CUBO, A Mathematical Journal

Vol. 23, no. 01, pp. 161–170, April 2021

DOI: 10.4067/S0719-06462021000100161

Idempotents in an ultrametric Banach algebra

Alain Escassut

Université Clermont Auvergne,

UMR CNRS 6620, LMBP,

F-63000 Clermont-Ferrand, France.

alain.escassut@uca.fr

ABSTRACT

Let IK be a complete ultrametric field and let A be a unital commu-

tative ultrametric Banach IK-algebra. Suppose that the multiplicative

spectrum admits a partition in two open closed subsets. Then there

exist unique idempotents u, v ∈ A such that φ(u) = 1, φ(v) = 0 ∀φ ∈
U, φ(u) = 0 φ(v) = 1 ∀φ ∈ V . Suppose that IK is algebraically closed.
If an element x ∈ A has an empty annulus r < |ξ − a| < s in its
spectrum sp(x), then there exist unique idempotents u, v such that

φ(u) = 1, φ(v) = 0 whenever φ(x − a) ≤ r and φ(u) = 0, φ(v) = 1
whenever φ(x−a) ≥ s.

RESUMEN

Sea IK un cuerpo ultramétrico completo y sea A una IK-algebra de

Banach ultramétrica unital conmutativa. Suponga que el espectro mul-

tiplicativo admite una partición en dos conjuntos abiertos y cerrados.

Luego, existen idempotentes únicos u, v ∈ A tales que φ(u) = 1, φ(v) =
0 ∀φ ∈ U, φ(u) = 0 φ(v) = 1 ∀φ ∈ V . Suponga que IK es algebraica-
mente cerrado. Si un elemento x ∈ A tiene un anillo vaćıo r < |ξ−a| < s
en su espectro sp(x), entonces existen idempotentes únicos u, v tales

que φ(u) = 1, φ(v) = 0 cada vez que φ(x−a) ≤ r y φ(u) = 0, φ(v) = 1
cada vez que φ(x−a) ≥ s.

Keywords and Phrases: ultrametric Banach algebras, multiplicative semi-norms, idempotents, affinoid algebras.

2020 AMS Mathematics Subject Classification: 12J25, 30D35, 30G06.

Accepted: 17 March, 2021

Received: 05 November, 2020

©2021 A. Escassut. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462021000100161
https://orcid.org/0000-0003-0002-0194


162 A. Escassut CUBO
23, 1 (2021)

1 Introduction and main Theorem

Ultrametric Banach algebras have been a topic of many resarch along the last years [1], [3], [4],

[5],[6], [10], [11], [12]. The following Theorem 1.1 (stated in [14]) corresponds in ultrametric Banach

algebras to a well known theorem in complex Banach algebra: if the spectrum of maximal ideals

admits a partition in two open closed subsets U and V with respect to the Gelfand topology, there

exist idempotents u and v such that χ(u) = 1, χ(v) = 0 ∀χ ∈ U and χ(u) = 0, χ(v) = 1 ∀χ ∈ V .

In an ultrametric Banach algebra, it is impossible to have a similar result because a partition

in two open closed subsets for the Gelfand topology on the spectrum of maximal ideals then

makes no sense, due to the total disconnection of the spectrum. B. Guennebaud first had the

idea to consider the set of continuous multiplicative semi-norms of an ultrametric Banach algebra,

denoted by Mult(A,‖ . ‖) instead of the spectrum of maximal ideals [14], an idea that later
suggested Berkovich theory [2]. Recall that Mult(A,‖ . ‖) is compact with respect to the topology
of pointwise convergence (Theorem 1.11 in [7]).

The proof of Theorem 1.1, stated in [14], was heavy and involved many particular notions in

a chapter of over 40 pages that was never published. We will use Propositions 2.10, 2.11, 2.12 in

order to assure the unicity. Finally, we will show that if the theorem is proven for affinoid algebras,

that may be generalised to all ultrametric Banach algebras (Proposition 2.12).

Notations: We denote by IK a complete ultrametric field. Given a IK-algebra A, we denote by

Mult(A) the set of multplicative semi-norms of A and if A is a normed IK-algebra, we denote by

Mult(A,‖ . ‖) the set of continuous multplicative semi-norms of A provided with the topology of
pointwise convergence. Next, we denote by Multm(A,‖ . ‖) the set of continuous multplicative
semi-norms of A whose kernel is a maximal ideal of A. Given φ ∈ Mult(A,‖ . ‖), we denote by
Ker(φ) the closed prime ideal of the x ∈ A such that φ(x) = 0.

It is well known that every maximal ideal is the kernel of at least one multiplicative semi-norm

on A (see for example [9]). The algebra A is said to be multbijective if for every maximal ideal M,
A

M
admits only one absolute value that is an expansion of this of IK. It is easily seen that if every

maximal ideal is of finite codimension, then the algebra A is multbijective.

Consider then a multbijective unital commutative ultrametric IK-Banach algebra A. We de-

note by X(A) the set of algebra homomorphisms from A onto a field extension of IK of the form
A

M
where M is a maximal ideal of A. So, for every χ ∈ X(A), the mapping |χ| defined on A by

|χ|(x) = |χ(x)| belongs to Multm(A,‖ . ‖) and this is the unique φ ∈ Multm(A,‖ .‖) such that
Ker(φ) = Ker(χ).

Theorem 1.1. Let A be a unital commutative ultrametric IK-Banach algebra such that Mult(A,‖ . ‖)
admits a partition in two compact subsets U, V . There exist unique idempotents u,v ∈ A such
that φ(u) = 1, φ(v) = 0, ∀φ ∈ U and φ(u) = 0, φ(v) = 1, ∀φ ∈ V .



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Idempotents in an ultrametric Banach algebra 163

Corollary 1.2. Let A be a unital commutative ultrametric IK-Banach algebra such that Mult(A,‖ . ‖)
admits a partition in two compact subsets U, V . Then A is isomorphic to a direct product of two

IK-Banach algebras AU ×AV such that Mult(AU,‖ . ‖) = U and Mult(AV ,‖ . ‖) = V . Given the
idempotent u ∈ A such that φ(u) = 1 ∀φ ∈ U, φ(u) = 0 ∀φ ∈ V , then AU = uA, AV = (1 −u)A.

As an easy consequence, we have Theorem 1.3. A few definitions are necessary:

Definitions and notations: Suppose that IK is algebraically closed. Let a ∈ IK and r, s ∈ IR+
with 0 < r < s. We denote by Γ(a,r,s) the set {x ∈ IK| r < |x−a| < s}. Let D be a subset of
IK, let a ∈ D be such that D ∩ Γ(a,r,s) = ∅ and that r = sup{|a − x|, x ∈ D, |a − x| ≤ r} and
s = inf{|a−x|, x ∈ D, |a−x| ≥ s}. The annulus Γ(a,r,s) is called an empty-annulus of D.

Let A be a unital commutative IK-algebra and let x ∈ A. We denote by sp(x) the set of all
λ ∈ IK such that x−λ is not invertible.

Theorem 1.3. Suppose that IK is algebraically closed. Let A be a unital commutative ultrametric

IK-Banach algebra such that Multm(A,‖ . ‖) is dense in Mult(A,‖ . ‖) and let x ∈ A be such
that sp(x) admits an empty-annulus Γ(a,r,s). Then there exist a unique idempotent u ∈ A and a
unique idempotent v ∈ A such that χ(u) = 1, χ(v) = 0 ∀χ ∈ X(A) satisfying |χ(x) −a| ≤ r and
χ(u) = 0, χ(v) = 1 ∀χ ∈X(A) satisfying |χ(x) −a| ≥ s.

2 The proofs

Proving theorem 1.1 requires some preparation. We will use Propositions 2.10, 2.11 and 2.12 and

mainly Theorem 2.7.

Definitions and notations: Let A be a unital commutative ultrametric IK-Banach algebra

whose norm is ‖ . ‖. We define the spectral semi-norm ‖ . ‖sp as ‖f‖sp = limn→+∞‖fn‖
1
n . By

[13] we have Theorem 2.1 (see also [9], theorem 6.19).

Theorem 2.1. ‖f‖sp = sup{φ(f) | φ ∈ Mult(A,‖ . ‖)}.

Affinoid algebras were introduced by John Tate in [17] who called them algebras topologically

of finite type and are now usually called affinoid algebras. As this first name suggests, such an

algebra is the completion of an algebra of finite type for a certain norm.

Definitions and notation: The IK-algebra of polynomials in n variables

IK[X1, . . . ,Xn] is equipped with the Gauss norm ‖ . ‖ defined as∣∣∣∣∣∣
∣∣∣∣∣∣
∑

i1,...,in

ai1,...,inX
i1
1 · · ·X

in
n

∣∣∣∣∣∣
∣∣∣∣∣∣ = supi1,...,in |ai1,...,in|.



164 A. Escassut CUBO
23, 1 (2021)

We denote by IK{X1, . . . ,Xn} the set of power series in n variables∑
i1,...,in

ai1,...,inX
i1
1 · · ·X

in
n such that lim

i1+...+in→∞
ai1,...,in = 0. The elements of such an algebra

are called the restricted power series in n variables, with coefficients in IK. Hence, by definition,

IK[X1, . . . ,Xn] is dense in IK{X1, . . . ,Xn}. Then IK{X1, . . . ,Xn} is a IK-Banach algebra which is
just the completion of IK[X1, . . . ,Xn] and is denoted by Tn.

By [16] (see also [9]): we have Theorem 2.2:

Theorem 2.2. Every algebra IK{X1, . . . ,Xn} is factorial and all ideals are closed.

A IK- affinoid algebra corresponds to a quotient of any algebra of the form IK{X1, . . . ,Xn}
by one of its ideals equipped with its quotient norm of Banach IK-algebra.

By Theorems 31.1 and 32.7 of [9] (see also [17] and [14]):

Theorem 2.3. Let A be a IK-affinoid algebra. Then A is noetherian and all its ideals are closed.

Each maximal ideal is of finite codimension. Moreover the nilradical of A is equal to its Jacobson

radical. Further, A has finitely many minimal prime ideals.

By Theorems 35.4 in [9] or Proposition 2.8 of III in [14], we have Theorem 2.4:

Theorem 2.4. Let A be a IK-affinoid algebra. Then Multm(A,‖ . ‖) is dense in Mult(A,‖ . ‖)
for the topology of pointwise convergence.

By Theorems 35.4 in [9] we can state Theorem 2.5:

Theorem 2.5. Let A be a reduced IK-affinoid algebra. Then the spectral norm ‖ . ‖ of A is a
norm and is equivalent to the norm of affinoid algebra.

Remark 2.6. The proofs given in [9] for Theorems 2.2, 2.3, 2.4, 2.5 are given for algebraically

closed complete ultrametric field but they hold on any complete ultrametric field.

By Corollary 2.2.7 in [2] we have Theorem 2.7:

Theorem 2.7. Let A be a reduced IK-affinoid algebra such that Mult(A,‖ . ‖) admits a partition
in two compact subsets U1 and U2. Then A is isomorphic to a direct product A1 ×A2 where Aj is
a IK-affinoid algebra such that Mult(Aj,‖ . ‖) is homeomorphic to Uj, j = 1, 2.

Proposition 2.8. Let A be a IK-affinoid algebra of Jacobson radical R and let w ∈ R. The
equation x2 −x + w = 0 has a solution in R.

Proof. Since A is affinoid, by Theorem 2.3, w is nilpotent, hence we can consider the element

u = −
1

2

+∞∑
n=1

(
1
2

n

)
(−4w)n.



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Idempotents in an ultrametric Banach algebra 165

Now we can check that (2u− 1)2 = 1 − 4w and then u2 −u−w = 0.

Proposition 2.9. Let A be a IK-affinoid algebra of Jacobson radical R and let w ∈ A be such that
w2 −w ∈R. There exists an idempotent u ∈ A such that w −u ∈R.

Proof. We will roughly follow the proof known in complex algebra [15]. Let r = w2 −w. We first
notice that 1 + 4r = (2w − 1)2. Next,

r

1 + 4r
belongs to R hence by Proposition 2.8, there exists

x ∈R such that x2 −x +
r

1 + 4r
= 0, and hence

((2w − 1)x)2 − (2w − 1)2x + r = 0.

Now set s = (2w − 1)x. Then s belongs to R, as x. Then we obtain

s2 − (2w − 1)s + r = 0.

Let us now put u = w −s and compute u2:

(w −s)2 = w2 − 2ws + s2 = w + r − 2ws + s2.

But s2 = −r + (2w − 1)s, hence finally:

(w + s)2 = w −r + 2ws + r − (2w − 1)s = w + s.

Thus u is an idempotent such that u−w ∈R.

Proposition 2.10. [14] Let A be a commutative unital ultrametric IK-Banach algebra and assume

that Mult(A,‖ . ‖) admits a partition in two compact subsets U, V . Suppose that there exist two
idempotents u and e such that ∀φ ∈ U, φ(u) = φ(e) = 1 and ∀φ ∈ V, φ(u) = φ(e) = 0 . Then
u = e.

Proof. Put e = u + r. Since e2 = e, we have (u + r)2 = u + 2ur + r2 hence u + r = u + 2ur + r2

and hence r = 2ur + r2, therefore r(2u + r − 1) = 0.

Suppose r 6= 0. Then 2u + r−1 is a divisor of zero. Now, when φ ∈ U, we have φ(1−u) = 0,
hence φ(−1 + 2u + r) = φ(u + r) = φ(e) = 1, and when φ ∈ V , we have φ(u) = φ(e) = 0, hence
φ(1−2u−r) = φ(1−u−r) = φ(1−e) = 1. Hence, ∀φ ∈ Mult(A,‖ . ‖), we have φ(1−2u−r) = 1.
Consequently, 1 − 2u− r does not belong to any maximal ideal of A and hence is invertible. But
then 1 − 2u−r is not a divisor of zero, which proves that r = 0 and hence e = u.

Proposition 2.11. [14] Let A be a IK-affinoid algebra such that Mult(A,‖ . ‖) admits a partition
in two compact subsets U1, U2. There exist unique idempotents e1,e2 ∈ A such that φ(e1) =
1,φ(e2) = 0 ∀φ ∈ U1 and φ(e1) = 0, φ(e2) = 1∀φ ∈ U2.



166 A. Escassut CUBO
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Proof. Suppose first that A is reduced. By Theorem 2.7, A is isomorphic to the direct product

A1 × A2 where Aj is a IK-affinoid algebra such that Mult(Aj,‖ . ‖) = Uj, j = 1, 2. Let Φ
be the isomorphism from A1 × A2 onto A, let uj be the unity of Aj, j = 1, 2 and let e1 =
Φ(u1, 0), e2 = Φ(0,u2). So e1, e2 are idempotents of A. Let A

′
1 = {Φ(x, 0) |x ∈ A1} and let

A′2 = {Φ(0,x) |x ∈ A2}.

Then, given ϕ ∈ Uj, it factorizes in the form ψ ◦ Φ−1 with ψ ∈ Mult(Aj, ‖ . ‖), (j = 1, 2)
and for ϕ ∈ U1, we have ϕ(e1) = 1, ϕ(e2) = 0, and given ϕ ∈ U2, we have ϕ(e1) = 0, ϕ(e2) = 1.
By Proposition 2.10, the idempotents e1, e2 are unique.

We can easily greneralize when A is no longer supposed to be reduced. Let R be the Jacobson
radical of A and let B =

A

R
. Let θ be the canonical surjection from A onto B. Every φ ∈

Mult(A,‖ . ‖) is of the form ϕ ◦ θ with ϕ ∈ Mult(B,‖ . ‖). Let U′1 = {ϕ ∈ Mult(B,‖ . ‖)} be
such that ϕ◦θ ∈ U1 and let U′2 = {ϕ ∈ Mult(B,‖ . ‖)} be such that ϕ◦θ ∈ U2. Then U′1 and U′2
are two compact subsets making a partition of Mult(B,‖ . ‖). Therefore, B has an idempotent
u1 such that ϕ(u1) = 1 ∀ϕ ∈ U′1 and ϕ(u1) = 0 ∀ϕ ∈ U′2. Let w ∈ A be such that θ(w) = u1.
Then we can check that φ(w) = 1 ∀φ ∈ U1 and φ(w) = 0 ∀φ ∈ U2. But by Proposition 2.9,
there exists an idempotent e1 ∈ A such that e1 − w ∈ R. Then χ(e1) = χ(w) ∀χ ∈ X(A) and
hence φ(e1) = φ(w) ∀φ ∈ Mult(A,‖ . ‖) because, by Theorem 2.4 Multm(A,‖ . ‖) is dense in
Mult(A,‖ . ‖). The unicity of e1 follows from Proposition 2.10. Similarly, there exists a unique
idempotent e2 ∈ A such that φ(e2) = 1 ∀φ ∈ U2 and φ(e2) = 0 ∀φ ∈ U1.

Definition and notations: We will denote by | . |∞ the Archimedean absolute value of IR. Given
a unital commutative ultrametric IK-normed algebra A and φ ∈ Mult(A,‖ . ‖), y1, . . .yq ∈ A and
� > 0, we will denote by W(φ,y1, . . . ,yq,�) the set of θ ∈ Mult(A,‖ .‖) such that |φ(yj)−θ(yj)|∞ ≤
� ∀j = 1, . . . ,q. Given a unital commutative ultrametric IK-normed algebra A and a subalgebra
B, we call canonical mapping from Mult(A,‖ . ‖) to Mult(B,‖ . ‖) the mapping Φ defined by
Φ(ϕ)(x) = ϕ(x) ∀x ∈ B, ϕ ∈ Mult(A,‖ . ‖).

Proposition 2.12. [14] Let A be a unital commutative ultrametric IK-Banach algebra and assume

that Mult(A,‖ . ‖) admits a partition in two compact subsets U, V . There exists a IK-affinoid
algebra B included in A, admitting for norm this of A, such that Mult(B,‖ . ‖) admits a partition
in two open subsets U′, V ′ where the canonical mapping Φ from Mult(A,‖ . ‖) to Mult(B,‖ . ‖)
satisfies Φ(U) ⊂ U′, Φ(V ) ⊂ V ′.

Proof. Since U and V are compact sets, we can easily define a covering of open sets (Oj)j∈J such

that Oj ∩V = ∅ ∀j ∈ J. From this, we can extract a finite covering (Ui)1≤i≤n of U where the Ui
are of the form W(fi,xi,1, . . . ,xi,mi,�i) with xi,j ∈ A, such that Ui ∩ V = ∅ ∀i = 1, . . . ,n. Let
à be the finite type IK-subalgebra generated by all the xi,j, 1 ≤ j ≤ mi, 1 ≤ i ≤ n. Consider
the image of Mult(A,‖ . ‖) in Mult(Ã,‖ . ‖) through the mapping Φ that associates to each



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Idempotents in an ultrametric Banach algebra 167

φ ∈ Mult(A,‖ . ‖) its restriction to à and let Ũ = Φ(U), Ṽ = Φ(V ). Then both Ũ, Ṽ are
compact with respect to the topology of Mult(Ã,‖ . ‖) and hence there exist open neighborhoods
U′ of Ũ and V ′ of Ṽ in Mult(Ã,‖ . ‖) such that U′ ∩V ′ = ∅. Let Y = U′ ∪V ′. By construction
we have Φ(U) ⊂ U′, Φ(V ) ⊂ V ′.

Let φ ∈ Mult(Ã,‖ . ‖) \ Y . There exists a finite type algebra Ãφ containing Ã, such that
the canonical image Hϕ of Mult(Ãφ,‖ . ‖) in Mult(Ã,‖ . ‖) does not contain φ. Since this
image Hφ is compact, there exists a neighborhood G(φ) of φ such that G(φ) ∩ Hφ = ∅. Next,
we notice that Mult(Ã,‖ . ‖) \Y is compact, hence we can find φ1, . . . ,φn ∈ Mult(Ã,‖ . ‖) \Y
and neighborhoods Z(φ1), . . . ,Z(φn) making a covering of Mult(Ã,‖ . ‖) \Y . Let E be the finite
type algebra generated by the Ãφi, 1 ≤ i ≤ n. Then E is a IK-subalgebra of A of finite type which
contains à and hence is equipped with the IK-algebra norm ‖ . ‖ of A. Moreover, by construction,
Mult(E,‖ . ‖) is equal to Y = U′ ∪V ′.

Let {x1, . . . ,xN} be a finite subset of the unit ball of E such that IK[x1, . . . ,xN ] = E. Let
T be the topologically pure extension IK{X1, . . . ,XN} and consider the canonical morphism Θ
from IK[X1, . . . ,XN ] equipped with the Gauss norm, into E, equipped with the norm ‖ . ‖ of A,
defined as Θ(F(X1, . . . ,XN )) = F(x1, . . . ,xN ). Since by hypotheses, ‖xj‖≤ 1 ∀j = 1, . . . ,N, Θ is
continuous and has expansion to a continuous morphism Θ from T into A. Let I be the closed ideal
of the elements F ∈ T such that Θ(F) = 0. Then Θ(T) is the IK-affinoid algebra B =

T

I
containing

E and included in A. By construction, the IK-affinoid norm of B is the restriction of the norm ‖ . ‖
of A. Since by construction E is dense in B, we have Mult(B,‖ . ‖) = Mult(E,‖ . ‖) = U′ ∪V ′.
Consequently, Φ(U) ⊂ U′, Φ(V ) ⊂ V ′, which ends the proof.

Remark 2.13. Proposition 2.12 was roughly stated in [14]. However, its proof was confusing

about subsets containing U and V and norms defined on an affinoid subalgebra B, which then puts

in doubt the conclusion.

We can now conclude.

Proof of Theorem 1.1. By Proposition 2.12, there exists a IK-affinoid algebra B included in A

such that Mult(B,‖ . ‖) admits a partition in two open disjoint subsets U′, V ′ and such that
the canonical mapping Φ from Mult(A,‖ .‖) to Mult(B,‖ . ‖) satisfies Φ(U) ⊂ U′, Φ(V ) ⊂ V ′.
Now, by Proposition 2.11, there exist idempotents u′, v′ ∈ B such that φ(u′) = 1 ∀φ ∈ U′ and
φ(u′) = 0 ∀φ ∈ V ′. Consequently, we have φ(u) = 1 ∀φ ∈ U, φ(u) = 0 ∀φ ∈ V and φ(v) = 0 ∀φ ∈ U,
φ(v) = 1 ∀φ ∈ V . The unicity follows from Proposition 2.11. That ends the proof.

Proof of Theorem 1.3. Without loss of generality, we can suppose a = 0. Let U = {φ ∈
Mult(A,‖ . ‖)} such that φ(x) ≤ r, and let V = {φ ∈ Mult(A,‖ . ‖)} such that φ(x) ≥ s.
Since Multm(A,‖ . ‖) is dense in Mult(A,‖ . ‖), it is clear that no φ ∈ Mult(A,‖ . ‖) can sat-
isfy r < φ(x) < s. Consequently, U,V make a partition of Mult(A,‖ . ‖). Next, one can easily



168 A. Escassut CUBO
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check that U and V are open and closed with respect to the pointwise convergence. Indeed, given

φ ∈ Mult(A,‖ . ‖), g1, . . . ,gt ∈ A and � > 0, we denote by W(φ,g1, . . . ,gt,�) the neighborhood of

φ defined as {θ ∈ Mult(A,‖ . ‖) |φ(gj) − θ(gj)|∞ ≤ � ∀j = 1, . . . , t}. So, let � ∈
]
0,
s−r

2

[
and consider the families of neighborhoods of U and V of the form W(φ,x,f1, . . . ,fm,�)φ∈U

and W(ψ,x,g1, . . . ,gn,�)ψ∈V respectively. Then given any W(φ,x,f1, . . . ,fm,�), φ ∈ U and
W(ψ,x,g1, . . . ,gn,�), ψ ∈ V we have W(φ,x,f1, . . . ,fm,�) ∩ W(ψ,x,g1, . . . ,gn,�) = ∅ hence U
and V are two open subsets such that U ∩V = ∅. By construction Mult(A,‖ . ‖) = U ∪V . Con-
sequently, U and V are two open subsets making a partition of Mult(A,‖ . ‖), which by Theorem
1.1, ends the proof.

Remark 2.14. The proof of Theorem 1.3 consists of injecting the Krasner-Tae algebra [8]

H(Γ(a,r,s)) into A.

Acknowledgement: I am grateful to the Referee for useful remarks.



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Idempotents in an ultrametric Banach algebra 169

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	 Introduction and main Theorem
	The proofs