Universal Thickening of the Field of Real Numbers

Abstract

We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield $\mathbb{R}^{\flat }$. Then, we prove that the archimedean Witt construction in the context of hyperfields allows one to recover a field from a hyperfield, and we obtain the universal pro-infinitesimal thickening $\mathbb{R}_{\infty }$ of $\mathbb{R}$. Finally, we provide the real analogues of several algebras used in the construction of the rings of p-adic periods. We supply the canonical decomposition of elements in terms of Teichmüller lifts, we make the link with the Mikusinski field of operational calculus and compute the Gelfand spectrum of the archimedean counterparts of the rings of p-adic periods. In the second part of the paper we also discuss the complex case and its relation with the theory of oscillatory integrals in quantum physics.

To the memory of M. Krasner, in recognition of his farsightedness.

Appendices

Appendix 1

The following table reports the archimedean structures that we have defined and discussed in this paper and their p-adic counterparts (cf. [9]).

Table 1

Appendix 2

In this appendix we give a short overview of the well-known construction of universal perfection in number theory: we refer to [10], Chap. V, Sect. 1.4; [11, 26], Sect. 2.1; [9], Sect. 2.4 for more details.

The universal perfection is a procedure which associates, in a canonical way, a perfect field F(L) of characteristic p to a p-perfect field L. This construction is particularly relevant when char(L) = 0, since it determines the first step toward the definition of a universal, Galois equivariant cover of L (cf. Appendix 4).

We recall that a field L is said to be p-perfect if it is complete with respect to a non archimedean absolute value |   |  _L , L has a residue field of characteristic p and the endomorphism of $\mathcal{O}_{L}/p\mathcal{O}_{L}$, x → x ^p is surjective. Furthermore, the field L is said to be strictly p-perfect if $\mathcal{O}_{L}$ is not a discrete valuation ring.

Starting with a p-perfect field L, one introduces the set

$$\displaystyle{ F(L) =\{ x = (x^{(n)})_{ n\in \mathbb{N}}\vert \ x^{(n)} \in L;\ (x^{(n+1)})^{p} = x^{(n)}\}. }$$

(153)

If x, y ∈ F(L), one sets

$$\displaystyle{ (x + y)^{(n)} =\lim _{ m\rightarrow \infty }(x^{(n+m)} + y^{(n+m)})^{p^{m} };\qquad (xy)^{(n)} = x^{(n)}y^{(n)}. }$$

(154)

We recall from [9] (cf. Sect. 2.4) the following result

Proposition 16.

Let L be a p-perfect field. Then F(L) with the above two operations is a perfect field of characteristic p, complete with respect to the absolute value defined by |x| = |x ⁽⁰⁾ | _L . Moreover if $\mathfrak{a} \subset \mathfrak{m}_{L}$ is a finite type (i.e. principal) ideal of $\mathcal{O}_{L}$ containing $p\mathcal{O}_{L}$ , then the map reduction mod.  $\mathfrak{a}$ induces an isomorphism of topological rings

$$\displaystyle{ \mathcal{O}_{F(L)}\stackrel{\sim }{\longrightarrow }\mathop{\lim }\limits_\longleftarrow _{n\in \mathbb{N}}\mathcal{O}_{L}/\mathfrak{a},\qquad x = (x^{(n)})_{ n\in \mathbb{N}}\mapsto \bar{x} = (x^{(n)}\text{mod.}\ \mathfrak{a})_{ n\in \mathbb{N}} }$$

(155)

where the transition maps in the projective limit are given by the ring homomorphism $\bar{x} \rightarrow \bar{ x}^{p}$ .

In other words, the bijection (155) allows one to transfer (uniquely) the natural (perfect) algebra structure on $\mathop{\lim }\limits_\longleftarrow _{v\rightarrow v^{p}}\mathcal{O}_{L}/\mathfrak{a}$ over the inverse limit set $\mathcal{O}_{F(L)} = \mathop{\lim }\limits_\longleftarrow _{x\mapsto x^{p}}\mathcal{O}_{L}$ of p-power compatible sequences $x = (x^{(n)})_{n\geq 0}$, $x^{(n)} \in \mathcal{O}_{L}$. Indeed, one shows that for any $v = (v_{n}) \in \mathop{\lim }\limits_\longleftarrow _{v\rightarrow v^{p}}\mathcal{O}_{L}/\mathfrak{a}$ and arbitrary lifts $x_{n} \in \mathcal{O}_{L}$ of $v_{n} \in \mathcal{O}_{L}/\mathfrak{a}\forall n \geq 0$, the limit $x^{(n)} =\lim _{m\rightarrow \infty }x_{n+m}^{p^{m} }$ exists in $\mathcal{O}_{L}\forall n \geq 0$ and is independent of the choice of the lifts x _n . This lifting process is naturally multiplicative, whereas the additive structure on $\mathop{\lim }\limits_\longleftarrow _{v\rightarrow v^{p}}\mathcal{O}_{L}/\mathfrak{a}$ lifts on $\mathcal{O}_{F(L)}$ as (154).

Appendix 3

In this appendix we provide, for completeness, a proof of Proposition 2. We recall that a pro-infinitesimal thickening of a ring R (cf. [12], Sect. 1.1.1 with $\varLambda = \mathbb{Z}$) is a surjective ring homomorphism $\theta: A \rightarrow R$, such that the ring A is Hausdorff and complete for the $\mathrm{Ker}(\theta )$-adic topology i.e. 

$$\displaystyle{ A = \mathop{\lim }\limits_\longleftarrow _{n}A/\mathrm{Ker}(\theta )^{n}. }$$

(156)

As a minor variant, we consider triples $(A,\theta,\tau )$, where $\theta: A \rightarrow R$ is a ring homomorphism with multiplicative section τ: R → A and condition (156) holds.

A morphism from the triple $(A_{1},\theta _{1},\tau _{1})$ to the triple $(A_{2},\theta _{2},\tau _{2})$ is given by a ring homomorphism α: A ₁ → A ₂ such that

$$\displaystyle{ \tau _{2} =\alpha \circ \tau _{1},\quad \theta _{1} =\theta _{2} \circ \alpha. }$$

(157)

Let R be a perfect ring of characteristic p and let W(R) be the p-isotypical Witt ring of R. Let ρ _R : W(R) → R be the canonical homomorphism and τ _R : R → W(R) the multiplicative section given by the Teichmüller lift.

By construction one has Ker(ρ _R ) = pW(R) and condition (156) holds.

We show that for any triple (A, ρ, τ) fulfilling (156), there exists a unique ring homomorphism from (W(R), ρ _R , τ _R ) to (A, ρ, τ) (Compare with Theorem 4.2 of [15] and Theorem 1.2.1 of [12]). The ring A with the sequence of ideals $\mathfrak{a}_{n} = \mathrm{Ker}(\rho )^{n}$ fulfills the hypothesis of [22] (II, Sect. 4, Proposition 8). Thus it follows from [22] (II, Sect. 5, Proposition 10) that there exists a (unique) ring homomorphism α: W(R) → A such that ρ ∘α = ρ _R . Moreover the uniqueness of the multiplicative section shown in [22] (II, Sect. 4, Proposition 8) proves that one has τ = α ∘τ _R . This completes the proof of Proposition 2.

Next we show that the notion of thickening involving a multiplicative section τ is in general different from the classical notion.

Consider $R = \mathbb{Z}$. Then, for any surjective ring homomorphism $\theta: A \rightarrow R$, the map $\mathbb{Z} \ni n\mapsto n1_{A}$ is the unique homomorphism from the pair $(\mathbb{Z},id)$ to the pair $(A,\theta )$. It follows that the pair $(\mathbb{Z},id)$ is the universal pro-infinitesimal thickening of  $\mathbb{Z}$. This no longer holds when one involves the multiplicative section τ.

Given a ring R, we consider R-triples (A, ρ, τ) where ρ: A → R is a ring homomorphism, τ: R → A is a multiplicative section (i.e. a morphism of monoids such that τ(0) = 0 and τ(1) = 1) and one also assumes (156). A morphism between two triples is a ring homomorphism α: A ₁ → A ₂ such that ρ ₁ = ρ ₂ ∘α and τ ₂ = α ∘τ ₁.

Proposition 2 shows that when R is a perfect ring of characteristic p there exists an initial object in the category of R-triples. For $R = \mathbb{Z}$ the triple $(\mathbb{Z},id,id)$ is a $\mathbb{Z}$-triple but it is not the universal one. The latter is in fact obtained using the ring $\mathbb{Z}[[\{\delta _{p}\}]] \otimes (\mathbb{Z} \oplus \mathbb{Z}_{2}e)$ of formal series with independent generators δ _p  = [p] − p, for each prime p and an additional generator e = [−1] + 1 such that e ² = 2e. The augmentation defines a surjection $\epsilon: A \rightarrow \mathbb{Z}$, ρ(e) = 0, and there exists a unique multiplicative section τ, τ(1) = 1, such that

$$\displaystyle{ \tau (p) = p +\delta _{p}\,,\ \forall p\ \text{prime},\ \tau (-1) = -1 + e. }$$

(158)

Proposition 17.

The triple $(\mathbb{Z}[[\{\delta _{p}\}]] \otimes (\mathbb{Z} \oplus \mathbb{Z}_{2}e),\epsilon,\tau )$ is the universal $\mathbb{Z}$ -triple. The map

$$\displaystyle{ D: \mathbb{Z} \rightarrow \mathrm{Ker}(\epsilon )/\mathrm{Ker}(\epsilon )^{2},\ \ D(n):=\tau (n) - n }$$

(159)

fulfills the Leibnitz rule and its component on δ _p coincides with the map $\frac{\partial } {\partial p}: \mathbb{Z} \rightarrow \mathbb{Z}$ defined in [18].

Proof.

By construction one has δ _p  ∈ Ker(ε) and thus ε ∘τ = i d. Consider first the subring $\mathbb{Z}[\{\delta _{p}\}][e]$ freely generated by the δ _p  = [p] − p for each prime p and an additional generator e = [−1] + 1 such that e ² = 2e. Given a $\mathbb{Z}$-triple (A, ρ, τ), there exists a unique ring homomorphism

$$\displaystyle{ \alpha: \mathbb{Z}[\{\delta _{p}\}][e] \rightarrow A\,,\ \ \alpha (\delta _{p}) =\tau (p) - p\,,\ \alpha (e) =\tau (-1) + 1. }$$

(160)

This ring homomorphism extends uniquely, by continuity, to a homomorphism

$$\displaystyle{ \alpha: \mathop{\lim }\limits_\longleftarrow _{n}\mathbb{Z}[\{\delta _{p}\}][e]/\mathrm{Ker}(\epsilon )^{n} \rightarrow A = \mathop{\lim }\limits_\longleftarrow _{ n}A/\mathrm{Ker}(\rho )^{n} }$$

and this shows that $(\mathbb{Z}[[\{\delta _{p}\}]] \otimes (\mathbb{Z} \oplus \mathbb{Z}_{2}e),\epsilon,\tau )$ is the universal $\mathbb{Z}$-triple.

The second assertion follows from [18] (cf. Theorem 1) and the identity

$$\displaystyle{ [nm] - nm = ([n] - n)m + n([m] - m) + ([n] - n)([m] - m)\,,\ \forall n,m \in \mathbb{Z}. }$$

⊓⊔

Appendix 4

In this appendix we shortly review some relevant constructions in p-adic Hodge theory which lead to the definition of the rings of p-adic periods. The main references are [8, 9].

We fix a non-archimedean locally compact field K of characteristic zero with a finite residue field k of characteristic p: q =  | k | . Let v _K be the (discrete) valuation of K normalized by $v_{K}(K^{{\ast}}) = \mathbb{Z}$.

Let F be any perfect field containing k. We assume that F is complete for a given (non-trivial) absolute value |   | . By W(F) and W(k) we denote the rings of isotypical Witt-vectors.

There exists a unique (up-to a unique isomorphism) field extension $\mathfrak{E}_{F,K}$ of K, complete with respect to a discrete valuation v extending v _K such that:

• $v(\mathfrak{E}_{F,K}^{{\ast}}) = v_{K}(K^{{\ast}}) = \mathbb{Z}$

• F is the residue field of $\mathfrak{E}_{F,K}$.

One sees that $\mathfrak{E}_{F,K}$ can be identified with K ⊗ _W(k) W(F). Thus, if π is a chosen uniformizing parameter of K, then an element of $\mathfrak{E}_{F,K}$ can be written uniquely as $\mathfrak{e} =\sum _{n\gg -\infty }[a_{n}]\pi ^{n}$, a _n  ∈ F. In particular $\mathfrak{e} \in K$ if and only if $a_{n} \in k\forall n$.

Let $\mathcal{O}_{\mathfrak{E}_{F,K}}$ be the (discrete) valuation ring of $\mathfrak{E}_{F,K}$. Each element of $\mathcal{O}_{\mathfrak{E}_{F,K}}$ can be written uniquely as $\sum _{n\geq 0}[a_{n}]\pi ^{n}$, a _n  ∈ F. The projection map $\mathcal{O}_{\mathfrak{E}_{F,K}} \twoheadrightarrow F$ has a unique multiplicative section i.e. the Teichmüller map a ↦ [a] = 1 ⊗ (a, 0, 0, …, 0, …).

There is a universal (local) subring $W_{\mathcal{O}_{K}}(\mathcal{O}_{F}) \subset \mathcal{O}_{\mathfrak{E}_{F,K}}$ which describes the unique π-adic torsion-free lifting of the perfect $\mathcal{O}_{K}$-algebra $\mathcal{O}_{F}$. If K ₀ denotes the maximal unramified extension of $\mathbb{Q}_{p}$ inside K, there is a canonical isomorphism:

$$\displaystyle{\mathcal{O}_{K} \otimes _{\mathcal{O}_{K_{ 0}}}W(\mathcal{O}_{F})\stackrel{\sim }{\longrightarrow }W_{\mathcal{O}_{K}}(\mathcal{O}_{F}),\ 1 \otimes [a]_{F}\mapsto [a].}$$

If A is any separated and complete π-adic $\mathcal{O}_{K}$-algebra with field of fractions L and F = F(L) (cf. Appendix 3 for notation), there is a ring homomorphism

$$\displaystyle{ \theta: W_{\mathcal{O}_{K}}(\mathcal{O}_{F(L)})\longrightarrow A,\qquad \sum _{n\geq 0}[x_{n}]\pi ^{n}\mapsto \sum _{ n\geq 0}x_{n}^{(0)}\pi ^{n}. }$$

(161)

In the particular case of the algebra $A = \mathcal{O}_{F} = \mathcal{O}_{F(\mathbf{C}_{K})}$ (C _K  = completion of a fixed algebraic closure of K), the surjective ring homomorphism $\theta _{0}: \mathcal{O}_{F} \twoheadrightarrow \mathcal{O}_{\mathbf{C}_{K}}/(p)$, $\theta _{0}((x^{(n)})_{n\geq 0}) = x^{(0)}$ lifts to a surjective ring homomorphism of $\mathcal{O}_{K}$-algebras

$$\displaystyle{ \theta: W_{\mathcal{O}_{K}}(\mathcal{O}_{F(\mathbf{C}_{K})}) \twoheadrightarrow \mathcal{O}_{\mathbf{C}_{K}},\qquad \sum _{n\geq 0}[x_{n}]\pi ^{n}\mapsto \sum _{ n\geq 0}x_{n}^{(0)}\pi ^{n} }$$

(162)

which is independent of the choice of the uniformizer π.

The valued field $(\mathfrak{E}_{F,K},\vert \cdot \vert )$ ( | ⋅ | non discrete) contains two further sub-$\mathcal{O}_{K}$-algebras which are also independent of the choice of a uniformizer $\pi \in \mathcal{O}_{K}$. They are

$$\displaystyle{ B^{b,+}:= W_{ \mathcal{O}_{K}}(\mathcal{O}_{F})[\frac{1} {\pi } ] =\{ x =\sum _{n\gg -\infty }[x_{n}]\pi ^{n} \in \mathfrak{E}_{ F,K}\vert x_{n} \in \mathcal{O}_{F},\forall n\} }$$

(163)

and if $a \in \mathfrak{m}_{F}\setminus \{0\} \subset \mathcal{O}_{F}$, the ring $B^{b}:= B^{b,+}[ \frac{1} {[a]}]$ which can be equivalently described as

$$\displaystyle{ B^{b} = B_{ F,K}^{b} =\{ f =\sum _{ n\gg -\infty }[x_{n}]\pi ^{n} \in \mathfrak{E}_{ F,K}\vert \exists C > 0,\vert x_{n}\vert \leq C,\forall n\}. }$$

(164)

If a p-perfect field L contains K as a closed subfield, the ring homomorphism (161) extends to a surjective homomorphism of K-algebras

$$\displaystyle{ \theta: B_{F(L),K}^{b} \rightarrow L,\qquad \theta (\sum _{ n\gg -\infty }[x_{n}]\pi ^{n}) =\sum _{ n\gg -\infty }x_{n}^{(0)}\pi ^{n} }$$

(165)

which is independent of the choice of π. If moreover L is a strictly p-perfect field, then | F(L) |  =  | L | and the kernel of the map $\theta$ in (165) is a prime ideal of $B_{F(L),K}^{b}$ of degree one. One has

$$\displaystyle{\theta (B_{F(L),K}^{b,+}) = L\quad \text{and}\quad \theta (W_{ \mathcal{O}_{K}}(\mathcal{O}_{F(L)})) = \mathcal{O}_{L}.}$$

Let $\mathfrak{E}_{0} = \mathfrak{E}_{k_{F},K}$. Then the projection $\mathcal{O}_{F} \rightarrow k_{F}$, $x \rightarrow \bar{ x}$ induces an augmentation map

$$\displaystyle{ \varepsilon: B^{b,+} \rightarrow \mathfrak{E}_{ 0},\qquad \varepsilon (\sum _{n\gg -\infty }[x_{n}]\pi ^{n}) =\sum _{ n\gg -\infty }[\bar{x}_{n}]\pi ^{n} }$$

(166)

with $\varepsilon (W_{\mathcal{O}_{K}}(\mathcal{O}_{F})) = \mathcal{O}_{\mathfrak{E}_{0}}$. $E =\{\sum _{n\gg -\infty }[x_{n}]\pi ^{n}\vert \ x_{n} \in k_{F},\forall n\}$ is a local sub-field of B ^b, +.

One introduces for $r \in \mathbb{R}_{\geq 0}$ the family of valuations on B ^b, +:

$$\displaystyle{x =\sum _{n\gg -\infty }[x_{n}]\pi ^{n},\qquad v_{ r}(x) = \text{inf}_{n\in \mathbb{Z}}\{v(x_{n}) + nr\} \in \mathbb{R} \cup \{ +\infty \}}$$

and defines B ⁺ as the completion of B ^b, + for the family of norms $(q^{-v_{r}})_{r>0}$ (q =  | k | ), r ∈ v(F).

An equivalent definition of these multiplicative norms is given as follows: for $\mathbb{R} \ni \rho \in [0,1]$ one defines

$$\displaystyle\begin{array}{rcl} \vert x\vert _{\rho }& =& \max _{n\in \mathbb{Z}}\vert x_{n}\vert \rho ^{n} \\ \vert x\vert _{0}& =& q^{-r},\ \mbox{ $r$ smallest integer, $x_{ r}\neq 0$};\qquad \vert x\vert _{1} = \sup _{n\in \mathbb{Z}}\vert x_{n}\vert.{}\end{array}$$

(167)

In view of the description of B ^b given in (164), the norms (167) are well-defined on the larger ring B ^b. The completion B of B ^b for these norms, as ρ ∈ (0, 1), contains $B^{+}[ \frac{1} {[a]}]$ for any chosen $a \in \mathfrak{m}_{F}\setminus \{0\}$.

B is the analogue in mixed characteristics of the ring of rigid analytic functions on the punctured unit disk in equal characteristics.

The subalgebra $B^{+} \subset B$ is characterized by the condition

$$\displaystyle{ B^{+} =\{ b \in B\mid \vert b\vert _{ 1} \leq 1\}. }$$

(168)

The extension of rings $B^{+} \subset B^{+}[ \frac{1} {[a]}]$ gives a perfect control of the divisibility as explained in [8], Theorem 6.55. A remarkable property of this construction is that the Frobenius endomorphism on B ^b, +

$$\displaystyle{\varphi: B^{b,+} \rightarrow B^{b,+},\quad \varphi (\sum _{ n\gg -\infty }[x_{n}]\pi ^{n}) =\sum _{ n\gg -\infty }[x_{n}^{q}]\pi ^{n}}$$

extends to a Frobenius automorphism on B ^b and on B ⁺ thus to a continuous Frobenius automorphism $\varphi: B\stackrel{\sim }{\longrightarrow }B$ (i.e. the unique K-automorphism which induces x ↦ x ^q on F) which satisfies $\vert \varphi (f)\vert _{\rho ^{q}} = (\vert f\vert _{\rho })^{q}$, $\forall \rho \in (0,1)$.

The homomorphism (165) (in particular for L = C _K ) extends to a canonical continuous universal (Galois equivariant) cover of L

$$\displaystyle{\theta: B \twoheadrightarrow L.}$$

Appendix 5

In this appendix we explain the connection of the above construction with the archimedean analogue of the Witt construction in the framework of perfect semi-rings of characteristic one [3, 5]. Given a multiplicative cancellative perfect semi-ring R of characteristic 1, one keeps the same multiplication but deforms the addition into the following operation

$$\displaystyle{ x + _{w}y =\sum _{\alpha \in I}w(\alpha )x^{\alpha }y^{1-\alpha }\,,\ I = (0,1) \cap \mathbb{Q}, }$$

(169)

which is commutative provided $w(1-\alpha ) = w(\alpha ),\ \forall \alpha \in I,$ and associative provided that the following equation holds

$$\displaystyle{ w(\alpha )w(\beta )^{\alpha } = w(\alpha \beta )w(\gamma )^{(1-\alpha \beta )}\,,\ \ \gamma = \frac{\alpha (1-\beta )} {1-\alpha \beta } \qquad \forall \alpha,\beta \in I. }$$

(170)

By applying Theorem 5.4 in [3], one sees that the positive symmetric solutions to (170) are parameterized by ρ ∈ R, ρ > 1, and they are given by the following formula involving the entropy S,

$$\displaystyle{ w(\alpha ) =\rho ^{S(\alpha )},\qquad S(\alpha ) = -\alpha \log (\alpha ) - (1-\alpha )\log (1-\alpha )\,,\ \forall \alpha \in I. }$$

(171)

We apply this result to the semi-field $\mathbb{R}_{+}^{\mathrm{max}}$ of tropical geometry. We write the elements $\rho \in \mathbb{R}_{+}^{\mathrm{max}}$, ρ > 1, in the convenient form ρ = e ^T for some T > 0. In this way, we can view w(α) as the function of T given by

$$\displaystyle{ w_{T}(\alpha ) = w(\alpha,T) = e^{TS(\alpha )}\qquad \forall \alpha \in I\,. }$$

(172)

By performing a direct computation one obtains for x, y > 0 that the perturbed sum $x + _{w_{T}}y$ is given by

$$\displaystyle{ x + _{w_{T}}y = \left (x^{1/T} + y^{1/T}\right )^{T}\,. }$$

(173)

The formula (173) shows that the sum of two elements of $\mathbb{R}_{+}^{\mathrm{max}}$, computed by using w _T , is a function which depends explicitly on the variable T. The functions [x](T) = x (for $x \in \mathbb{R}_{+}^{\mathrm{max}}$) which are constant in T describe the Teichmüller lifts. The sum of such functions is no longer constant in T. In particular one can compute the sum of n constant functions all equal to 1:

$$\displaystyle{ 1 + _{w_{T}}1 + _{w_{T}}\cdots + _{w_{T}}1 = n^{T} }$$

(174)

which shows that the sum of n terms equal to the unit of the structure is given by the function of the variable T: T ↦ n ^T.

Proposition 18.

The following map χ is a homomorphism from the semi-ring R generated by the functions T ↦ α ^T , $\alpha \in \mathbb{Q}_{+}$ , and the Teichmüller lifts to the algebra of real-valued functions from $(0,\infty )$ to $\mathbb{R}_{+}$ with pointwise sum and product

$$\displaystyle{ \chi (f)(T) = f(T)^{1/T}\,,\ \forall T > 0. }$$

(175)

The range of the map χ is the semi-ring of finite linear combinations, with positive rational coefficients, of Teichmüller lifts of elements $x \in \mathbb{R}_{+}^{\mathrm{max}}$ given in the χ-representation by

$$\displaystyle{ \chi ([x])(T) = x^{1/T}\,,\ \forall T > 0\,,\ \forall x \in \mathbb{R}_{ +}^{\mathrm{max}} }$$

(176)

The following defines a homomorphism from R to $W_{\mathbb{Q}}(\mathbb{R}^{\flat })$ ,

$$\displaystyle{ f\mapsto \beta (f),\ \ \beta (f)(z) =\chi (f)(\frac{1} {z})\,. }$$

(177)

Proof.

The proof is straightforward using [3]. ⊓⊔

Thus (177) gives the translation from the framework of [3, 5] to the framework of the present paper.