Remark 8.2.

Let $\theta:=p^{-1/(p-1)}$ be the radius of convergence of the $p$-adic
exponential function. Note that $\theta\in p^{\mathbb{Q}}$ and
$\frac{1}{p}\leq\theta<1$. Keep the setup of Lemma 8.1, but assume $d=1$ for
simplicity. Then:

$\lim_{\alpha\rightarrow\infty}|m_{\alpha}|s^{\alpha}=0$ for some
$s>\theta^{-1}$ $\Longrightarrow$
$\lim_{\alpha\rightarrow\infty}\frac{|m_{\alpha}|}{|\alpha!|}r^{\alpha}=0$ for
some $r>1$ $\Longrightarrow$
$\lim_{\alpha\rightarrow\infty}|m_{\alpha}|s^{\alpha}=0$ for $s=\theta^{-1}$.

This is easily checked, by using the asymptotic formula
$v(\alpha!)=\frac{\alpha}{p-1}+O(\log_{p}\alpha)$ as
$\alpha\rightarrow\infty$. Recall that, by Amice’s theorem, $f$ is locally
analytic on $\mathbb{Z}_{p}$ if and only if
$\lim_{\alpha\rightarrow\infty}|m_{\alpha}|s^{\alpha}=0$ for some $s>1$.

###  8.2. Distributions on polydiscs

We keep the setup from Lemma 8.1 and
$\mathcal{C}^{\text{rig}}(\mathbb{B}(r_{1},\ldots,r_{d}),K)$ denotes the space
of rigid functions on the polydisc of polyradius $(r_{1},\ldots,r_{d})$. Here
we will give a description of the strong dual
$\mathcal{C}^{\text{rig}}(\mathbb{B}(r_{1},\ldots,r_{d}),K)_{b}^{\prime}$ (the
space of continuous linear forms with the operator norm topology).

First, for each $\alpha\in\mathbb{N}_{0}^{d}$ we introduce the linear form
${\bf{m}}_{\alpha}(f)=m_{\alpha}$. This is clearly continuous, and it has
operator norm $\|{\bf{m}}_{\alpha}\|\leq|\alpha!|r^{-|\alpha|}$. All the
distributions on $\mathbb{B}(r_{1},\ldots,r_{d})$ are suitable linear
combinations of $\\{{\bf{m}}_{\alpha}\\}_{\alpha\in\mathbb{N}_{0}^{d}}$.

######  Lemma 8.3.

Every
$\delta\in\mathcal{C}^{\text{rig}}(\mathbb{B}(r_{1},\ldots,r_{d}),K)^{\prime}$
has a unique expansion as a weakly convergent infinite series
$\delta=\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{m}}_{\alpha}$ where
the coefficients $d_{\alpha}\in K$ satisfy the following boundedness
condition:

$\|\delta\|={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|\alpha!d_{\alpha}|r^{-|\alpha|}<\infty.$

###### Proof.

This follows straightforwardly from the duality between null sequences and
bounded sequences. See the example in [Sch02, p. 12] for example, which gives
an isometry
$c_{0}(\mathbb{N}_{0}^{d})\overset{\sim}{\longrightarrow}\ell^{\infty}(\mathbb{N}_{0}^{d})$.
With Lemma 8.1 in hand, a simple reparametrization of the coefficients gives
the result. ∎

We will mainly be interested in the polyradii
$(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}})$ as $N\rightarrow\infty$. In this case

$(B_{N})_{b}^{\prime}=\mathcal{C}^{\text{rig}}(\mathbb{B}(p^{\tau_{N,1}},\ldots,p^{\tau_{N,d}}),K)_{b}^{\prime}$

is the distribution algebra appearing in the description of $D^{\dagger}(G,K)$
as a Fréchet algebra.

###  8.3. Contact with the Schneider-Teitelbaum algebras

We return to our saturated group $(G,\omega)$ with a choice of ordered basis
$(g_{1},\ldots,g_{d})$. In particular this gives a homeomorphism

$\displaystyle\psi:\mathbb{Z}_{p}^{d}$
$\displaystyle\overset{\sim}{\longrightarrow}G$ $\displaystyle
x=(x_{1},\ldots,x_{d})$ $\displaystyle\longmapsto g_{1}^{x_{1}}\cdots
g_{d}^{x_{d}}.$

Via $\psi$ we view functions on $G$ as functions on $\mathbb{Z}_{p}^{d}$, and
vice versa. More formally, we have an isomorphism of $K$-Banach spaces

$\displaystyle\psi^{*}:\mathcal{C}(G,K)$
$\displaystyle\overset{\sim}{\longrightarrow}\mathcal{C}(\mathbb{Z}_{p}^{d},K)$
$\displaystyle f$ $\displaystyle\longmapsto f\circ\psi.$

Similarly for the space $\mathcal{C}^{\text{an}}(G,K)$ of locally analytic
functions.

Following [ST03, Sect. 4] we embed
$K[\\![G]\\!]:=K\otimes_{\mathbb{Q}_{p}}\mathbb{Z}_{p}[\\![G]\\!]$ in the
locally analytic distribution algebra $D(G,K)$ by sending
$g\mapsto\delta_{g}$. We introduce the elements
$b_{i}=g_{i}-1\in\mathbb{Z}_{p}[\\![G]\\!]$, and for each
$\alpha\in\mathbb{N}_{0}^{d}$ we let
${\bf{b}}^{\alpha}=b_{1}^{\alpha_{1}}\cdots b_{d}^{\alpha_{d}}$. Elements of
$K[\\![G]\\!]$ have the form
$\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{b}}^{\alpha}$ with bounded
coefficients $d_{\alpha}\in K$. Similarly, a distribution $\lambda\in D(G,K)$
has a unique expansion

$\lambda=\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{b}}^{\alpha},\hskip
6.0pt\hskip 6.0pt\hskip
6.0pt\|\lambda\|_{s}^{\prime}={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|d_{\alpha}|s^{|\alpha|}<\infty,\hskip
6.0pt\hskip 6.0pt\hskip 6.0pt\forall s\in(0,1).$

The sum converges in $D(G,K)$ essentially by (the proof of) [ST02, Lem. 2.5].
Note that ${\bf{b}}^{\alpha}(f)=m_{\alpha}$ is the $\alpha^{\text{th}}$ Mahler
coefficient of $f$. We recall from [ST03, Prop. 4.2] that the norm

$\|\lambda\|_{s}={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|d_{\alpha}|s^{\tau\alpha},\hskip
6.0pt\hskip 6.0pt\hskip
6.0pt\tau\alpha:=\sum_{i=1}^{d}\omega(g_{i})\alpha_{i},$

is submultiplicative for $s\in[\frac{1}{p},1)$. Note that the collection of
norms $\\{\|\cdot\|_{s}:0<s<1\\}$ define the same topology as
$\\{\|\cdot\|_{s}^{\prime}:0<s<1\\}$. Indeed
$\|\lambda\|_{s^{\min\omega(g_{i})}}^{\prime}\leq\|\lambda\|_{s}\leq\|\lambda\|_{s^{\max\omega(g_{i})}}^{\prime}$.

In [ST03] the $K$-Banach algebra $D_{s}(G,K)$ is defined as the completion of
$D(G,K)$ with respect to the norm $\|\cdot\|_{s}$, where $s\in[\frac{1}{p},1)$
is now fixed. The elements of $D_{s}(G,K)$ have unique convergent expansions

$\lambda=\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{b}}^{\alpha},\hskip
6.0pt\hskip 6.0pt\hskip 6.0pt|d_{\alpha}|s^{\tau\alpha}\longrightarrow 0\text{
as }|\alpha|\longrightarrow\infty.$

The Fréchet-Stein structures in [ST03, Sect. 4] arise from the isomorphism
$D(G,K)\overset{\sim}{\longrightarrow}\varprojlim_{s\in[\frac{1}{p},1)}D_{s}(G,K)$.

######  Lemma 8.4.

Any distribution $\lambda\in D^{\dagger}(G,K)$ has a unique expansion as a
weakly convergent sum
$\lambda=\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{b}}^{\alpha}$ with
coefficients $d_{\alpha}\in K$ satisfying:

$\|\lambda\|_{s}^{\dagger}:={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|\alpha!d_{\alpha}|s^{|\alpha|}<\infty,\hskip
6.0pt\hskip 6.0pt\hskip 6.0pt\forall s\in(0,1).$

###### Proof.

This follows from Lemma 8.3 with $r_{1}=\cdots=r_{d}=\frac{1}{s}$. ∎

Recall that $\omega(g)>\frac{1}{p-1}$ for all $g\in G$, and $\theta<1$.
Therefore $\frac{1}{p}<\theta^{1/\min\omega(g_{i})}$. The next result at least
gives a continuous algebra homomorphism

$D^{\dagger}(G,K)\longrightarrow{\varprojlim}_{s\in[\frac{1}{p},\theta^{1/\min\omega(g_{i})})}D_{s}(G,K).$

Under favorable circumstances we will show this is an isomorphism, and then
deduce from [ST03] that $D^{\dagger}(G,K)$ is Fréchet-Stein.

######  Lemma 8.5.

We have a continuous inclusion of algebras $D^{\dagger}(G,K)\hookrightarrow
D_{s}(G,K)$ for $\frac{1}{p}\leq s<\theta^{1/\min\omega(g_{i})}$.

###### Proof.

Let $\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{b}}^{\alpha}$ be a
weakly convergent sum representing an element of $D^{\dagger}(G,K)$. Then for
all tuples $\alpha$ we have the estimate

$\displaystyle|d_{\alpha}|s^{\tau\alpha}$
$\displaystyle=|\alpha!d_{\alpha}|s^{\tau\alpha}\frac{1}{|\alpha!|}$
$\displaystyle\leq|\alpha!d_{\alpha}|s^{\min\omega(g_{i})|\alpha|}p^{|\alpha|/(p-1)}$
$\displaystyle=|\alpha!d_{\alpha}|(s^{\min\omega(g_{i})}\theta^{-1})^{|\alpha|}$

which shows that $|d_{\alpha}|s^{\tau\alpha}\longrightarrow 0$ as
$|\alpha|\longrightarrow\infty$ provided $s^{\min\omega(g_{i})}\theta^{-1}<1$.
∎

###  8.4. Comparison with the Fréchet algebra structure

We compare the inverse system $(B_{N})_{b}^{\prime}$ to the system
$D_{s}(G,K)$ from [ST03]. Recall that
$D^{\dagger}(G,K)\overset{\sim}{\longrightarrow}\varprojlim_{N}(B_{N})_{b}^{\prime}$
by (the proof of) Theorem 7.1.

######  Lemma 8.6.

The following holds.

  * (1)

Let $\frac{1}{p}\leq s<\theta^{1/\min\omega(g_{i})}$. Then for all
$N>\\!\\!>0$ there is a continuous linear map

$\displaystyle(B_{N})_{b}^{\prime}$ $\displaystyle\longrightarrow D_{s}(G,K)$
$\displaystyle{\bf{m}}_{\alpha}$ $\displaystyle\longmapsto{\bf{b}}^{\alpha}.$

  * (2)

Let $N\geq 1$. Then for all $s\geq\frac{1}{p}$ sufficiently close to
$\theta^{1/\max\omega(g_{i})}$ there is a continuous linear map

$\displaystyle D_{s}(G,K)$ $\displaystyle\longrightarrow(B_{N})_{b}^{\prime}$
$\displaystyle{\bf{b}}^{\alpha}$ $\displaystyle\longmapsto{\bf{m}}_{\alpha}.$

###### Proof.

The proof of part (1) is very similar to that of Lemma 8.5. Fix an $s$ in the
given range. We must show $|d_{\alpha}|s^{\tau\alpha}\rightarrow 0$ as
$|\alpha|\rightarrow 0$, provided
$\sum_{\alpha\in\mathbb{N}_{0}^{d}}d_{\alpha}{\bf{m}}_{\alpha}$ lies in
$(B_{N})_{b}^{\prime}$ where $N$ is large enough. So, assume
$A:={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|\alpha!d_{\alpha}|r^{-|\alpha|}<\infty$
where $r_{i}=p^{\tau_{N,i}}$. Then

$\displaystyle|d_{\alpha}|s^{\tau\alpha}$
$\displaystyle=|\alpha!d_{\alpha}|r^{-|\alpha|}r^{|\alpha|}s^{\tau\alpha}\frac{1}{|\alpha!|}$
$\displaystyle\leq
A\cdot\prod_{j=1}^{d}(r_{j}s^{\min\omega(g_{i})}\theta^{-1})^{\alpha_{j}}.$

By assumption $s^{-\min\omega(g_{i})}\theta>1$. Since
$p^{\tau_{N,j}}\rightarrow 1^{+}$ as $N\rightarrow\infty$, we can arrange that
$r_{j}s^{\min\omega(g_{i})}\theta^{-1}<1$ for all $j$ by choosing $N$
sufficiently large. This shows $|d_{\alpha}|s^{\tau\alpha}$ goes to zero –
plus the fact that the resulting map is continuous (in fact a contraction).

For the proof of part (2) we now start with an $N\geq 1$ and seek conditions
on $s$ which guarantee that $|\alpha!d_{\alpha}|r^{-|\alpha|}$ is bounded when
${\lim}_{|\alpha|\rightarrow\infty}|d_{\alpha}|s^{\tau\alpha}=0$. Here we keep
the polyradius $r_{i}=p^{\tau_{N,i}}$.

$\displaystyle|\alpha!d_{\alpha}|r^{-|\alpha|}$
$\displaystyle=|d_{\alpha}|s^{\tau\alpha}s^{-\tau\alpha}r^{-|\alpha|}|\alpha!|$
$\displaystyle\leq
B\cdot(\alpha_{1}\cdots\alpha_{d})^{C}\cdot\prod_{j=1}^{d}(s^{-\max\omega(g_{i})}r_{j}^{-1}\theta)^{\alpha_{j}}$

where $B:={\sup}_{\alpha\in\mathbb{N}_{0}^{d}}|d_{\alpha}|s^{\tau\alpha}$, and
$C>0$ is a large enough constant arising from the asymptotics
$v(n!)=\frac{n}{p-1}+O(\log_{p}n)$. Now the $r_{j}>1$ are fixed, and if we
choose $s$ close enough to $\theta^{1/\max\omega(g_{i})}$ we can arrange that
$s^{-\max\omega(g_{i})}r_{j}^{-1}\theta<1$ for all $j$. This implies that
$|\alpha!d_{\alpha}|r^{-|\alpha|}$ indeed stays bounded, and moreover that the
map taking ${\bf{b}}^{\alpha}\mapsto{\bf{m}}_{\alpha}$ is continuous (but not
necessarily a contraction). ∎

Altogether, this gives the commutative diagram of continuous linear maps:

${\varprojlim_{s<\theta^{1/\max\omega(g_{i})}}D_{s}(G,K)}$${\varprojlim_{N}(B_{N})_{b}^{\prime}}$${\varprojlim_{s<\theta^{1/\min\omega(g_{i})}}D_{s}(G,K)}$${D^{\dagger}(G,K)}$inclusion$\scriptstyle{\simeq}$

If $(G,\omega)$ is equi-$p$-valued, which means $\omega(g_{i})=\omega_{0}$ is
constant for all $i=1,\ldots,d$, then we deduce an isomorphism of topological
algebras

(8.1)
$D^{\dagger}(G,K)\overset{\sim}{\longrightarrow}{\varprojlim}_{s<\theta^{1/\omega_{0}}}D_{s}(G,K).$

The condition (HYP) in [ST03, p. 163] here amounts to the inequality
$2\omega_{0}>\frac{p}{p-1}$ (and saturation). If this is satisfied [ST03, Thm.
4.5, Thm. 4.9] imply the following properties for any two
$\frac{1}{p}<s^{\prime}\leq s<1$ in $p^{\mathbb{Q}}$.

  * •

$D_{s}(G,K)$ is a (left and right) Noetherian integral domain;

  * •

The transition map $D_{s}(G,K)\longrightarrow D_{s^{\prime}}(G,K)$ is flat.

In particular, having established the isomorphism (8.1) the main results of
[ST03] immediately give the Fréchet-Stein structure on $D^{\dagger}(G,K)$ –
under the assumption that $(G,\omega)$ is equi-$p$-valued and
$2\omega_{0}>\frac{p}{p-1}$.

######  Example 8.7.

Suppose $G$ is a uniform pro-$p$-group. By [DdSMS, Thm. 4.5] this means $G$ is
topologically finitely generated, torsion-free, and powerful – which means
$G/\overline{G^{p\varepsilon}}$ is abelian, where
$\varepsilon:=\begin{cases}1&\text{if $p>2$}\\\ 2&\text{if $p=2$}\end{cases}$.
Let $P_{i}(G)$ be the lower $p$-series, as defined in [DdSMS, Df. 1.15]. One
can easily check that

$\omega^{\text{can}}(g)=\sup\\{i\geq 1:g\in
P_{i}(G)\\}+\begin{cases}0&\text{if $p>2$}\\\ 1&\text{if $p=2$}\end{cases}$

defines a $p$-valuation on $G$, and $(G,\omega^{\text{can}})$ is a saturated
group which admits a basis $(g_{1},\ldots,g_{d})$ with
$\omega^{\text{can}}(g_{i})=\varepsilon$ for all $i$. For instance, these
claims are checked in [HKN11, p. 249]. In particular $(G,\omega^{\text{can}})$
satisfies the condition (HYP) from [ST03] since $2\varepsilon>\frac{p}{p-1}$
for all primes $p\geq 2$.

######  Remark 8.8.

In [ST03, Sect. 5] the authors proceed to remove the hypothesis (HYP) and show
that the locally analytic distribution algebra $D(G,K)$ is Fréchet-Stein for
any compact $p$-adic Lie group $G$. See [ST03, Thm. 5.1]. To do so, they
select an open normal subgroup $U\vartriangleleft G$ which is uniform, along
with coset representatives $y_{1},\ldots,y_{\ell}$. The Dirac distributions
$\delta_{y_{1}},\ldots,\delta_{y_{\ell}}$ then form a basis for $D(G,K)$ as a
$D(U,K)$-module, and this is used to define a Fréchet-Stein on $D(G,K)$ in
terms of that on $D(U,K)$.

This approach does not work in the overconvergent setting. The coset
decomposition of the space of locally analytic functions
$\mathcal{C}^{\text{an}}(G,K)=\bigoplus_{j=1}^{\ell}\mathcal{C}^{\text{an}}(y_{j}U,K)$
does not have an analogue for $\mathcal{C}^{\dagger}(G,K)$. The
overconvergence of a function $f:G\longrightarrow K$ cannot be detected
locally on the cosets of $U$.

##  9\. Overconvergent representations

###  9.1. The space of $V$-valued overconvergent functions

Let $V$ be a locally convex $K$-vector space of compact type. In this section
we will define a space $\mathcal{C}^{\dagger}(G,V)$ of locally analytic
functions $f:G\longrightarrow V$ which is canonically topologically isomorphic
to the completed tensor product
$\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V$. (Since
$\mathcal{C}^{\dagger}(G,K)$ is also of compact type, the inductive and
projective tensor product topologies coincide. We adopt the convention in
[Em17, Prop. 1.1.32] and omit the corresponding subscripts $\iota$ and $\pi$.)

######  Lemma 9.1.

Suppose $V$ is a locally convex $K$-vector space of compact type. Then the
following holds.

  * (1)

The inclusion
$\mathcal{C}^{\dagger}(G,K)\hookrightarrow\mathcal{C}^{\text{an}}(G,K)$ is
continuous;

  * (2)

The induced map
$\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V\longrightarrow\mathcal{C}^{\text{an}}(G,K)\widehat{\otimes}_{K}V$
is a continuous injection.

###### Proof.

After choosing a basis for $(G,\omega)$ we have to check the continuity of the
inclusion $T_{d,r}\hookrightarrow\mathcal{C}^{\text{an}}(G,K)$ for all
$r\in\Gamma_{>1}$. The latter factors as $T_{d,r}\hookrightarrow T_{d}$
composed with $T_{d}\hookrightarrow\mathcal{C}^{\text{an}}(G,K)$. Both of
these maps are continuous. (In the terminology of [ST02, p. 447] take a
$V$-index consisting of a single global chart
$G\overset{\sim}{\longrightarrow}\mathbb{Z}_{p}^{d}$.)

In part (2) continuity is clear. The fact that the map is an injection follows
from [Em17, Prop. 1.1.26], which applies since compact type spaces are
bornological and complete ([ST02, Thm. 1.1]). ∎

The dual of the inclusion in part (1) is a continuous algebra homomorphism
$D(G,K)\longrightarrow D^{\dagger}(G,K)$ which is injective. (Indeed, if
$\lambda\in D(G,K)$ vanishes on all polynomial functions, such as
$x\mapsto\binom{x}{\alpha}$, then clearly $\lambda=0$.)

By [Em17, Prop. 2.1.28] there is a topological isomorphism

(9.1)
$\mathcal{C}^{\text{an}}(G,V)\overset{\sim}{\longrightarrow}\mathcal{C}^{\text{an}}(G,K)\widehat{\otimes}_{K}V.$

We define $\mathcal{C}^{\dagger}(G,V)$ to be the subspace of
$\mathcal{C}^{\text{an}}(G,V)$ which corresponds to
$\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V$ under (9.1).

######  Definition 9.2.

$\mathcal{C}^{\dagger}(G,V)$ is the space of functions $f:G\longrightarrow V$
arising from $\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V$, where the
latter is viewed as a subset of $\mathcal{C}^{\text{an}}(G,V)$ via the inverse
of (9.1) and the injection in part (2) of Lemma 9.1. We endow
$\mathcal{C}^{\dagger}(G,V)$ with a topology by transferring the topology from
$\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V$.

We thus have a commutative diagram

${\mathcal{C}^{\dagger}(G,V)}$${\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V}$${\mathcal{C}^{\text{an}}(G,V)}$${\mathcal{C}^{\text{an}}(G,K)\widehat{\otimes}_{K}V}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the horizontal maps are topological isomorphisms, and the vertical maps
are continuous injections.

###  9.2. The notion of an overconvergent representation

Now we assume $V$ is a locally analytic $G$-representation of compact type.
Let $\rho:V\longrightarrow\mathcal{C}^{\text{an}}(G,V)$ be the orbit map
$v\mapsto\rho_{v}$, where $\rho_{v}(g)=gv$. This is a continuous injection by
[Em17, Thm. 3.6.12] (and the paragraph right below its proof). Obviously
$\rho$ is $G$-equivariant if we let $G$ act by right translations on
$\mathcal{C}^{\text{an}}(G,V)$.

######  Definition 9.3.

We say $V$ is overconvergent if there is a continuous map
$V\longrightarrow\mathcal{C}^{\dagger}(G,V)$ such that the diagram

${\mathcal{C}^{\dagger}(G,V)}$${V}$${\mathcal{C}^{\text{an}}(G,V)}$$\scriptstyle{\rho}$

commutes. (The dashed arrow is necessarily unique if it exists.)

When $V$ is overconvergent we will equip $V_{b}^{\prime}$ with a
$D^{\dagger}(G,K)$-module structure. We start by reviewing how
$V_{b}^{\prime}$ becomes a $D(G,K)$-module. Composing the isomorphism (9.1)
with the orbit map yields

$c:V\overset{\rho}{\longrightarrow}\mathcal{C}^{\text{an}}(G,V)\overset{\sim}{\longrightarrow}\mathcal{C}^{\text{an}}(G,K)\widehat{\otimes}_{K}V.$

Passing to strong duals gives a continuous map

$\displaystyle D(G,K)\widehat{\otimes}_{K}V_{b}^{\prime}$
$\displaystyle\overset{\sim}{\longrightarrow}(\mathcal{C}^{\text{an}}(G,K)\widehat{\otimes}_{K}V)_{b}^{\prime}\overset{c^{\prime}}{\longrightarrow}V_{b}^{\prime}$
$\displaystyle\lambda\otimes v^{\prime}$ $\displaystyle\longmapsto(\lambda\ast
v^{\prime})(v)=\lambda\big{(}g\mapsto\langle gv,v^{\prime}\rangle\big{)}.$

(The isomorphism is discussed in part (ii) of [Em17, Prop. 1.1.32].) As a
special case, temporarily taking $V=C^{\text{an}}(G,K)$ gives a jointly
continuous multiplication map $\ast$ on $D(G,K)$ which is the opposite of the
usual convolution product. For instance
$\delta_{x}\ast\delta_{y}=\delta_{yx}$, as is easily checked. Since $D(G,K)$
is isomorphic to its opposite algebra via $g\mapsto g^{-1}$ this subtle
difference will not play a role.

A standard and straightforward calculation shows that $V_{b}^{\prime}$ thus
becomes a left module over $D(G,K)$, and the action $D(G,K)\times
V_{b}^{\prime}\longrightarrow V_{b}^{\prime}$ is jointly continuous (by
definition of the projective tensor product topology).

When $V$ is an overconvergent representation we have a commutative diagram of
continuous maps

${\mathcal{C}^{\dagger}(G,V)}$${\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}V}$${V}$${\mathcal{C}^{\text{an}}(G,V)}$${\mathcal{C}^{\text{an}}(G,K)\widehat{\otimes}_{K}V.}$$\scriptstyle{\sim}$$\scriptstyle{\rho}$$\scriptstyle{c}$$\scriptstyle{\sim}$

The dual diagram implies the $D(G,K)$-action on $V_{b}^{\prime}$ factors
through $D^{\dagger}(G,K)$. This argument is reversible. The dual of a
$D^{\dagger}(G,K)$-module is an overconvergent representation. As in [ST02,
Cor. 3.4] the strong duality functor $V\mapsto V_{b}^{\prime}$ restricts to an
anti-equivalence of categories

${\\{\text{overconvergent $G$-representations of compact
type}\\}}$${\\{\text{continuous $D^{\dagger}(G,K)$-modules on nuclear
Fr\'{e}chet spaces}\\}.}$

###  9.3. Examples of overconvergent representations

Our first example is $\mathcal{C}^{\dagger}(G,K)$ on which $G$ acts by right
translations. (By a small modification of the proof of [LS23, Lem. 4.6] this
is a topological $G$-action in the sense of [Em17, Sect. 0.3.11], which means
each $g\in G$ acts via a topological automorphism of
$\mathcal{C}^{\dagger}(G,K)$.)

We have already noted that $\mathcal{C}^{\dagger}(G,K)$ is of compact type
(since $W_{d}$ with the fringe topology is of compact type). Moreover, the
orbit map may be viewed as a map into $\mathcal{C}^{\dagger}(G\times G,K)$ as
follows:

$\displaystyle\widetilde{\rho}:\mathcal{C}^{\dagger}(G,K)$
$\displaystyle\longrightarrow\mathcal{C}^{\dagger}(G,\mathcal{C}^{\dagger}(G,K))\overset{\sim}{\longrightarrow}\mathcal{C}^{\dagger}(G,K)\widehat{\otimes}_{K}\mathcal{C}^{\dagger}(G,K)\underset{\Psi}{\overset{\sim}{\longrightarrow}}\mathcal{C}^{\dagger}(G\times
G,K)$ $\displaystyle f$ $\displaystyle\longmapsto\big{[}(x,y)\mapsto
f(yx)\big{]}.$

(See Remark 4.2 for the relation between $\otimes_{K}^{\dagger}$ and
$\widehat{\otimes}_{K}$. The isomorphism $\Psi$ was defined in Section 5.4.)
This map $\widetilde{\rho}$ is clearly a homomorphism of dagger algebras, and
it is therefore automatically continuous with respect to the fringe topology
by [GK00, Lem. 1.8]. This shows that indeed $\mathcal{C}^{\dagger}(G,K)$ is an
overconvergent representation.

More generally, let $W\subset\mathcal{C}^{\dagger}(G,K)$ be a closed
$G$-invariant subspace (with the induced topology). Note that $W$ is of
compact type by part (i) of [ST02, Prop. 1.2]. Then $W$ is an overconvergent
representation by the following result.

######  Lemma 9.4.

Let $V$ be an overconvergent representation. Then any closed $G$-invariant
subspace $W\subset V$ is an overconvergent representation (relative to the
induced topology).

###### Proof.

We construct the dual of the orbit map
$W\longrightarrow\mathcal{C}^{\dagger}(G,W)$. Passing to strong duals gives a
strict surjection $p:V_{b}^{\prime}\twoheadrightarrow W_{b}^{\prime}$ by part
(i) of [ST02, Prop. 1.2], whose kernel $\\{v^{\prime}\in
V^{\prime}:v^{\prime}|_{W}=0\\}$ is a $D^{\dagger}(G,K)$-submodule (since $W$
is $G$-invariant). This equips $W^{\prime}$ with an abstract
$D^{\dagger}(G,K)$-module structure for which $p$ is
$D^{\dagger}(G,K)$-linear. What remains is verifying the action map
$D^{\dagger}(G,K)\times W_{b}^{\prime}\longrightarrow W_{b}^{\prime}$ is
jointly continuous. As $D^{\dagger}(G,K)$ and $W_{b}^{\prime}$ are both
Fréchet spaces, it suffices to show separate continuity by [Sch02, Prop.
17.6].

First, for a fixed $w^{\prime}\in W^{\prime}$ the map
$D^{\dagger}(G,K)\rightarrow W_{b}^{\prime}$ which takes
$\lambda\mapsto\lambda\ast w^{\prime}$ is clearly continuous (write
$w^{\prime}=p(v^{\prime})$ and compose $p$ with $\lambda\mapsto\lambda\ast
v^{\prime}$, which is continuous since $V$ is assumed overconvergent).

On the other hand, for a fixed $\lambda\in D^{\dagger}(G,K)$ the map
$a_{W}:W_{b}^{\prime}\rightarrow W_{b}^{\prime}$ sending
$w^{\prime}\mapsto\lambda\ast w^{\prime}$ is continuous by the following
argument: For an open subset $U\subset W_{b}^{\prime}$, in order to show
$a_{W}^{-1}(U)$ is open it suffices to show $p^{-1}(a_{W}^{-1}(U))$ is open –
since $p$ is strict. This is now obvious from the commutative diagram:

${W_{b}^{\prime}}$${W_{b}^{\prime}}$${V_{b}^{\prime}}$${V_{b}^{\prime}.}$$\scriptstyle{a_{W}}$$\scriptstyle{a_{V}}$$\scriptstyle{p}$$\scriptstyle{p}$

This essentially finishes the proof: By the completeness of $W_{b}^{\prime}$,
the action map extends uniquely to a continuous map
$D^{\dagger}(G,K)\widehat{\otimes}_{K}W_{b}^{\prime}\longrightarrow
W_{b}^{\prime}$, the dual of which is the desired continuous orbit map. ∎

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