S}{\mathchoice{\left\|r_{s}\right\|}{\|r_{s}\|}{\|r_{s}\|}{\|r_{s}\|}}_{R}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}\in[0,\infty)$
when $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$. We put $\text{\rm
Ch}^{\text{\rm iso}-}_{\mathscr{C}}(R)\coloneqq\text{\rm Ch}^{-}(\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R))\subset\text{\rm
Ch}^{-}_{\mathscr{C}}(R)$.

######  Proposition 6.7.

Let $S\in\text{\rm NSet}$.

  * (i)

If $S$ and $R$ are isolated at $0$, then the natural $R$-linear homomorphism
$R^{\oplus S_{+}}\to\ell^{\mathscr{C}}(S,R)$ is bijective, and
$\ell^{\mathscr{C}}(S,R)$ is a projective object of $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$.

  * (ii)

Let $A\in\text{\rm Alg}_{\mathscr{C}}(R)$. The natural morphism
$\varphi\colon\ell^{\mathscr{C}}(S,R)\widehat{\otimes}{}_{R}A\to\ell^{\mathscr{C}}(S,A)$
in $\text{\rm Mod}_{\mathscr{C}}(A)$ is an isomorphism. In addition, if
${\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}=1$ and
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{A}$ is
submultiplicative over $R$, then it is an isometric $A$-linear isomorphism.

  * (iii)

If $S$ and $R$ are isolated at $0$, then for any $A\in\text{\rm
Alg}^{\text{\rm iso}}_{\mathscr{C}}(R)$, the natural $A$-linear homomorphism
$\ell^{\mathscr{C}}(S,R)\otimes_{R}A\to\ell^{\mathscr{C}}(S,A)$ is bijective.

###### Proof.

The assertion (iii) follows from the assertions (i) and (ii), and by
Proposition 6.4 (i). We show the assertion (i). For any $(r_{s})_{s\in S}\in
R^{\oplus S}$, the seminorm of $(r_{s})_{s\in S}$ is given as $\sum_{s\in
S}{\mathchoice{\left\|r_{s}\right\|}{\|r_{s}\|}{\|r_{s}\|}{\|r_{s}\|}}_{R}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}=\sum_{s\in
S_{+}}{\mathchoice{\left\|r_{s}\right\|}{\|r_{s}\|}{\|r_{s}\|}{\|r_{s}\|}}_{R}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}$,
and hence the completion $\ell^{\mathscr{C}}(S,R)$ of $R^{\oplus S}$ is
naturally identified with the completion of $R^{\oplus S_{+}}$ equipped with
the norm given by the restriction of the seminorm. By
${\mathchoice{\left\\{r\in R^{\oplus
S_{+}}\middle|{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}_{R^{\oplus
S_{+}}}<\epsilon^{2}\right\\}}{\\{r\in R^{\oplus
S_{+}}\mid{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}_{R^{\oplus
S_{+}}}<\epsilon^{2}\\}}{\\{r\in R^{\oplus
S_{+}}\mid{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}_{R^{\oplus
S_{+}}}<\epsilon^{2}\\}}{\\{r\in R^{\oplus
S_{+}}\mid{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}_{R^{\oplus
S_{+}}}<\epsilon^{2}\\}}}={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$,
$R^{\oplus S_{+}}$ is complete and isolated at $0$. This implies that the
morphism in the assertion (i) is an isomorphism in $\text{\rm
Mod}_{\mathscr{C}}(R)$, and hence $\ell^{\mathscr{C}}(S,R)$ is isolated at
$0$. The bijectivity of the natural $R$-linear homomorphism follows from
Proposition 2.2. The projectivity of $\ell^{\mathscr{C}}(S,R)$ in $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ follows from the projectivity in
$\text{\rm Mod}_{\mathscr{C}}(R)$, because the inclusion $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)\hookrightarrow\text{\rm
Mod}_{\mathscr{C}}(R)$ preserves the strict exactness of a null sequence by
Proposition 6.1 (ii).

The morphism $\varphi$ in the assertion (ii) is a bounded isomorphism by the
universality of $\widehat{\otimes}{}_{R}$ for bounded $R$-bilinear
homomorphisms (cf. [BM21] p. 8) and the adjoint property of
$\ell^{\mathscr{C}}(\cdot,R)$ for bounded maps (cf. [BM21] p. 12). Therefore,
it suffices to show that $\varphi$ is an isometry under the assumption that
${\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}=1$ and
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{A}$ is
submultiplicative over $R$. For this purpose, it suffices to show that for any
$(a_{s})_{s\in S_{+}}\in A^{\oplus S_{+}}$, the tensor seminorm $C$ of an
element $\sum_{s\in S_{+}}\delta_{s}\otimes a_{s}\in R^{\oplus
S_{+}}\otimes_{R}A$ with respect to the restriction of
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\ell^{\mathscr{C}}(S,R)}$
to $R^{\oplus S_{+}}$ coincides with $\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}$,
where $\delta_{s}\in R^{\oplus S_{+}}$ denotes the image of $s\in S_{+}$ by
the canonical embedding $S_{+}\hookrightarrow R^{\oplus S_{+}}$, and
$\sum^{\mathscr{C}}$ denotes $\sum$ when $\mathscr{C}=\text{\rm Ab}^{\text{\rm
A}}$ and $\sup$ when $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$. We have
$C\leq\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|\delta_{s}\right\|}{\|\delta_{s}\|}{\|\delta_{s}\|}{\|\delta_{s}\|}}_{\ell^{\mathscr{C}}(S,R)}{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}=\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}=\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}$.

Let $((r_{i,s})_{s\in S_{+}},b_{i})_{i=0}^{n}\in(R^{\oplus S_{+}},A)^{n+1}$
with $n\in\mathbb{N}$ and $\sum_{i=0}^{n}(r_{i,s})_{s\in S_{+}}\otimes
b_{i}=\sum_{s\in S_{+}}\delta_{s}\otimes a_{s}$. For any $s\in S_{+}$, we have
$\sum_{i=0}^{n}r_{i,s}b_{i}=a_{s}$, and hence
$\sum^{\mathscr{C},n}_{i=0}{\mathchoice{\left\|r_{i,s}\right\|}{\|r_{i,s}\|}{\|r_{i,s}\|}{\|r_{i,s}\|}}_{R}{\mathchoice{\left\|b_{i}\right\|}{\|b_{i}\|}{\|b_{i}\|}{\|b_{i}\|}}_{A}\geq\sum^{\mathscr{C},n}_{i=0}{\mathchoice{\left\|r_{i,s}b_{i}\right\|}{\|r_{i,s}b_{i}\|}{\|r_{i,s}b_{i}\|}{\|r_{i,s}b_{i}\|}}_{A}\geq{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}$
by the submultiplicativity of
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{A}$
over $R$ and by the triangle inequality when $\mathscr{C}=\text{\rm
Ab}^{\text{\rm A}}$ and the strong triangle inequality when
$\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$. We obtain

$\displaystyle\sum^{\mathscr{C},n}_{i=0}{\mathchoice{\left\|(r_{i,s})_{s\in
S_{+}}\right\|}{\|(r_{i,s})_{s\in S_{+}}\|}{\|(r_{i,s})_{s\in
S_{+}}\|}{\|(r_{i,s})_{s\in
S_{+}}\|}}_{\ell^{\mathscr{C}}(S,R)}{\mathchoice{\left\|b_{i}\right\|}{\|b_{i}\|}{\|b_{i}\|}{\|b_{i}\|}}_{A}\geq\sum^{\mathscr{C},n}_{i=0}\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|r_{i,s}\right\|}{\|r_{i,s}\|}{\|r_{i,s}\|}{\|r_{i,s}\|}}_{R}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}{\mathchoice{\left\|b_{i}\right\|}{\|b_{i}\|}{\|b_{i}\|}{\|b_{i}\|}}_{A}$
$\displaystyle=$ $\displaystyle\sum^{\mathscr{C}}_{s\in
S_{+}}\left(\sum^{\mathscr{C},n}_{i=0}{\mathchoice{\left\|r_{i,s}\right\|}{\|r_{i,s}\|}{\|r_{i,s}\|}{\|r_{i,s}\|}}_{R}{\mathchoice{\left\|b_{i}\right\|}{\|b_{i}\|}{\|b_{i}\|}{\|b_{i}\|}}_{A}\right){\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}\geq\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}.$

This implies $C\geq\sum^{\mathscr{C}}_{s\in
S_{+}}{\mathchoice{\left\|a_{s}\right\|}{\|a_{s}\|}{\|a_{s}\|}{\|a_{s}\|}}_{A}{\mathchoice{\left|s\right|}{|s|}{|s|}{|s|}}_{S}$.
∎

By Proposition 6.7 (i), we obtain the following.

######  Corollary 6.8.

  * (i)

If $R$ is isolated at $0$, then $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$ has enough projective objects.

  * (ii)

If $R$ is isolated at $0$, then $\text{\rm Ch}^{\text{\rm
iso}-}_{\mathscr{C}}(R)$ is stable under the functor
$\mathscr{P}_{\mathscr{C}}^{R}$, and the restriction of
$\mathscr{P}_{\mathscr{C}}^{R}$ gives a projective resolution functor on
$\text{\rm Ch}^{\text{\rm iso}-}_{\mathscr{C}}(R)$.

We denote by $\text{\rm Der}^{\text{\rm iso}-}_{\mathscr{C}}(R)$ the derived
category of $\text{\rm Ch}^{\text{\rm iso}-}_{\mathscr{C}}(R)$, i.e. the
localisation of the homotopy category of $\text{\rm Ch}^{\text{\rm
iso}-}_{\mathscr{C}}(R)$ by the multiplicative system of morphisms represented
by quasi-isomorphisms. By the universality of localisation, the strongly exact
inclusion $\text{\rm Ch}^{\text{\rm
iso}-}_{\mathscr{C}}(R)\hookrightarrow\text{\rm Ch}^{-}_{\mathscr{C}}(R)$
induces a functor $F^{\text{\rm iso}}\colon\text{\rm Der}^{\text{\rm
iso}-}_{\mathscr{C}}(R)\to\text{\rm Der}^{-}_{\mathscr{C}}(R)$.

Suppose that $R$ is isolated at $0$. Then, for any $(M_{0},M_{1})\in\text{\rm
Ch}^{\text{\rm iso}-}_{\mathscr{C}}(R)^{2}$, the total complex $\text{\rm
Tot}(\mathscr{P}_{\mathscr{C}}^{R}(M_{0})\widehat{\otimes}{}_{R}\mathscr{P}_{\mathscr{C}}^{R}(M_{1}))$
of $M_{0}\widehat{\otimes}{}_{R}M_{1}$ with respect to the direct sum in
$\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ makes sense as an object of
$\text{\rm Ch}^{\text{\rm iso}-}_{\mathscr{C}}(R)$, because the direct sums
are essentially finite and we can apply Proposition 6.4 (i), Proposition 6.1
(iii), and Proposition 6.7 (i). The correspondence
$(M_{0},M_{1})\mapsto\text{\rm
Tot}(M_{0}\widehat{\otimes}{}_{R}^{\mathbb{L}}M_{1})$ gives a functor
$\widehat{\otimes}{}_{R}^{\text{\rm iso}\mathbb{L}}\colon\text{\rm
Der}^{\text{\rm iso}-}_{\mathscr{C}}(R)^{2}\to\text{\rm Der}^{\text{\rm
iso}-}_{\mathscr{C}}(R)$, which is an isolated analogue of
$\widehat{\otimes}{}_{R}^{\mathbb{L}}$, by Corollary 6.8 (ii).

######  Proposition 6.9.

If $R$ is isolated at $0$, then the tuple $(\text{\rm Der}^{\text{\rm
iso}-}_{\mathscr{C}}(R),\widehat{\otimes}{}_{R}^{\text{\rm iso}\mathbb{L}},R)$
naturally forms a symmetric monoidal category, and $F^{\text{\rm
iso}}\colon\text{\rm Der}^{\text{\rm iso}-}_{\mathscr{C}}(R)\to\text{\rm
Der}^{-}_{\mathscr{C}}(R)$ is a symmetric monoidal fully faithful functor.

###### Proof.

The first assertion follows from the symmetry and the associativity of the
total complex, and the fact that $F^{\text{\rm iso}}$ is a symmetric monoidal
functor immediately follows from the construction of
$\widehat{\otimes}{}_{R}^{\text{\rm iso}\mathbb{L}}$ using the same
representative as a chain complex. It remains to show that $F^{\text{\rm
iso}}$ is fully faithful. Using the dual of [Büh10] Corollary 10.22 (ii), it
is enough to check that whenever we have a strictly exact sequence

$\displaystyle 0\to M\to N\to L\to 0$

of $\text{\rm Mod}_{\mathscr{C}}(R)$ with $L\in\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$, then there there exists a commutative diagram

in $\text{\rm Mod}_{\mathscr{C}}(R)$ such that
$(M^{\prime},N^{\prime})\in\text{\rm Mod}_{\mathscr{C}}^{\text{\rm
iso}}(R)^{2}$, the top row is strictly exact, and the bottom row is the given
strict exact sequence. By Corollary 6.8 (ii), there exists a pair
$(\mathscr{P}_{L},\pi_{L})$ of a projective object $\mathscr{P}_{L}$ of
$\text{\rm Mod}_{\mathscr{C}}(R)$ that is isolated at $0$ and a strict
epimorphism $\pi_{L}\colon\mathscr{P}_{L}\to L$. By the lifting property of
projective objects, we have a commutative diagram of strictly exact sequences

and by Proposition 6.1 (i) we have that $\ker(\pi_{L})$ is isolated at $0$,
concluding the proof. ∎

######  Remark 6.10.

Although we have not studied the question in depth, we think that it is
probably false that the canonical functors $\text{\rm Der}^{\text{\rm
iso}}_{\mathscr{C}}(R)\to\text{\rm Der}_{\mathscr{C}}(R)$ and $\text{\rm
Der}^{\text{\rm iso}+}_{\mathscr{C}}(R)\to\text{\rm Der}^{+}_{\mathscr{C}}(R)$
are fully faithful.

Next, we study the relations between $\text{\rm C}_{\text{\rm fin}}^{\text{\rm
iso}}$ and $\widehat{\otimes}{}_{R}$. For any $(M_{0},M_{1})\in\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)^{2}$, we denote by
$A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ the natural morphism $\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}\to\text{\rm
C}_{\text{\rm fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})$ in $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$.

######  Theorem 6.11.

Let $(M_{0},M_{1})\in\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)^{2}$.
Then, the following hold:

  * (i)

The natural $R$-linear homomorphism $\varphi\colon\text{\rm C}_{\text{\rm
fin}}(X,M_{0})\otimes_{R}M_{1}\to\text{\rm C}_{\text{\rm
fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})$ is bijective.

  * (ii)

If $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$, then
$A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ is an isometric isomorphism.

  * (iii)

If $R$ is isolated at $0$ and there exists a finite subset $S\subset M_{1}$
such that the natural morphism $\ell^{\mathscr{C}}(S,R)\to M_{1}$ in
$\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ is a strict epimorphism,
then $A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ is an isomorphism.

  * (iv)

If $\mathscr{C}=\text{\rm Ab}^{\text{\rm A}}$, $X$ is $\mathbb{Z}$ equipped
with the discrete topology,
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, and $R$
is isolated at $0$, then
$A^{\mathscr{D}_{3}}_{R,\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)}$ is
not an isomorphism (cf. Example 6.6 (ii)).

###### Proof.

We show the assertion (i). The surjectivity of the $R$-linear homomorphism
$\varphi$ in the assertion (i) follows from Proposition 2.4 (i) and
Proposition 6.4 (i). We show the injectivity of $\varphi$. Let $f\in\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\otimes_{R}M_{1}$ with $\varphi(f)=0$. Take an
$(f_{i},m_{i})_{i=0}^{n}\in(\text{\rm C}_{\text{\rm fin}}(X,M_{0})\times
M_{1})^{n+1}$ with $n\in\mathbb{N}$ and $f=\sum_{i=0}^{n}f_{i}\otimes m_{i}$.
By Proposition 2.4 (ii), there exists a finite subset
$\mathscr{U}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$
such that $\mathscr{U}$ is a disjoint cover of $X$ and for any
$(i,U)\in\mathbb{N}\times\mathscr{U}$ with $i\leq n$, $f_{i}|_{U}$ is
constant. For each $(i,U)\in\mathbb{N}\times\mathscr{U}$ with $i\leq n$, we
denote by $m_{i,U}\in M_{0}$ the value of $f_{i}|_{U}$, which makes sense by
$U\neq\emptyset$. Since $\varphi(f)|_{U}$ is a constant map with value
$\sum_{i=0}^{n}m_{i,U}\otimes m_{i}$, we have $\sum_{i=0}^{n}m_{i,U}\otimes
m_{i}=0$ for any $U\in\mathscr{U}$ by $\varphi(f)=0$. For each
$U\in\mathscr{U}$, we denote by $i_{U}$ the $R$-linear homomorphism
$M_{0}\to\text{\rm C}_{\text{\rm fin}}(X,M_{0})$ assigning $1_{X,R,U}m$ to
each $m\in M_{0}$. We obtain

$\displaystyle f=\sum_{i=0}^{n}f_{i}\otimes
m_{i}=\sum_{i=0}^{n}\sum_{U\in\mathscr{U}}i_{U}(m_{i,U})\otimes
m_{i}=\sum_{U\in\mathscr{U}}\sum_{i=0}^{n}i_{U}(m_{i,U})\otimes m_{i}$
$\displaystyle=$
$\displaystyle\sum_{U\in\mathscr{U}}(i_{U}\otimes_{R}\text{\rm
id}_{M_{1}})\left(\sum_{i=0}^{n}m_{i,U}\otimes
m_{i}\right)=\sum_{U\in\mathscr{U}}(i_{U}\otimes_{R}\text{\rm
id}_{M_{1}})(0)=0.$

We show the assertion (ii). If $X=\emptyset$, then both objects of $\text{\rm
Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ are zero objects, which have unique
norms. Therefore, we may assume $X\neq\emptyset$. First, we show that
${\mathchoice{\left\|\varphi(f)\right\|}{\|\varphi(f)\|}{\|\varphi(f)\|}{\|\varphi(f)\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})}<{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}+\epsilon$ for any
$(\epsilon,f)\in(0,\infty)\times(\text{\rm C}_{\text{\rm
fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1})$. By the definition of the tensor
seminorm for $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$, there exists an
$(f_{i},m_{i})_{i=0}^{n}\in(\text{\rm C}_{\text{\rm fin}}(X,M_{0})\times
M_{1})^{n+1}$ with $n\in\mathbb{N}$ and $f=\sum_{i=0}^{n}f_{i}\otimes m_{i}$
such that
$\max_{i=0}^{n}{\mathchoice{\left\|f_{i}\right\|}{\|f_{i}\|}{\|f_{i}\|}{\|f_{i}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0})}{\mathchoice{\left\|m_{i}\right\|}{\|m_{i}\|}{\|m_{i}\|}{\|m_{i}\|}}_{M_{1}}<{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}+\epsilon$. Then, we
have

$\displaystyle{\mathchoice{\left\|\varphi(f)(x)\right\|}{\|\varphi(f)(x)\|}{\|\varphi(f)(x)\|}{\|\varphi(f)(x)\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}={\mathchoice{\left\|\sum_{i=0}^{n}f_{i}(x)\otimes
m_{i}\right\|}{\|\sum_{i=0}^{n}f_{i}(x)\otimes
m_{i}\|}{\|\sum_{i=0}^{n}f_{i}(x)\otimes
m_{i}\|}{\|\sum_{i=0}^{n}f_{i}(x)\otimes
m_{i}\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}\leq\max{}_{i=0}^{n}{\mathchoice{\left\|f_{i}(x)\right\|}{\|f_{i}(x)\|}{\|f_{i}(x)\|}{\|f_{i}(x)\|}}_{M_{0}}{\mathchoice{\left\|m_{i}\right\|}{\|m_{i}\|}{\|m_{i}\|}{\|m_{i}\|}}_{M_{1}}$
$\displaystyle\leq$
$\displaystyle\max{}_{i=0}^{n}{\mathchoice{\left\|f_{i}\right\|}{\|f_{i}\|}{\|f_{i}\|}{\|f_{i}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0})}{\mathchoice{\left\|m_{i}\right\|}{\|m_{i}\|}{\|m_{i}\|}{\|m_{i}\|}}_{M_{1}}<{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}+\epsilon$

for any $x\in X$, and hence
${\mathchoice{\left\|\varphi(f)\right\|}{\|\varphi(f)\|}{\|\varphi(f)\|}{\|\varphi(f)\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})}<{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}+\epsilon$. This
implies that $\varphi$ is a contraction.

Next, we show that
${\mathchoice{\left\|\varphi^{-1}(f)\right\|}{\|\varphi^{-1}(f)\|}{\|\varphi^{-1}(f)\|}{\|\varphi^{-1}(f)\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}<{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})}+\epsilon$ for any
$(\epsilon,f)\in(0,\infty)\times\text{\rm C}_{\text{\rm
fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})$. By Proposition 2.4 (ii), there
exists a pair $(\mathscr{U},e)$ of a finite subset
$\mathscr{U}\subset\text{\rm
CO}(X)\setminus{\mathchoice{\left\\{\emptyset\right\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}{\\{\emptyset\\}}}$
and a map $e\colon\mathscr{U}\to M_{0}\widehat{\otimes}{}_{R}M_{1}$ such that
$\mathscr{U}$ is a disjoint cover of $X$ and
$\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)=f$. Then, we have
${\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm C}_{\text{\rm
fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})}=\sup_{U\in\mathscr{U}}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}$.

For any $U\in\mathscr{U}$, there exists an
$(m_{U,0,i},m_{U,1,i})_{i=0}^{n_{U}}\in(M_{0}\times M_{1})^{n_{U}+1}$ with
$n_{U}\in\mathbb{N}$ such that $\sum_{i=0}^{n_{U}}m_{U,0,i}\otimes
m_{U,1,i}=e(U)$ by Proposition 6.4 (i). By the definition of the tensor
seminorm for $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$, we choose $n_{U}$
and $(m_{U,0,i},m_{U,1,i})_{i=0}^{n_{U}}$ so that the inequality
$\max_{i=0}^{n_{U}}{\mathchoice{\left\|m_{U,0,i}\right\|}{\|m_{U,0,i}\|}{\|m_{U,0,i}\|}{\|m_{U,0,i}\|}}_{M_{0}}{\mathchoice{\left\|m_{U,1,i}\right\|}{\|m_{U,1,i}\|}{\|m_{U,1,i}\|}{\|m_{U,1,i}\|}}_{M_{1}}<{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}+\epsilon$
holds. Then, we have

$\displaystyle{\mathchoice{\left\|\varphi^{-1}(f)\right\|}{\|\varphi^{-1}(f)\|}{\|\varphi^{-1}(f)\|}{\|\varphi^{-1}(f)\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}={\mathchoice{\left\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)\right)\right\|}{\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)\right)\|}{\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)\right)\|}{\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}e(U)\right)\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}$ $\displaystyle=$
$\displaystyle{\mathchoice{\left\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}\sum_{i=0}^{n_{U}}m_{U,0,i}\otimes
m_{U,1,i}\right)\right\|}{\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}\sum_{i=0}^{n_{U}}m_{U,0,i}\otimes
m_{U,1,i}\right)\|}{\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}\sum_{i=0}^{n_{U}}m_{U,0,i}\otimes
m_{U,1,i}\right)\|}{\|\varphi^{-1}\left(\sum_{U\in\mathscr{U}}1_{X,R,U}\sum_{i=0}^{n_{U}}m_{U,0,i}\otimes
m_{U,1,i}\right)\|}}_{\text{\rm C}_{\text{\rm
fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}$ $\displaystyle=$
$\displaystyle{\mathchoice{\left\|\sum_{U\in\mathscr{U}}\sum_{i=0}^{n_{U}}(1_{X,R,U}m_{U,0,i})\otimes
m_{U,1,i}\right\|}{\|\sum_{U\in\mathscr{U}}\sum_{i=0}^{n_{U}}(1_{X,R,U}m_{U,0,i})\otimes
m_{U,1,i}\|}{\|\sum_{U\in\mathscr{U}}\sum_{i=0}^{n_{U}}(1_{X,R,U}m_{U,0,i})\otimes
m_{U,1,i}\|}{\|\sum_{U\in\mathscr{U}}\sum_{i=0}^{n_{U}}(1_{X,R,U}m_{U,0,i})\otimes
m_{U,1,i}\|}}_{\text{\rm C}_{\text{\rm
fin}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}}$ $\displaystyle\leq$
$\displaystyle\sup_{U\in\mathscr{U}}\textstyle\max_{i=0}^{n_{U}}\displaystyle{\mathchoice{\left\|1_{X,R,U}m_{U,0,i}\right\|}{\|1_{X,R,U}m_{U,0,i}\|}{\|1_{X,R,U}m_{U,0,i}\|}{\|1_{X,R,U}m_{U,0,i}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,M_{0})}{\mathchoice{\left\|m_{U,1,i}\right\|}{\|m_{U,1,i}\|}{\|m_{U,1,i}\|}{\|m_{U,1,i}\|}}_{M_{1}}$
$\displaystyle=$
$\displaystyle\sup_{U\in\mathscr{U}}\textstyle\max_{i=0}^{n_{U}}\displaystyle{\mathchoice{\left\|m_{U,0,i}\right\|}{\|m_{U,0,i}\|}{\|m_{U,0,i}\|}{\|m_{U,0,i}\|}}_{M_{0}}{\mathchoice{\left\|m_{U,1,i}\right\|}{\|m_{U,1,i}\|}{\|m_{U,1,i}\|}{\|m_{U,1,i}\|}}_{M_{1}}<\sup_{U\in\mathscr{U}}{\mathchoice{\left\|e(U)\right\|}{\|e(U)\|}{\|e(U)\|}{\|e(U)\|}}_{M_{0}\widehat{\otimes}{}_{R}M_{1}}+\epsilon$
$\displaystyle<$
$\displaystyle{\mathchoice{\left\|f\right\|}{\|f\|}{\|f\|}{\|f\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})}+\epsilon$

by $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$. This implies that
$\varphi^{-1}$ is a contraction.

We show the assertion (iii). We may assume $0\notin S$. We denote by $\pi$ the
natural morphism $\ell^{\mathscr{C}}(S,R)\to M$ in $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$. We denote by $E$ the null sequence

$\displaystyle\ker(\pi)\hookrightarrow\ell^{\mathscr{C}}(S,R)\stackrel{{\scriptstyle\pi}}{{\to}}M\to
0$

of $\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$, which is strictly
coexact by the assumption. We denote by $\varphi$ the natural morphism
$\text{\rm C}_{\text{\rm fin}}(X,R)\widehat{\otimes}{}_{R}E\to\text{\rm
C}_{\text{\rm fin}}(X,E)$ of null sequences of $\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$, which is termwise bijective by the assertion (i). By
Proposition 6.3, $\text{\rm C}_{\text{\rm fin}}(X,E)$ is strictly coexact. By
the strong right exactness of $\text{\rm C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}_{R}(\cdot)$, $\text{\rm C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}_{R}E$ is strictly coexact. For each
$r\in(0,\infty)$, we denote by $Rr$ the regular $R$-module object rescaled by
$r$. We consider the commutative diagram

of $\text{\rm Mod}_{\mathscr{C}}(R)$.The horizontal arrows are isomorphisms by
the additivity of the functors and the right vertical arrow is an isomorphism
by the definition of rescaling. This implies that the left vertical arrow is
an isomorphism, and hence $\varphi$ is an isomorphism at the second term.
Since $\varphi$ is a termwise bijective morphism between strictly coexact
sequences and is an isomorphism at the second term, it is an isomorphism at
the first term. By the universality of cokernels, $\varphi$ is an isomorphism
at the third term, i.e. $A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ is an isomorphism.

We show the assertion (iv). For this purpose, it suffices to show that the
inverse of the underlying $R$-linear homomorphism $\psi$ of
$A^{\mathscr{D}_{3}}_{R,\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)}$ is
unbounded. We denote by $\iota$ the canonical embedding
$(\mathbb{Z},1_{\mathbb{Z}})\hookrightarrow\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)$.
Since $R$ is isolated at $0$, there exists an $\epsilon\in(0,\infty)$ such
that ${\mathchoice{\left\\{r\in
R\middle|{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}<\epsilon\right\\}}{\\{r\in
R\mid{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}<\epsilon\\}}{\\{r\in
R\mid{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}<\epsilon\\}}{\\{r\in
R\mid{\mathchoice{\left\|r\right\|}{\|r\|}{\|r\|}{\|r\|}}<\epsilon\\}}}={\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$.
Since $R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, we
have ${\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}\neq 0$. It
suffices to show that $\psi^{-1}$ is of operator norm
$\geq{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}^{-1}\epsilon^{2}n$
for any $n\in\mathbb{N}$. Put
$f_{n}\coloneqq\sum_{i=0}^{n}1_{X,R,{\mathchoice{\left\\{n\right\\}}{\\{n\\}}{\\{n\\}}{\\{n\\}}}}\otimes\iota(n)\in\text{\rm
C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}_{R}\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)$.
Since $\psi(f_{n})$ assigns $\iota(i)$ to each $i\in X$ with $0\leq i\leq n$
and $0$ to each $i\in X$ with $i<0$ or $i>n$, we have

$\displaystyle{\mathchoice{\left\|\psi(f_{n})\right\|}{\|\psi(f_{n})\|}{\|\psi(f_{n})\|}{\|\psi(f_{n})\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R))}=\textstyle\max_{i=0}^{n}\displaystyle{\mathchoice{\left\|\iota(i)\right\|}{\|\iota(i)\|}{\|\iota(i)\|}{\|\iota(i)\|}}_{\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)}=\textstyle\max_{i=0}^{n}\displaystyle{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}1_{\mathbb{Z}}(i)={\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}\neq
0.$

For any $(g_{h},a_{h})_{h=0}^{m}\in(\text{\rm C}_{\text{\rm
fin}}(X,R),\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R))^{m+1}$ with
$m\in\mathbb{N}$ and $\sum_{h=0}^{m}g_{h}\otimes a_{h}=f_{n}$, we have $m\geq
n$ by the computation of the rank of the $(n\times n)$-matrix corresponding to
$\sum_{h=0}^{m}(g_{h}(i))_{i=0}^{n}\otimes(a_{h}(j))_{j=0}^{n}\in\mathbb{Z}^{n+1}\otimes_{\mathbb{Z}}\mathbb{Z}^{n+1}$.
This implies
${\mathchoice{\left\|f_{n}\right\|}{\|f_{n}\|}{\|f_{n}\|}{\|f_{n}\|}}_{\text{\rm
C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}_{R}\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)}\geq\epsilon^{2}n$,
because every non-zero element of $\text{\rm C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}_{R}\ell^{\mathscr{C}}((\mathbb{Z},1_{\mathbb{Z}}),R)$
is of norm $\geq\epsilon^{2}$. This implies that $\psi^{-1}$ is of operator
norm
$\geq{\mathchoice{\left\|1\right\|}{\|1\|}{\|1\|}{\|1\|}}_{R}^{-1}\epsilon^{2}n$.
∎

######  Remark 6.12.

Theorem 6.11 (i) just implies that $A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ is
bijective but does not imply that it is an isomorphism, as the open mapping
theorem does not necessarily hold for Banach $R$-modules. Indeed, although it
is an isometric isomorphism when $\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$
by Theorem 6.11 (ii), it is not necessarily an isomorphism when
$\mathscr{C}=\text{\rm Ab}^{\text{\rm A}}$ by Theorem 6.11 (iv). The condition
on $M_{1}$ in Theorem 6.11 (iii) is not equivalent to the condition that
$M_{1}$ is finitely generated as an $R$-module. Indeed, if
$R=\mathbb{Z}_{\infty}$ and $M_{1}=\mathbb{Z}_{\text{\rm triv}}$ (cf. Example
2.1 (i) and (iii)), then $M_{1}$ is finitely generated but does not satisfy
the condition. It is also remarkable that the bijectivity is a special feature
of the isolated setting and the local field setting (cf. [Mih14] Theorem
4.12). For example, when $\mathscr{C}=\text{\rm Ab}^{\text{\rm A}}$, $X$ is
the residue field of $\mathbb{C}_{p}$, $R=M_{0}=\mathbb{Z}_{\infty}$, and
$M_{1}=\mathbb{C}_{p}$, then the morphism $\text{\rm C}_{\text{\rm
bd}}(X,M_{0})\widehat{\otimes}{}_{R}M_{1}\to\text{\rm C}_{\text{\rm
bd}}(X,M_{0}\widehat{\otimes}{}_{R}M_{1})$ in $\text{\rm
Mod}_{\mathscr{C}}(\mathbb{Z}_{\infty})$ analogous to
$A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ is not surjective. Indeed, every function
$X\to M_{0}\widehat{\otimes}{}_{R}M_{1}\cong\mathbb{C}_{p}$ in the image of
$A^{\mathscr{D}_{3}}_{M_{0},M_{1}}$ can be approximated by functions whose
images are contained in finitely generated subfields of $\mathbb{C}_{p}$, but
the Teichmüller embedding $X\hookrightarrow\mathbb{C}_{p}$ cannot be
approximated by such functions.

As an immediate consequence of Proposition 2.2 and Theorem 6.11 (iii), we
obtain the following corollary.

######  Corollary 6.13.

Suppose that $R$ is isolated at $0$. Let $M\in\text{\rm Mod}^{\text{\rm
iso}}_{\mathscr{C}}(R)$. For any ideal $I\subset R$, $R/I$ is an object of
$\text{\rm Mod}^{\text{\rm iso}}_{\mathscr{C}}(R)$ with respect to the
quotient seminorm, and $A^{\mathscr{D}_{3}}_{M,R/I}$ is an isomorphism. In
particular, for any $x\in\mathscr{M}(R)$, if ${\mathchoice{\left\\{a\in
R\middle|x(a)=0\right\\}}{\\{a\in R\mid x(a)=0\\}}{\\{a\in R\mid
x(a)=0\\}}{\\{a\in R\mid x(a)=0\\}}}$ is a maximal ideal, then
$A^{\mathscr{D}_{3}}_{M,\kappa(x)}$ is an isomorphism, where $\kappa(x)$
denotes the completed residue field at $x$.

##  7 Schauder bases and $\mathbb{Z}$-forms

In this last section, we study some further properties of the algebras
$\text{\rm C}_{\text{\rm fin}}(X,R)$ for any topological space $X$ and
specific choices of the Banach ring $R$. We show in Proposition 7.1 that
$\text{\rm C}(\zeta(X),R)$ is free as an $R$-module if $R$ is isolated at $0$,
and topologically free as a Banach $R$-module if $R$ is non-Archimedean by
constructing (algebraic and topological) bases in both cases consisting of
characteristic functions of clopen subsets. This extends well-known results
about the existence of an orthonormal Schauder basis of $\text{\rm
C}(\zeta(X),R)$ when $R$ is a complete non-Archimedean valued field (cf.
[Mon70] IV 3 Corollaire 1, [BGR84] 2.5.2 Lemma 2, and the proof of [Sch02]
Proposition 10.1). We also show in Example 7.9 in the specific case
$X=\mathbb{Z}_{p}$ that the algebra $\text{\rm C}_{\text{\rm
fin}}(\mathbb{Z}_{p},R)$ has another orthonormal Schauder basis, called the
Mahler basis, whose properties under base change are very different from the
ones of the orthonormal Schauder basis coming from the characteristic
functions of clopen subsets. We conclude by proving the Stone–Weierstrass
Theorem for $\text{\rm C}_{\text{\rm fin}}(X,R)$, under a suitable hypothesis
on $R$ in Theorem 7.11.

Let $M\in\text{\rm Mod}_{\mathscr{C}}(R)$. A subset $S\subset M$ is said to be
a $\mathscr{C}$-orthonormal Schauder $R$-linear basis of $M$ if $S$ consists
of elements of norm $1$ and the natural $R$-linear homomorphism $R^{\oplus
S}\to M$ extends to an isometric $R$-linear isomorphism
$\ell^{\mathscr{C}}(S,R)\to M$. We recall that for a $U\in\text{\rm CO}(X)$,
$\text{\rm cl}_{\zeta,X}(U)$ denotes the unique clopen subset of $\zeta(X)$
satisfying $\iota_{\zeta,X}^{-1}(\text{\rm cl}_{\zeta,X}(U))=U$ (cf. Corollary
2.7). A monoid object $A$ of $\text{\rm Ab}^{\text{\rm nA}}$ is said to be
with submultiplicative norm (in the sense of [BM21] p. 6) if the canonical
morphisms $\mathbb{Z}_{\text{\rm triv}}\to A$ and $A\widehat{\otimes}{}A\to A$
defining the monoid structure are contractions, or equivalently, $A$ is a
monoid object of the symmetric subcategory of $\text{\rm Ab}^{\text{\rm nA}}$
with the same class of objects whose class of morphisms consists of
contracting morphisms. As a consequence of Theorem 6.11 (i), we obtain a
generalisation of [Nöb68] Satz 1 (cf. [Sch19] Theorem 5.4).

######  Proposition 7.1.

There exists an $\mathscr{F}\subset\text{\rm CO}(X)$ satisfying the following:

  * (i)

For any monoid object $A$ of $\text{\rm Ab}^{\text{\rm A}}$ isolated at $0$,
i.e. a Banach ring in the sense in [BM21] p. 6 isolated at $0$, $\text{\rm
C}_{\text{\rm fin}}(X,A)$ is a free $A$-module with $A$-linear basis
${\mathchoice{\left\\{1_{X,A,U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}}$.

  * (ii)

For any monoid object $A$ of $\text{\rm Ab}^{\text{\rm nA}}$ with
submultiplicative norm isolated at $0$, the $A$-linear basis
${\mathchoice{\left\\{1_{X,A,U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}}$ of $\text{\rm C}_{\text{\rm fin}}(X,A)$ is an $\text{\rm
Ab}^{\text{\rm nA}}$-orthonormal Schauder $A$-linear basis of $\text{\rm
C}_{\text{\rm fin}}(X,A)$.

  * (iii)

For any complete non-Archimedean valued ring (resp. field) $A$,
${\mathchoice{\left\\{1_{\zeta(X),A,V}\middle|V\in\mathscr{F}^{\prime}\right\\}}{\\{1_{\zeta(X),A,V}\mid
V\in\mathscr{F}^{\prime}\\}}{\\{1_{\zeta(X),A,V}\mid
V\in\mathscr{F}^{\prime}\\}}{\\{1_{\zeta(X),A,V}\mid
V\in\mathscr{F}^{\prime}\\}}}$ is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $A$-linear basis of $\text{\rm C}(\zeta(X),A)$,
where $\mathscr{F}^{\prime}$ denotes the image of $\mathscr{F}$ by $\text{\rm
cl}_{\zeta,X}$.

###### Proof.

We show the assertion (i). When $X=\emptyset$, then $\mathscr{F}$ can be taken
as $\emptyset$. Therefore, we may assume $X\neq\emptyset$. By $X\neq\emptyset$
and [Nöb68] Satz 1, there exists an $\mathscr{F}\subset\text{\rm CO}(X)$ such
that
${\mathchoice{\left\\{1_{X,\mathbb{Z}_{\infty},U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,\mathbb{Z}_{\infty},U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\infty},U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\infty},U}\mid U\in\mathscr{F}\\}}}$
is a $\mathbb{Z}$-linear basis of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\infty})$. Since $\mathbb{Z}_{\infty}$ is an initial
object in the category of monoids of $\text{\rm Ab}^{\text{\rm A}}$ and monoid
homomorphisms,
${\mathchoice{\left\\{1_{X,A,U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}}$ is an $A$-linear basis of $\text{\rm C}_{\text{\rm
fin}}(X,A)$ for any monoid $A$ of $\text{\rm Ab}^{\text{\rm A}}$ by Theorem
6.11 (i) applied to
$(R,M_{0},M_{1})=(\mathbb{Z}_{\infty},\mathbb{Z}_{\infty},A)$.

We show the assertion (ii). By the assertion (i), there exists an
$\mathscr{F}\subset\text{\rm CO}(X)$ such that
${\mathchoice{\left\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\mid U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm triv}},U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm triv}},U}\mid
U\in\mathscr{F}\\}}}$ is a $\mathbb{Z}$-linear basis of $\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})$. Since
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})}$ is the trivial norm, any
$\mathbb{Z}$-linear basis of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $\mathbb{Z}_{\text{\rm triv}}$-linear basis. By
Theorem 6.11 (ii), the canonical morphism
$\mathscr{A}^{\mathscr{D}_{3}}_{\mathbb{Z}_{\text{\rm triv}},A}\colon\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm
triv}})\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm triv}}}A\to\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm
triv}}\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm triv}}}A)$ is an isometric
isomorphism. Since the unitor $\mathbb{Z}_{\text{\rm
triv}}\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm triv}}}A\to A$ is an isometry
by the assumption of the submultiplicativity, the $A$-linear basis
${\mathchoice{\left\\{1_{X,A,U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid U\in\mathscr{F}\\}}{\\{1_{X,A,U}\mid
U\in\mathscr{F}\\}}}$ of $\text{\rm C}_{\text{\rm fin}}(X,A)$ is an $\text{\rm
Ab}^{\text{\rm nA}}$-orthonormal Schauder $A$-linear basis.

We show the assertion (iii). Let $k$ be a complete non-Archimedean valued
field. Put $O_{k}\coloneqq{\mathchoice{\left\\{c\in
k\middle|{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}\leq 1\right\\}}{\\{c\in
k\mid{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}\leq 1\\}}{\\{c\in
k\mid{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}\leq 1\\}}{\\{c\in
k\mid{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}\leq 1\\}}}$,
$m_{k}\coloneqq{\mathchoice{\left\\{c\in
O_{k}\middle|{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}<1\right\\}}{\\{c\in
O_{k}\mid{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}<1\\}}{\\{c\in
O_{k}\mid{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}<1\\}}{\\{c\in
O_{k}\mid{\mathchoice{\left|c\right|}{|c|}{|c|}{|c|}}<1\\}}}$, and
$\kappa_{k}\coloneqq O_{k}/m_{k}$. By the assertion (i),
${\mathchoice{\left\\{1_{X,\kappa_{k},U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,\kappa_{k},U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,\kappa_{k},U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,\kappa_{k},U}\mid U\in\mathscr{F}\\}}}$ forms a
$\kappa_{k}$-linear basis of $\text{\rm C}_{\text{\rm fin}}(X,\kappa_{k})$.
Since $\kappa_{k}$ is isolated at $0$,
${\mathchoice{\left\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\middle|U\in\mathscr{F}\right\\}}{\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}}$ forms a $\kappa_{k}$-linear basis
of $\text{\rm C}(\zeta(X),\kappa_{k})$ by Proposition 2.6 and Corollary 2.7.
For any $U_{0}\in\text{\rm CO}(X)$, $1_{X,k,U_{0}}$ belongs to the Abelian
subgroup of $k^{X}$ generated by
${\mathchoice{\left\\{1_{X,k,U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,k,U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,k,U}\mid U\in\mathscr{F}\\}}{\\{1_{X,k,U}\mid
U\in\mathscr{F}\\}}}$ by the definition of $\mathscr{F}$. Therefore, for any
$\overline{U}{}_{0}\in\text{\rm CO}(\zeta(X))$,
$1_{\zeta(X),k,\overline{U}{}_{0}}$ belongs to the Abelian subgroup of
$k^{\zeta(X)}$ generated by ${\mathchoice{\left\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\middle|U\in\mathscr{F}\right\\}}{\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}}$ by Corollary 2.7. This implies
that ${\mathchoice{\left\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\middle|U\in\mathscr{F}\right\\}}{\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}}$ generates the Abelian subgroup of
$\text{\rm C}(\zeta(X),k)$ generated by
${\mathchoice{\left\\{1_{\zeta(X),k,\overline{U}{}}\middle|\overline{U}{}\in\text{\rm
CO}(\zeta(X))\right\\}}{\\{1_{\zeta(X),k,\overline{U}{}}\mid\overline{U}{}\in\text{\rm
CO}(\zeta(X))\\}}{\\{1_{\zeta(X),k,\overline{U}{}}\mid\overline{U}{}\in\text{\rm
CO}(\zeta(X))\\}}{\\{1_{\zeta(X),k,\overline{U}{}}\mid\overline{U}{}\in\text{\rm
CO}(\zeta(X))\\}}}$ again by Corollary 2.7, and hence generates a dense
$k$-subalgebra of $\text{\rm C}(\zeta(X),k)$ by [Mih14] Theorem 4.12.

Therefore, it suffices to show
${\mathchoice{\left\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\right\|}{\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\|}{\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\|}{\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\|}}_{\text{\rm
C}(\zeta(X),k)}=\sup_{U\in\mathscr{F}}{\mathchoice{\left|a_{U}\right|}{|a_{U}|}{|a_{U}|}{|a_{U}|}}$
for any $(a_{U})_{U\in\mathscr{F}}\in k^{\oplus\mathscr{F}}$. By the
finiteness of
${\mathchoice{\left\\{a_{U}\middle|U\in\mathscr{F}\right\\}}{\\{a_{U}\mid
U\in\mathscr{F}\\}}{\\{a_{U}\mid U\in\mathscr{F}\\}}{\\{a_{U}\mid
U\in\mathscr{F}\\}}}\subset k$, there exists a $U_{0}\in\mathscr{F}$ such that
${\mathchoice{\left|a_{U}\right|}{|a_{U}|}{|a_{U}|}{|a_{U}|}}\leq{\mathchoice{\left|a_{U_{0}}\right|}{|a_{U_{0}}|}{|a_{U_{0}}|}{|a_{U_{0}}|}}$
for any $U\in\mathscr{F}$. If $a_{U_{0}}=0$, then both hand sides are $0$.
Therefore, we may assume $a_{U_{0}}\neq 0$. We have
$(a_{U_{0}}^{-1}a_{U})_{U\in\mathscr{F}}\in O_{k}^{\oplus\mathscr{F}}\setminus
m_{k}^{\oplus\mathscr{F}}$, and hence
$((a_{U_{0}}^{-1}a_{U})+m_{k})_{U\in\mathscr{F}}$ is a non-zero element of
$\kappa_{k}^{\oplus\mathscr{F}}$. Since
${\mathchoice{\left\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\middle|U\in\mathscr{F}\right\\}}{\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),\kappa_{k},\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}}$ is a $\kappa_{k}$-linear basis of
$\text{\rm C}(\zeta(X),\kappa_{k})$, we have
$\sum_{U\in\mathscr{F}}((a_{U_{0}}^{-1}a_{U})+m_{k})1_{\zeta(X),\kappa_{k},\text{\rm
cl}{\zeta,X}(U)}\neq 0$. Therefore, there exists an $x\in\zeta(X)$ such that
$\sum_{U\in\mathscr{F}}((a_{U_{0}}^{-1}a_{U})+m_{k})1_{\zeta(X),\kappa_{k},\text{\rm
cl}{\zeta,X}(U)}(x)\neq 0$, or equivalently, $\sum_{U\in\mathscr{F}\cap
x}((a_{U_{0}}^{-1}a_{U})+m_{k})\neq 0$ by the definition of $\zeta(X)$ as the
Stone space of $\text{\rm CO}(X)$. This implies $\sum_{U\in\mathscr{F}\cap
x}a_{U_{0}}^{-1}a_{U}\in O_{k}\setminus m_{k}$, and hence
${\mathchoice{\left|\sum_{U\in\mathscr{F}\cap
x}a_{U}\right|}{|\sum_{U\in\mathscr{F}\cap
x}a_{U}|}{|\sum_{U\in\mathscr{F}\cap x}a_{U}|}{|\sum_{U\in\mathscr{F}\cap
x}a_{U}|}}={\mathchoice{\left|a_{U_{0}}\right|}{|a_{U_{0}}|}{|a_{U_{0}}|}{|a_{U_{0}}|}}$.
We obtain

$\displaystyle{\mathchoice{\left\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\right\|}{\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\|}{\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\|}{\|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}\|}}_{\text{\rm
C}(\zeta(X),k)}\geq{\mathchoice{\left|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}(x)\right|}{|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}(x)|}{|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}(x)|}{|\sum_{U\in\mathscr{F}}a_{U}1_{\zeta(X),k,\text{\rm
cl}_{\zeta,X}(U)}(x)|}}={\mathchoice{\left|\sum_{U\in\mathscr{F}\cap
x}a_{U}\right|}{|\sum_{U\in\mathscr{F}\cap
x}a_{U}|}{|\sum_{U\in\mathscr{F}\cap x}a_{U}|}{|\sum_{U\in\mathscr{F}\cap
x}a_{U}|}}$ $\displaystyle=$
$\displaystyle{\mathchoice{\left|a_{U_{0}}\right|}{|a_{U_{0}}|}{|a_{U_{0}}|}{|a_{U_{0}}|}}=\sup_{U\in\mathscr{F}}{\mathchoice{\left|a_{U}\right|}{|a_{U}|}{|a_{U}|}{|a_{U}|}}.$

∎

Suppose that $R$ is isolated at $0$. For an $A\in\text{\rm
Alg}_{\mathscr{C}}(R)$, we denote by $B^{\mathscr{D}_{3}}_{A}$ the composition
$\text{\rm C}_{\text{\rm fin}}(X,R)\widehat{\otimes}{}_{R}A\to\text{\rm
C}(\zeta(X),A)$ of the inverse of the restriction map $\text{\rm
C}(\zeta(X),R)\widehat{\otimes}{}_{R}A\to\text{\rm C}_{\text{\rm
fin}}(X,R)\widehat{\otimes}{}_{R}A$, which is an isometric isomorphism in
$\text{\rm Alg}_{\mathscr{C}}(A)$ by Proposition 2.6, and the natural morphism
$\text{\rm C}(\zeta(X),R)\widehat{\otimes}{}_{R}A\to\text{\rm C}(\zeta(X),A)$
in $\text{\rm Alg}_{\mathscr{C}}(A)$.

######  Corollary 7.2.

Let $R$ be a monoid object of $\text{\rm Ab}^{\text{\rm nA}}$ with
submultiplicative norm isolated at $0$ or a complete non-Archimedean valued
ring (resp. field). Then $B^{(\text{\rm Ab}^{\text{\rm
nA}},X,\mathbb{Z}_{\text{\rm triv}})}_{R}$ is an isometric $R$-linear
isomorphism.

###### Proof.

The claim for the isolated case follows from Proposition 2.6 and Proposition
7.1 (ii), because $B^{(\text{\rm Ab}^{\text{\rm nA}},X,\mathbb{Z}_{\text{\rm
triv}})}_{R}$ coincides with the composite of
$A^{\mathscr{D}_{3}}_{\mathbb{Z}_{\text{\rm triv}},R}$ and the inverse of the
restriction map $\text{\rm C}(\zeta(X),R)\to\text{\rm C}_{\text{\rm
fin}}(X,R)$. Suppose that $R$ is a complete valued ring (resp. field). By
Proposition 7.1 (iii), there exists an $\mathscr{F}\subset\text{\rm CO}(X)$
such that ${\mathchoice{\left\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\mid U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm triv}},U}\mid
U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm triv}},U}\mid
U\in\mathscr{F}\\}}}$ is a $\mathbb{Z}$-linear basis of $\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ and
${\mathchoice{\left\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\middle|U\in\mathscr{F}\right\\}}{\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}}$ is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $R$-linear basis of $\text{\rm C}(\zeta(X),R)$.
Since
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})}$ is the trivial norm, any
$\mathbb{Z}$-linear basis of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $\mathbb{Z}_{\text{\rm triv}}$-linear basis.
Therefore, the natural $\mathbb{Z}_{\text{\rm triv}}$-linear homomorphism
$\mathbb{Z}_{\text{\rm triv}}^{\oplus\mathscr{F}}\to\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ gives an isometric
$\mathbb{Z}_{\text{\rm triv}}$-linear isomorphism $\ell^{\text{\rm
Ab}^{\text{\rm nA}}}(\mathscr{F},\mathbb{Z}_{\text{\rm triv}})\to\text{\rm
C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})$, and
${\mathchoice{\left\\{1_{X,\mathbb{Z}_{\text{\rm triv}},U}\otimes
1\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes 1\mid U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes 1\mid U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes 1\mid U\in\mathscr{F}\\}}}$ is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $R$-linear basis of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm
triv}})\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm triv}}}R$ by Proposition 6.7
(ii) applied to $(\text{\rm Ab}^{\text{\rm nA}},\mathbb{Z}_{\text{\rm
triv}},R)$. Since $B^{(\text{\rm Ab}^{\text{\rm nA}},X,\mathbb{Z}_{\text{\rm
triv}})}_{R}$ sends the $\text{\rm Ab}^{\text{\rm nA}}$-orthonormal Schauder
$R$-linear basis ${\mathchoice{\left\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes
1\middle|U\in\mathscr{F}\right\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes 1\mid U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes 1\mid U\in\mathscr{F}\\}}{\\{1_{X,\mathbb{Z}_{\text{\rm
triv}},U}\otimes 1\mid U\in\mathscr{F}\\}}}$ to the $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $R$-linear basis
${\mathchoice{\left\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\middle|U\in\mathscr{F}\right\\}}{\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}{\\{1_{\zeta(X),R,\text{\rm
cl}_{\zeta,X}(U)}\mid U\in\mathscr{F}\\}}}$, it is an isometric $R$-linear
isomorphism. ∎

The existence of an $\text{\rm Ab}^{\text{\rm nA}}$-orthonormal Schauder
$R$-linear basis for $\text{\rm C}_{\text{\rm fin}}(X,R)$ has the following
remarkable consequence:

######  Corollary 7.3.

Let $R$ be a monoid object of $\text{\rm Ab}^{\text{\rm nA}}$ isolated at $0$.
Then, for all topological spaces $X$, the Banach $R$-module $\text{\rm
C}_{\text{\rm fin}}(X,R)$ is strongly flat as an object of $\text{\rm
Mod}_{\text{\rm Ab}^{\text{\rm nA}}}(R)$.

###### Proof.

We may assume that the canonical morphisms $\mathbb{Z}_{\text{\rm triv}}\to R$
and $R\widehat{\otimes}{}R\to R$ are contractions by a standard argument (cf.
[BM21] p. 6). Therefore, by Proposition 7.1 (ii), $\text{\rm C}_{\text{\rm
fin}}(X,R)$ admits an $\text{\rm Ab}^{\text{\rm nA}}$-orthonormal Schauder
$R$-linear basis, and hence we have that $\text{\rm C}_{\text{\rm
fin}}(X,R)\cong\ell^{\text{\rm Ab}^{\text{\rm nA}}}((S,1_{S}),R)$ for a set
$S$ (cf. Example 6.6 (ii)). This implies that $\text{\rm C}_{\text{\rm
fin}}(X,R)$ is a projective object of $\text{\rm Mod}_{\text{\rm
Ab}^{\text{\rm nA}}}(R)$ and projective objects are strongly flat by [BK20]
Proposition 3.11. ∎

Let us fix two monoid objects $R$ and $R_{0}$ of $\mathscr{C}$ and a morphism
$R_{0}\to R$ of monoids. For an $A\in\text{\rm Alg}_{\mathscr{C}}(R)$, a
$(\mathscr{C},R_{0})$-form of $A$ over $R_{0}$ is an object
$A_{R_{0}}\in\text{\rm Alg}_{\mathscr{C}}(R_{0})$ equipped with an isomorphism
$\varphi_{A}\colon A_{R_{0}}\widehat{\otimes}{}_{R_{0}}R\to A$ in $\text{\rm
Alg}_{\mathscr{C}}(R)$. The most interesting case is when
$(\mathscr{C},R_{0})=(\text{\rm Ab}^{\text{\rm A}},\mathbb{Z}_{\infty})$ or
$(\text{\rm Ab}^{\text{\rm nA}},\mathbb{Z}_{\text{\rm triv}})$, in which case
the $(\mathscr{C},R_{0})$-forms of monoid objects of $\mathscr{C}$ have an
absolute nature due to the initiality of $R_{0}$. Indeed, the symmetric
monoidal structure $\widehat{\otimes}{}$ of $\mathscr{C}$ is naturally
identified with $\widehat{\otimes}{}_{\mathbb{Z}_{\infty}}$ when
$\mathscr{C}=\text{\rm Ab}^{\text{\rm A}}$, and with
$\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm triv}}}$ when
$\mathscr{C}=\text{\rm Ab}^{\text{\rm nA}}$. We will talk generically of
$\mathbb{Z}$-forms of a Banach algebra when both $(\text{\rm Ab}^{\text{\rm
A}},\mathbb{Z}_{\infty})$ or $(\text{\rm Ab}^{\text{\rm
nA}},\mathbb{Z}_{\text{\rm triv}})$ are considered or allowed.

######  Example 7.4.

  * (i)

By the proof of Corollary 7.2, we know that $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ is a canonical $(\text{\rm
Ab}^{\text{\rm nA}},\mathbb{Z}_{\text{\rm triv}})$-form of $\text{\rm
C}(\zeta(X),k)$ over $k$ for any complete non-Archimedean valued field $k$.

  * (ii)

On the contrary, $\text{\rm C}_{\text{\rm fin}}(X,\mathbb{Z}_{\infty})$ is not
necessarily a $(\text{\rm Ab}^{\text{\rm A}},\mathbb{Z}_{\infty})$-form of
$\text{\rm C}_{\text{\rm fin}}(X,\mathbb{R})$ of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{C})$ if $X$ is a connected compact Hausdorff topological space
with at least two points. For example, the image of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\infty})\widehat{\otimes}{}_{\mathbb{Z}_{\infty}}\mathbb{R}$
is a proper subalgebra of $\text{\rm C}_{\text{\rm fin}}(X,\mathbb{R})$.

  * (iii)

The $\mathbb{Z}$-forms of a fixed Banach algebra $A\in\text{\rm
Alg}_{\mathscr{C}}(R)$ are not necessarily unique. Later on in this section we
will present an important example of a Banach algebra admitting two non-
isomorphic $\mathbb{Z}$-forms. These amounts to two different Schauder bases
for the Banach algebra $\text{\rm C}(\zeta(X),R)$. See Example 7.9 for
details.

It is important to recognise when a Banach algebra admits a $\mathbb{Z}$-form.
We show that the notion of a $\mathbb{Z}$-form is essentially equivalent to
that of an orthonormal Schauder basis whose multiplication table is a matrix
with coefficients in $\mathbb{Z}$.

######  Proposition 7.5.

Let $A\in\text{\rm Alg}_{\text{\rm Ab}^{\text{\rm nA}}}(R)$.

  * (i)

Let $A_{\mathbb{Z}}$ be an $(\text{\rm Ab}^{\text{\rm
nA}},\mathbb{Z}_{\text{\rm triv}})$-form of $A$ over $R$ such that
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{A_{\mathbb{Z}}}$
is the trivial norm. For any $\mathbb{Z}$-linear basis $S$ of
$A_{\mathbb{Z}}$, $\varphi_{A}(S)$ is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal Schauder $R$-linear basis of $A$ whose multiplication table
is a matrix with coefficients in $\mathbb{Z}$.

  * (ii)

Let $S_{R}\subset A$ be an $\text{\rm Ab}^{\text{\rm nA}}$-orthonormal
Schauder $R$-linear basis of $A$ such that the $\mathbb{Z}$-submodule
$\mathbb{Z}S_{R}\subset A$ generated by $S_{R}$ is a subring. Then,
$(\mathbb{Z}S_{R})_{\text{\rm triv}}$ is an $(\text{\rm Ab}^{\text{\rm
nA}},\mathbb{Z}_{\text{\rm triv}})$-form of $A$ over $R$ with respect to the
morphism $(\mathbb{Z}S_{R})_{\text{\rm
triv}}\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm triv}}}R\to A$ in $\text{\rm
Alg}_{\text{\rm Ab}^{\text{\rm nA}}}(R)$ induced by the inclusion
$\mathbb{Z}S_{R}\hookrightarrow A$.

###### Proof.

The assertion (i) immediately follows from Proposition 6.7 (ii), because the
natural morphism $\ell^{\text{\rm Ab}^{\text{\rm
nA}}}((S,1_{S}),\mathbb{Z}_{\text{\rm triv}})\to A_{\mathbb{Z}}$ is an
isometric isomorphism (cf. Example 6.6 (ii)). The assertion (ii) also
immediately follows from Proposition 6.7 (ii), because the natural morphism
$\ell^{\text{\rm Ab}^{\text{\rm nA}}}(S_{R},(\mathbb{Z}_{R})_{\text{\rm
triv}})\to(\mathbb{Z}S_{R})_{\text{\rm triv}}$ is an isometric isomorphism,
where $\mathbb{Z}_{R}$ denote the set theoretic image of the canonical ring
homomorphism $\mathbb{Z}\to R$. ∎

As an immediate consequence of Corollary 7.2 and Proposition 7.5 (i), we
obtain a partial generalisation of Proposition 7.1.

######  Corollary 7.6.

Let $S$ be a $\mathbb{Z}$-linear basis of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$, and $R$ a monoid object of $\text{\rm
Ab}^{\text{\rm nA}}$ with submultiplicative norm isolated at $0$ or a complete
non-Archimedean valued ring (resp. field). Then
${\mathchoice{\left\\{f_{R}\middle|f\in S\right\\}}{\\{f_{R}\mid f\in
S\\}}{\\{f_{R}\mid f\in S\\}}{\\{f_{R}\mid f\in S\\}}}$ is an $\text{\rm
Ab}^{\text{\rm nA}}$-orthonormal Schauder $R$-linear basis of $\text{\rm
C}(\zeta(X),R)$ whose multiplication table is a matrix with coefficients in
$\mathbb{Z}$, where $f_{R}$ denotes the composite of the unique extension
$\zeta(X)\to\mathbb{Z}$ of $f$ (cf. Proposition 2.6) and the canonical ring
homomorphism $\mathbb{Z}\to R$.

By Proposition 2.3 (ii) and Corollary 7.6, we obtain the following:

######  Corollary 7.7.

Suppose $X\in\text{\rm TDCH}$. Let $S$ be a $\mathbb{Z}$-linear basis of
$\text{\rm C}(X,\mathbb{Z}_{\text{\rm triv}})$. Let $R$ be a monoid object $R$
of $\text{\rm Ab}^{\text{\rm nA}}$ with submultiplicative norm isolated at $0$
or a complete non-Archimedean valued ring (resp. field). Then
${\mathchoice{\left\\{f_{R}\middle|f\in S\right\\}}{\\{f_{R}\mid f\in
S\\}}{\\{f_{R}\mid f\in S\\}}{\\{f_{R}\mid f\in S\\}}}$ is an $\text{\rm
Ab}^{\text{\rm nA}}$-orthonormal Schauder $R$-linear basis of $\text{\rm
C}(X,R)$ whose multiplication table is a matrix with coefficients in
$\mathbb{Z}$, where $f_{R}$ denotes the composite of $f$ and the canonical
ring homomorphism $\mathbb{Z}\to R$.

One of the benefits of choosing a $\mathbb{Z}$-form is that analysis of $A$ is
reduced to a combinatorial computation based on a $\mathbb{Z}$-linear basis of
$A_{\mathbb{Z}}$. Indeed, Corollary 7.7 ensures that non-Archimedean
functional analysis on $X$ is completely reduced to the multiplication table
for a collection $S$ of functions $X\to\mathbb{Z}_{\text{\rm triv}}$ if
$X\in\text{\rm TDCH}$.

By Proposition 7.1, we can choose $S$ to be a set of idempotents, in which
case the multiplication table will be simpler, as the product just corresponds
to the intersection of the corresponding clopen subsets. We say that $S$ is a
generalised van der Put basis of $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ if it is a $\mathbb{Z}$-linear basis of
$\text{\rm C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ such that
$e_{0}e_{1}\in{\mathchoice{\left\\{e_{0},e_{1},0\right\\}}{\\{e_{0},e_{1},0\\}}{\\{e_{0},e_{1},0\\}}{\\{e_{0},e_{1},0\\}}}$
for any $(e_{0},e_{1})\in S^{2}$. Since $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ is reduced, a generalised van der Put
basis of $\text{\rm C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})$
consists of idempotents, and the multiplication table for a generalised van
der Put basis is even simpler than a generic multiplication table by
intersections of clopen subsets, because it is reduced to the monoid structure
of $S\cup{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$ or the
partial ordering of clopen subsets with respect to the inclusion. We give a
criterion for the existence of a generalised van der Put basis.

######  Proposition 7.8.

If $X$ is ultrametrisable, then $\text{\rm C}_{\text{\rm
fin}}(X,\mathbb{Z}_{\text{\rm triv}})$ admits a generalised van der Put basis.

###### Proof.

Fix an ultrametric $d_{X}$ on $X$ compatible with its topology. For each
$(x,r)\in X\times(0,\infty)$, we denote by $\chi_{x,r}$ the characteristic
function
$X\to{\mathchoice{\left\\{0,1\right\\}}{\\{0,1\\}}{\\{0,1\\}}{\\{0,1\\}}}$ of
${\mathchoice{\left\\{y\in X\middle|d_{X}(x,y)<r\right\\}}{\\{y\in X\mid
d_{X}(x,y)<r\\}}{\\{y\in X\mid d_{X}(x,y)<r\\}}{\\{y\in X\mid
d_{X}(x,y)<r\\}}}\in\text{\rm CO}(X)$. The assertion immediately follows from
the proof of [Sch19] Theorem 5.4 applied to the embedding
$X\hookrightarrow{\mathchoice{\left\\{0,1\right\\}}{\\{0,1\\}}{\\{0,1\\}}{\\{0,1\\}}}^{X\times(0,\infty)}$
given by $(\chi_{x,r})_{(x,r)\in X\times(0,\infty)}$. ∎

A typical example of a generalised van der Put basis is the van der Put basis
of $\text{\rm C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm triv}})$ (cf. p. 263
[Mah73] Chapter 16 §9 Theorem 4), as the name indicates. It is the set of the
idempotents corresponding to clopen subsets of $\mathbb{Z}_{p}$ of the form

$\displaystyle{\mathchoice{\left\\{x\in\mathbb{Z}_{p}\middle|{\mathchoice{\left|x-n\right|}{|x-n|}{|x-n|}{|x-n|}}_{p}<n^{-1}\right\\}}{\\{x\in\mathbb{Z}_{p}\mid{\mathchoice{\left|x-n\right|}{|x-n|}{|x-n|}{|x-n|}}_{p}<n^{-1}\\}}{\\{x\in\mathbb{Z}_{p}\mid{\mathchoice{\left|x-n\right|}{|x-n|}{|x-n|}{|x-n|}}_{p}<n^{-1}\\}}{\\{x\in\mathbb{Z}_{p}\mid{\mathchoice{\left|x-n\right|}{|x-n|}{|x-n|}{|x-n|}}_{p}<n^{-1}\\}}}$

with
$n\in\mathbb{N}\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$,
where the $p$-adic valuation on $\mathbb{Z}_{p}$ is normalised by
${\mathchoice{\left|p\right|}{|p|}{|p|}{|p|}}_{p}=p^{-1}$. There are many
classical results on $\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ based on
the presentation by the van der Put basis (cf. [Sch84] §62). We now show that
$\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ admits two non-isomorphic
$\mathbb{Z}$-forms, providing the example mentioned in Example 7.4 (iii).

######  Example 7.9.

The set of binomial polynomials is an $\text{\rm Ab}^{\text{\rm
nA}}$-orthonormal $\mathbb{Q}_{p}$-linear basis of $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$, by [Mah58] Theorem 1, called the Mahler
basis. This basis generates a $\mathbb{Z}$-submodule $\text{\rm
Int}(\mathbb{Z}_{\text{\rm triv}})\subset\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ which is a subring and a $\mathbb{Z}$-form
of $\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ over $\mathbb{Q}_{p}$, that
is isomorphic to the so-called _ring of integrally valued polynomials_. It is
easy to check that $\text{\rm Int}(\mathbb{Z}_{\text{\rm triv}})$ is not
contained in $\text{\rm C}_{\text{\rm fin}}(\mathbb{Z}_{p},\mathbb{Z})$, but
is contained in

$\displaystyle{\mathchoice{\left\\{f(T)\in\mathbb{Q}[T]\middle|\forall
x\in\mathbb{Z},f(x)\in\mathbb{Z}\right\\}}{\\{f(T)\in\mathbb{Q}[T]\mid\forall
x\in\mathbb{Z},f(x)\in\mathbb{Z}\\}}{\\{f(T)\in\mathbb{Q}[T]\mid\forall
x\in\mathbb{Z},f(x)\in\mathbb{Z}\\}}{\\{f(T)\in\mathbb{Q}[T]\mid\forall
x\in\mathbb{Z},f(x)\in\mathbb{Z}\\}}}$

under the identification of a polynomial and the corresponding polynomial
function. Although the Mahler basis does not consist of idempotent elements,
it has plenty of good combinatorial properties (cf. [Mah73] §9 – §10). For
example, the Mahler basis is the dual basis of the topological basis
$(([1]-[0])^{n})_{n\in\mathbb{N}}\in\mathbb{Z}_{p}[\\![\mathbb{Z}_{p}]\\!]$
associated to the topological basis
$(T^{n})_{n\in\mathbb{N}}\in\mathbb{Z}_{p}[\\![T]\\!]\cong\text{\rm
H}^{0}(\mathbb{G}_{\text{\rm m}},\mathscr{O}_{\mathbb{G}_{\text{\rm m}}})$
through the Iwasawa isomorphism
$\mathbb{Z}_{p}[\\![\mathbb{Z}_{p}]\\!]\cong\mathbb{Z}_{p}[\\![T]\\!]$, where
$\mathbb{G}_{\text{\rm m}}$ denotes the commutative formal group scheme over
$\text{\rm Spf}(\mathbb{Z}_{p})$ given as the formal completion of the
multiplicative group $\text{\rm Spec}(\mathbb{Z}_{p}[T,(1+T)^{-1}])$ at the
unit. Indeed, let $(n,i)\in\mathbb{N}^{2}$. If $i<n$, then we have
$\sum_{j=0}^{i}(-1)^{n-j}\binom{i}{j}\binom{j}{n}=0$. If $i\geq n$, then we
have

$\displaystyle\sum_{j=0}^{i}(-1)^{n-j}\binom{i}{j}\binom{j}{n}=\sum_{j=n}^{i}(-1)^{n-j}\binom{i}{j}\binom{j}{n}=\sum_{j=n}^{i}(-1)^{n-j}\frac{i!}{(i-j)!(j-n)!n!}$
$\displaystyle=$
$\displaystyle\binom{i}{n}\sum_{j=n}^{i}(-1)^{n-j}\binom{i-n}{j-n}=\binom{i}{n}\sum_{j=0}^{i-n}(-1)^{j}\binom{i-n}{j}=\binom{i}{n}(1-1)^{i-n}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}1&(i=n)\\\
0&(i>n)\end{array}\right..$

This implies that the canonical pairing $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Q}_{p})\times\mathbb{Z}_{p}[\\![\mathbb{Z}_{p}]\\!]\to\mathbb{Q}_{p}$
sends the pair of $\binom{x}{n}\in\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$
and $([1]-[0])^{i}\in\mathbb{Z}_{p}[\\![\mathbb{Z}_{p}]\\!]$ to $1$ if $i=n$
and to $0$ if $i\neq n$. Since the algebra structure of the topological Hopf
$\mathbb{Z}_{p}$-algebra $\mathbb{Z}_{p}[\\![\mathbb{Z}_{p}]\\!]$ corresponds
to the coalgebra structure of the Banach Hopf $\mathbb{Q}_{p}$-algebra
$\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$, the Mahler basis also possesses
information of the comultiplication helpful in the study of Banach comodules
over $\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$, which correspond to
$p$-adic representations of $\mathbb{G}_{\text{\rm m}}$. Further observations
of an $\text{\rm Ab}^{\text{\rm nA}}$-orthonormal Schauder
$\mathbb{Q}_{p}$-linear basis of the dual non-commutative Banach Hopf
$\mathbb{Q}_{p}$-algebra of the non-cocommutative topological Hopf
$\mathbb{Z}_{p}$-algebra associated to a non-commutative $p$-adic Lie group of
higher dimension are given in [Mih21] §4.1. Concerning this work, the
interesting observation is that the non-Archimedean Banach rings $\text{\rm
Int}(\mathbb{Z}_{\text{\rm triv}})$ and $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm triv}})$ are both $(\text{\rm
Ab}^{\text{\rm nA}},\mathbb{Z}_{\text{\rm triv}})$-forms of $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ but they are non-isomorphic as Banach
rings, and the interplay between these two different presentations of
$\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ is an important theme of its
study. To appreciate the difference between the algebras $\text{\rm
Int}(\mathbb{Z}_{\text{\rm triv}})$ and $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm triv}})$ one can notice that they
behave very differently for the base change functors. Indeed, consider a prime
$\ell\neq p$, then $\text{\rm Int}(\mathbb{Z}_{\text{\rm
triv}})\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm
triv}}}\mathbb{Q}_{\ell}\cong\text{\rm
C}(\mathbb{Z}_{\ell},\mathbb{Q}_{\ell})$ and $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm
triv}})\widehat{\otimes}{}_{\mathbb{Z}_{\text{\rm
triv}}}\mathbb{Q}_{\ell}\cong\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{\ell})$.

Similarly, it is reasonable to expect applications of the simpler
$\mathbb{Z}$-form associated to a generalised van der Put basis. As the van
der Put basis can be used to characterise Lipschitz functions (cf. [Sch84]
Theorem 63.2), $\text{\rm C}^{1}$-functions with zero derivatives (cf. [Sch84]
Theorem 63.3), functions with higher differentiabilities (cf. [Sch84] Exercise
63.D), and monotonous functions (cf. [Sch84] Exercise 63.E), one can formally
generalise those notions of smoothness to notions for a bounded continuous
function $X\to\mathbb{Q}_{p}$, which can be identified with a continuous
function $\zeta(X)\to\mathbb{Q}_{p}$ by the universality of Banaschewski
compactification, with respect to a fixed generalised van der Put basis $S$ of
$\text{\rm C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm triv}})$. Similarly,
one can imitate the computation of the integration using the van der Put basis
(cf. [Sch84] Exercise 62.E (ii)) to formally define an integration of a
bounded continuous function $X\to\mathbb{Q}_{p}$ with respect to $S$ and a
fixed assignment $S\to\mathbb{Q}_{p}$ analogous to a $p$-adic measure.

A subring $A\subset\text{\rm C}(\mathbb{Z}_{p},\mathbb{Q}_{p})$ is said to
satisfy the $p$-adic Weierstrass approximation theorem if
$A\cap\mathbb{Q}_{p}[T]$ is dense in $A$ with respect to the restriction of
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Q}_{p})}$. By the relations $\text{\rm
Int}(\mathbb{Z}_{\text{\rm triv}})\subset\mathbb{Q}[T]$ and $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm
triv}})\cap\mathbb{Q}_{p}[T]=\mathbb{Z}$, it follows that $\text{\rm
Int}(\mathbb{Z}_{\text{\rm triv}})$ satisfies the $p$-adic Weierstrass
approximation theorem, but $\text{\rm C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm
triv}})$ does not satisfy the $p$-adic Weierstrass approximation theorem.
Then, a natural question is: what approximation theorem should we expect for
$\text{\rm C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm triv}})$ (or more
generally, for $\text{\rm C}_{\text{\rm fin}}(X,\mathbb{Z}_{\text{\rm
triv}})$) to hold? One natural candidate is the Stone–Weierstrass theorem.

A subring $A\subset\text{\rm C}_{\text{\rm bd}}(X,R)$ is said to separate
points of $X$ if for any $(x_{0},x_{1})\in X^{2}$ with $x_{0}\neq x_{1}$,
there exists an $f\in A$ such that $f(x_{0})\neq f(x_{1})$. An $R$-subalgebra
$A\subset\text{\rm C}_{\text{\rm bd}}(X,R)$ is said to satisfy the
Stone–Weierstrass theorem over $R$ if for any $R$-subalgebra $B\subset A$
separating points of $X$, ${\mathchoice{\left\\{f\in A\middle|Rf\cap
B\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\right\\}}{\\{f\in
A\mid Rf\cap
B\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\\}}{\\{f\in
A\mid Rf\cap
B\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\\}}{\\{f\in
A\mid Rf\cap
B\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}\\}}}$ is
dense in $A$. We note that requiring $B$ itself to be dense in $A$ is not
sufficient, as $\mathbb{Z}+p\text{\rm C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm
triv}})$ and
$\sum_{n\in\mathbb{N}}\mathbb{Z}p^{n}1_{\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm
triv}},n+p^{n}\mathbb{Z}_{p}}$ are subrings of $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm triv}})$ separating points of
$\mathbb{Z}_{p}$ but are not dense in $\text{\rm
C}(\mathbb{Z}_{p},\mathbb{Z}_{\text{\rm triv}})$. We say that $X$ is zero
dimensional if $\text{\rm CO}(X)$ generates the topology of $X$. We show an
extension of [Mih14] Lemma 4.17.

######  Proposition 7.10.

Suppose that
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$, and $X$
is zero dimensional and $T_{0}$.

  * (i)

Every dense subring $A\subset\text{\rm C}_{\text{\rm bd}}(X,R)$ separates
points of $X$.

  * (ii)

For any $\mathbb{Z}$-form $(A,\varphi)$ of $\text{\rm C}_{\text{\rm bd}}(X,R)$
over $\mathbb{Z}$, the set ${\mathchoice{\left\\{\varphi(f\otimes
1)\middle|f\in A\right\\}}{\\{\varphi(f\otimes 1)\mid f\in
A\\}}{\\{\varphi(f\otimes 1)\mid f\in A\\}}{\\{\varphi(f\otimes 1)\mid f\in
A\\}}}$ is a subring of $\text{\rm C}_{\text{\rm bd}}(X,R)$ separating points
of $X$.

###### Proof.

We show the assertion (i). Let $(x_{0},x_{1})\in X^{2}$ with $x_{0}\neq
x_{1}$. Since $X$ is $T_{0}$, there exists an open subset $V\subset X$ such
that
$V\cap{\mathchoice{\left\\{x_{0},x_{1}\right\\}}{\\{x_{0},x_{1}\\}}{\\{x_{0},x_{1}\\}}{\\{x_{0},x_{1}\\}}}$
is a singleton. Since $X$ is zero-dimensional, there exists a $U\in\text{\rm
CO}(X)$ such that
$V\cap{\mathchoice{\left\\{x_{0},x_{1}\right\\}}{\\{x_{0},x_{1}\\}}{\\{x_{0},x_{1}\\}}{\\{x_{0},x_{1}\\}}}\subset
U\subset V$. Then, $1_{X,R,U}\in\text{\rm C}_{\text{\rm bd}}(X,R)$ separates
$x_{0}$ and $x_{1}$ by
$R\neq{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$. Since $A$
is dense in $\text{\rm C}_{\text{\rm bd}}(X,R)$, there exists an $f\in A$ such
that
${\mathchoice{\left\|1_{X,R,U}-f\right\|}{\|1_{X,R,U}-f\|}{\|1_{X,R,U}-f\|}{\|1_{X,R,U}-f\|}}_{\text{\rm
C}_{\text{\rm bd}}(X,R)}<2^{-1}$. We have
${\mathchoice{\left\|f(x_{0})-f(x_{1})\right\|}{\|f(x_{0})-f(x_{1})\|}{\|f(x_{0})-f(x_{1})\|}{\|f(x_{0})-f(x_{1})\|}}_{R}>{\mathchoice{\left\|1_{X,R,U}(x_{0})-1_{X,R,U}(x_{1})\right\|}{\|1_{X,R,U}(x_{0})-1_{X,R,U}(x_{1})\|}{\|1_{X,R,U}(x_{0})-1_{X,R,U}(x_{1})\|}{\|1_{X,R,U}(x_{0})-1_{X,R,U}(x_{1})\|}}-2^{-1}\times
2=0$ and hence $f(x_{0})\neq f(x_{1})$.

We show the assertion (ii). Let $(x_{0},x_{1})\in X^{2}$ with $x_{0}\neq
x_{1}$. If $\varphi(f\otimes 1)(x_{0})=\varphi(f\otimes 1)(x_{1})$ for any
$f\in A$, then $f(x_{0})=f(x_{1})$ for any $f\in A_{R}$, where
$A_{R}\subset\text{\rm C}_{\text{\rm bd}}(X,R)$ is the $R$-subalgebra
generated by ${\mathchoice{\left\\{\varphi(f\otimes 1)\middle|f\in
A\right\\}}{\\{\varphi(f\otimes 1)\mid f\in A\\}}{\\{\varphi(f\otimes 1)\mid
f\in A\\}}{\\{\varphi(f\otimes 1)\mid f\in A\\}}}$. This contradicts the
assertion (i), because $A_{R}$ is dense in $\text{\rm C}_{\text{\rm
bd}}(X,R)$. ∎

We note that every zero dimensional $T_{0}$ space is Hausdorff, as the proof
of Proposition 7.10 shows. We recall that Gel’fand’s theory ensures that if
$X$ is completely regular and Hausdorff, then closed $\mathbb{R}$-subalgebras
of $\text{\rm C}_{\text{\rm bd}}(X,\mathbb{R})$ separating points of $X$
precisely correspond to equivalence classes of Hausdorff compactifications of
$X$, and its non-Archimedean counterpart (cf. [Mih14] Theorem 4.19) ensures
that if $X$ is zero-dimensional and Hausdorff, then closed $k$-subalgebras of
$\text{\rm C}_{\text{\rm bd}}(X,k)$ separating points of $X$ precisely
correspond to equivalence classes of totally disconnected Hausdorff
compactifications of $X$ for any local field $k$. Therefore, when we expect
the Stone–Weierstrass theorem to hold, we need to assume $X\in\text{\rm CH}$.

######  Theorem 7.11.

If $X\in\text{\rm CH}$, then $\text{\rm C}(X,R)$ satisfies the
Stone–Weierstrass theorem over $R$ in the following cases:

  * (i)

There exists an isomorphism
$R\to(\mathbb{R},{\mathchoice{\left|\cdot\right|}{|\cdot|}{|\cdot|}{|\cdot|}}_{\mathbb{R}}^{\epsilon})$
of monoid objects in $\text{\rm Ab}^{\text{\rm A}}$ for some
$\epsilon\in[0,\infty)$.

  * (ii)

There exists an isomorphism $R\to k$ of monoid objects in $\text{\rm
Ab}^{\text{\rm A}}$ for some complete non-Archimedean valued field $k$.

  * (iii)

There exists an isomorphism $R\to\mathbb{Z}_{\infty}$ or
$R\to\mathbb{Z}_{\text{\rm triv}}$ of monoid objects in $\text{\rm
Ab}^{\text{\rm A}}$.

  * (iv)

There exists a total order $\leq$ on the underlying ring of $R$ such that
$(R,\leq)$ is an ordered ring, and $R$ is isolated at $0$.

###### Proof.

The claim for the condition (i) restricted to the case $\epsilon>0$ is
equivalent to the classical Stone–Weierstrass theorem, the claim for the
condition (ii) is equivalent to non-Archimedean Stone–Weierstrass theorem (cf.
[Ber90] 9.2.5 Theorem), and the claims for the condition (i) restricted to the
case $\epsilon=0$ and the condition (iii) immediately follow from the claim
for the condition (iv). We show the statement for the condition (iv). Let
$B\subset\text{\rm C}(X,R)$ be a subring separating points of $X$. It suffices
to show that for any $U\in\text{\rm CO}(X)$, there exists an $a_{U}\in
R\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$ such
that $a_{U}1_{X,R,U}\in B$. If $U=\emptyset$, then $a_{U}$ can be taken as
$1$. Therefore, we may assume $U\neq\emptyset$.

First, we show that for any $x\in X\setminus U$, there exists an $f_{x}\in B$
such that $x\in
f_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})\subset
X\setminus U$. Let $y\in U$. Since $B$ separates points of $X$, there exists
an $f_{x,y}\in B$ such that $f_{x,y}(x)\neq f_{x,y}(y)$. Replacing $f_{x,y}$
by $f_{x,y}-f_{x,y}(x)\in B$, we may assume $f_{x,y}(x)=0$. By
$U\subset\bigcup_{y\in
U}f_{x,y}^{-1}(R\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$,
the compactness of $X$ ensures that there exists a finite subset $S_{x}\subset
U$ such that $U\subset\bigcup_{y\in
S_{x}}f_{x,y}^{-1}(R\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$.
Put $f_{x}\coloneqq\sum_{y\in S_{x}}f_{x,y}^{2}\in B$. We have
$S_{x}\neq\emptyset$ by $U\neq\emptyset$. For any $z\in X$, we have
$f_{x}(z)=\sum_{y\in S_{x}}f_{x,y}(z)^{2}\geq 0$, and hence $f_{x}(z)=0$ if
and only if $f_{x,y}(z)=0$ for any $y\in S_{x}$ because $(R,\leq)$ is an
ordered ring. This implies $x\in
f_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})=\bigcap_{y\in
S_{x}}f_{x,y}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})\subset
X\setminus U$.

Secondly, we show that for any $x\in X\setminus U$, there exists an
$(a_{x},e_{x})\in R\times B$ such that $a_{x}>0$, $x\in
e_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})\subset
X\setminus U$, and $e_{x}^{2}=a_{x}e_{x}$. We have that $f_{x}(X)$ is a finite
subset of $R$ by Proposition 2.3 (ii), contains $0$ by $f_{x}(x)=0$, and does
not coincide with
${\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$ by
$U\neq\emptyset$. Put $a_{x}\coloneqq\prod_{a\in
f_{x}(X)\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}}a\in
R$ and $e_{x}\coloneqq a_{x}-\prod_{a\in
f_{x}(X)\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}}(a-f_{x})\in
B$. Since every element of $f_{x}(X)$ is the sum of square elements,
$f_{x}(X)\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}}$
consists of positive elements. This implies $a_{x}>0$. We have $e_{x}(y)=0$
for any $y\in
f_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$ and
$e_{x}(y)=a_{x}$ for any $y\in
f_{x}^{-1}(f_{x}(X)\setminus{\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})=X\setminus
f_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$.
This implies $x\in
e_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})\subset
X\setminus U$ and $e_{x}^{2}=a_{x}e_{x}$.

Finally, we show that there exists an $a_{U}\in R$ such that $a_{U}>0$ and
$a_{U}1_{X,R,U}\in B$. By $X\setminus U=\bigcup_{x\in X\setminus
U}e_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$,
the compactness of $X$ ensures that there exists a finite subset $S\subset
X\setminus U$ such that $X\setminus U=\bigcup_{x\in
S}e_{x}^{-1}({\mathchoice{\left\\{0\right\\}}{\\{0\\}}{\\{0\\}}{\\{0\\}}})$.
Put $a_{U}\coloneqq\prod_{x\in S}a_{x}\in R$. Then, we have $a_{U}>0$ and
$a_{U}1_{X,R,U}=\prod_{x\in S}e_{x}\in B$. ∎

We note that the total order $\leq$ in the condition (iv) in Theorem 7.11 is
used only to restrict the ring structure of $R$, and does not have to be
compatible with
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{R}$ in
any sense. For example, $\mathbb{R}_{\text{\rm triv}}$ and
$\mathbb{R}_{\text{\rm hyb}}$ satisfy the condition (iv), although the
canonical total order on $\mathbb{R}$ is irrelevant to
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
triv}}$ or
${\mathchoice{\left\|\cdot\right\|}{\|\cdot\|}{\|\cdot\|}{\|\cdot\|}}_{\text{\rm
hyb}}$.

Acknowledgements

The first listed author thanks Kobi Kremnizer and Oren Ben-Bassat for
important discussions concerning derived analytic geometry. During the
preparation of this paper the first listed author has been supported by the
DFG research grant BA 6560/1-1 “ _Derived geometry and arithmetic_ ”.

The second listed author thanks Frédéric Paugam for informing me of the
interesting result in [BK20] by the first listed author. We have started the
joint work [BM21] thanks to this information, and is continuing this joint
work. The second listed author also thanks Takuma Hayashi for instructing him
on elementary facts on derived categories when he was working on our preceding
paper [BM21], and colleagues in universities for daily discussions. The second
listed author is also thankful to his family for their deep affection. During
the preparation of this paper, the second listed author has been supported by
JSPS KAKENHI Grant Number 21K13763.

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