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Vs an<^ hence, £ o 7 has follower v such that ip < v. Let 5 be a subsemigroup of B such that v = (£ o 7 ) s and let z/' = C ( s ) . Then i / ^ v ' o ) . On the other hand, v' is the follower of £ in the set X' = X U {V}, hence X ' is a constructible set, so X' clearly belongs to Q, a contradiction with the hypothesis X maximal. This proves that for every u £ Z(B, 9) there exists 1/ e Z(B', 9') such that v = v' o 7. Now, we can show that Min(B,9) C j(Min(B',9')). Let < e Min(B,9). As we just showed, there exists £ € Z(B',9') such that C = £ o 7. Let F = 7(B) \ £ - 1 (°)- Then Y is a subsemi-group of (B',.) and we can check that £ is ^-multiplicative. By Lemma 5.13 there exists (' e Min(B',£) such that C'(x) = £(x) Vrr e Y", hence ( = ( ' 0 7 . But of course Min(B',^) C Min(B',9'), hence / belongs to j(Min(B',9')). D
Multiplicative
Semi-Norms
and Shilov Boundary
29
We must recall two classical results [4,5]: Theorem 5.16 (Urysohn): D be a compact and let a, f3 be two different points of D. There exists a continuous mapping f from D into K + such that f (a) = 0 and f {13) = 1. Lemma 5.17: Let D be a compact. The Gelgand mapping UJ from D into X(C(D,R),R) defined as uj(a) = <j>a, with <j>a{f) = f(a), f e C(D,R) actu ally is a bijection from D onto X(C(D, R), C) and the mapping u) defined as u>(a) = \<j>a\, with <j>a(f) = |/(fl)|, / € C(D,R) is a bijection from D onto Mult(C(D,R),\\ ■ \\D). Lemma 5.18: Let D be a compact, and let
(a)(f) = |/(a)|oo- Then <j> is a bijection from D onto Mult(C(D,R), \\ . \\D) which is equal to Min(C(D,R), || . | | 0 ) Proof: By Lemma 5.17 the mapping (j> from D into Mult(C(D, R), \\ . \\D) defined as 0(a) = ipa, with iba(f) = | f(a)| / G C(D,R), is a bijection from D onto Mult(C(D,R),\\ . \\D). By Corollary 5.12 Min(C(D,R),\\ . \\D) is included in Mult(C(D, R),\\ . \\D), so we just have to show that Mult{C(D,R),\\ isabijection from F onto Min(C(F,R), || . \\p). Consequently, ^ 0 7 is of the form («). And finally, for every x € A we have */(x) = 7(0(a)(x) = (0(a) o 7 )(x) = |7(x)(a)|oo = |a(x)|oo = a(x), because by definition a lies in Mult(A, || . ||). Thus f belongs to F. There fore Min(A, || . ||) is included in F and then the closure of Min(A, || . ||) is the smallest of all closed boundaries for (^4, || . ||). □ Corollary 5.20: Let (E, \ . \) be ultrametric and let || . \\ be a semimultiplicative E-algebra semi-norm. For every x G A, there exists ip G Mult(A, || . ||) such that ip(x) = ||x||. R e m a r k : In Chapter 22 we shall see that a minimal boundary for the spectral semi-norm of an ultrametric algebra is not necessarily closed. belongs to Mult(A). Lemma 6.5: If is surjective then <j>* is injective. If <j> is an isomor phism, then * is a homeomorphism from Mult(A') onto Mult(A) with respect to topologies of simple convergence on both sets. Proof: Suppose <j> is surjective and let ipi, ip2 € Mult(A') be such that j(xj) is a bijectionfrom\J9j=1Mult(Aj) onto Mult (A, || . ||). Proof: * is a bijection from Mult(A', || . ||') onto Mult(A, || . ||). We will check that <j>* is bicontinuous. Let ip e Mult(A',\\ . ||') and let V = , <> / ° / 1 , ..., o / „ , e) is a neighborhood of ■)> / whose image by Q. r £ Mult(R(D), || . \\D). First we will show that every neighborhood of ip^, with respect to the simple convergence topology, which is of the form Mult{R{EC\D), || . HEI-ID), where E is an affinoid set, contains a neighborhood of 0, and let h\,...,hq £ A be such that \\fj — hj\\ < | Vj = l,...,q. Then V( (y) = fy is an isometry from B onto fB. Therefore, jB is complete, and closed in B. Consequently, B', provided with its quotient norm, is a Banach K-algebra, and therefore is a K-affinoid algebra. Let 9 be the canonical surjection from B onto B', and for each i = 1 , . . . , r, let Xj = 9(Xi). Then, B' = K{x\,..., xr}. Moreover in B' we have f(x\,... ,xr) = 0. Now, for every x e B', we denote by x the residue class of x. Let / € /C[Yi,... Yr] be the residue class of / . 187 is a K-algebra automorphism of K[Xi,.. .Xn}- More over, it also induces on IC[Xi,... ,Xn] a /C-algebra automorphism, so it satisfies ||0(/)|| = ||/||. Therefore 0 has continuation to Tn and pre serves Vn. Now, let ui be the /^-algebra automorphism of Tn defined as w(f(Xi,..., Xn)) = f{Xn, X2,..., X n _ i , Xi). Then w o (Pn) - i>(Pn)U n*
. \\D) C Min(C(D,R),\\
. \\D).
Let a € D, and let us show that
30
Ultrametric Banach Algebra
On one hand, by Lemma 5.6 every element ip of Mult(A, || . ||) by definition satisfies ||x|| > sup{i/;(x) j rp € F}. On the other hand, since F is a boundary for (A, || . ||) there exists ip € F such that ||x|| = sup{ip(x) | ip € F}, and consequently, | | 7 ( X ) | | F = ||x||. Now, since || • ||F ° 7 = || • || and since 7 is a semi-group homomorphism from (A,.) into (C(F, R),.), we can apply Lemma 5.15 and we have Min(A,\\
. | | ) C 7 ( M m ( C ( F , R ) , | | . \\F)).
Let v € Min(A, || . ||). So, there exists ip e Min(C(F,R), || . \\F) such that v = ip o 7. Now, let
Chapter 6
Spectral Semi-Norm
We shall recall the basic properties of continuous multiplicative semi-norms in an ultrametric normed algebra: the set of continuous multiplicative seminorms is compact for the topology of simple convergence, and its superior envelope is a semi-multiplicative semi-norm called the spectral semi-norm. Applying Chapter 5, we will show that the set of continuous multiplicative semi-norms admits a Shilov boundary. N o t a t i o n : Throughout the chapter, L will denote a field provided with a non trivial ultrametric absolute value, and we will denote by (A, || . |j), (A', || . ||') commutative L-algebras with unity. We will show the existence of a spectral semi-norm equal to the supremum of continuous multiplicative semi-norms. 4> will be a L-algebra homomorphism from A into A' and we will denote by
31
32
Ultrametric Banach
Algebra
According to Lemma 1.11 in [30], we have Lemma 6.2: Lemma 6.2: Mult(A, || . \\) is compact with respect to the topology of simple convergence. Corollary 6.3: Let t € A. Then {<j>{t) \<j> € Mult{A, || . \\)}) is a closed bounded subset ofR. Lemma 6.4 is obvious: Lemma 6.4: Let
By Lemma 6.5 it is an injection. Putting
ui = ( 1 , 0 , . . . , 0), . . . , uq = ( 0 , , . . . , 0,1), for each j = 1 , . . . , q, we check that for every tp € Mult(A, || . ||) we have tp(uk) = 1 for a certain k and
of <j>*
to
Notation: In the hypothesis of Lemma 6.7 we will denote by the restric tion of 4>* to MuZt^MI . II')-
Spectral Semi-Norm
33
Theorem 6.8: Suppose that A' is a dense L-subalgebra of A and assume that
V(y> o 0, / l t . . . , / n ) e) = (^)-1(V(<^,gi, ...,<£, e)), which finishes proving that (*)" * also is bicontinuous.
□
N o t a t i o n : When A' is a dense L-subalgebra of A henceforth, we will consider that Mult(A, \\ . ||) and Mult(A', || . ||) are equal. Theorem 6.9: For every x e A the sequence (|| xn | | " ) n G N has a limit denoted by || x \\Si, satisfying ||x|| S j < ||x|| Vx € A and the mapping f defined in A as f(x) = \\ x ||Sj belongs to SM(A, \\ . ||) and is ultrametric. Moreover || x ||Sj < 1 if and only i/lim n _ ) 0 0 ||x™|| = 0. Proof: The statement is well known and proven, for instance, in [30] as Theorem 1.13 when the semi-norm || . || is a norm. Actually the proof holds all the same when it is just a semi-norm. □
34
Ultrametric Banach
Algebra
By Theorem 5.19 we have Corollary 6.10: Corollary 6.10: There exists a Shilov boundary for {A, \\ . \\si) which is included in Mult(A, || . ||) and is equal to the closure of Min(A, \\ . ||Sj) in Mult(A,\\ . ||). Theorem 6.11: Let f £ limn^oo||(A/)n||=0VA€L.
A.
Then \\f\\si
= 0 if and only if
Proof: If ||/|| s i = 0, then of course ||A/|U = 0, hence l i m ^ ^ ||(A/)"|| = 0 VA £ L. Now, suppose lim^oo ||(A/) n || = 0 VA £ L. Suppose ||/|| s i ^ 0, and let A € L be such that |A| > ,, }„ ,. Then linin^oo j|(A/)"j| S j = +oo, which contradicts the hypothesis limjj-xx, ||(A/) n || = 0 . D Definition: (A, || . ||) will be said to be uniform if the norm || . || is equivalent to || . || s i. In the same way as in archimedean analysis, we must also mention this proposition: Proposition 6.12: || . || and || . ||5, are two equivalent norms on A if and only if there exists C £ R+ such that \\x2\\ < C||a:||2 Vx £ A. And they are equal if and only if ||x"|| < ||x||" Vn £ N*, Vx £ A. Theorem 6.13: Let \\ . \\ be a norm of L-algebra on A. Then || . || is semi-multiplicative if and only if the semi-norm |J . ||Si associated to || . || is equal to || . ||. On the other hand, if A is complete for its norm, then for every x £ A satisfying \\x\\Si < I, 1 — x is invertible in A, the set of invertible elements in A is open, each maximal ideal of A is closed and for every ip £ X{A, L), the mapping \ip\ from A into M+ defined as \ip\(x) = \ip{x)\ belongs to Mult{A,\\ . \\si). Proof: By Theorem 6.9, if || . || s , is the i-algebra norm, then it is semimultiplicative. Conversely, if the L-algebra norm is semi-multiplicative, it is equal to || . ||Si because || xn ||« = ||x|| for every x £ A. Now assume that A is complete. Let x £ A satisfy ||x||Si < 1. By Theorem 6.9 we have lim^-x^ xn — 0 and then the series Xl^Lo xU converges in A to a limit S that obviously satisfies S(l — x) = 1. As a consequence, the set of invertible elements in A is open. Since a maximal ideal is either closed or dense, here each maximal ideal of A is closed. Moreover, for every A £ L satisfying ||:r||Si < |A|, 1 — f is invertible in A. Now, |i/;| obviously belongs to Mult(A) . Assume that \i>(x)\ > \\ x \\Si
Spectral Semi-Norm
35
for some x € A. Let A = ip(x) € L and let y = 1 — f. Then ip(y) = 0, hence 1 - | G Ker(ip). But |A| > || x \\si hence | | | || si < 1, so 1 - f is invertible. This just contradicts 1 — f G Ker(tp). Finally |T/>(X)| < || x \\si for every x G ^4, hence \4>\ € Mult(A, \\ . \\si). D Theorem 6.14 was given in several works: Theorem 1.16].
[33], [44] and [30,
Theorem 6.14: Let F be a field extension of L provided with a L-algebra semi-norm of L-algebra \\ . \\. Then || . || is a L-algebra norm and there exists an ultrametric absolute value (p on F extending that of L, such that
Given x
S
A,
then \sp(x)\
C
{
Theorem 6.17: Let F,G be a partition of Multm(A, \\ . ||) and suppose that there exists an idempotent u such that
=0VxeG.
□
36
Ultrametric Banach
Algebra
Proposition 6.18 is classical (in particular, we can obtain it by Proposi tion 1.17, and Proposition 4.1 in [30]): Proposition 6.18: Let A be a normed L-algebra whose norm || • || is ultrametric, let r e]0, +oo[ and let B be the set of the series ]C^Lo anXn such that limn_»oo Il a n|k n = 0. Then B, provided with the multiplication of series, is a commutative algebra which contains A and admits 1 for unity. Let || ■ || r be defined on B by \\ Y^=oan.xn\\r - sup n € N ||an|k n . Then || • ||r is a norm of L-algebra on B. Moreover, if the norm of E is multiplicative, then so is || . || r . Further, if A is complete, then so is B. Theorem 6.19: Given ip e Mult(A), tp belongs to Mult(A, || . ||) if and only if it satisfies (p(t) < \\t\\si whenever t £ A, and we have Mult(A, || . ||) = Mult(A, || . || 5 i ). Further, for every t e A we have
11*11* = supMt) |
Chapter 7
Hensel L e m m a
Definition and notation: Recall that L denotes a field provided with a non trivial ultrametric absolute value | . |. Let E be a commutative ultrametric normed L-algebra with unity. Let n € N and let || . ||o be the norm of E. The S-algebra of polynomials in n variables E[X\,... ,Xn) is provided with the Gauss norm || . || denned as /
,
a
»i,-.»„-^i: •••^n"
sup
\\ailt..
By Proposition 6.18 this norm is a L-algebra norm. In particular, if the norm of E is multiplicative, so is the norm || . ||. We denote by E{X\,..., Xn} the set of power series in n variables / ^—J
a
h,...,in^i1 ' '
■ ---^n"
s u c n
t n a t
u
m
a
*i,—,t„ — 0-
ii + ...+in—*oo
The elements of such an algebra are called the restricted power series in n variables, with coefficients in E. Hence by definition E[X\,..., Xn] is dense in E{X\,..., Xn}. By Proposition 6.18, if E is complete, E{Xi,...,Xn} is just the completion of E[Xi,...,Xn]. Throughout the chapter, A will denote a commutative L-algebra with provided with an absolute value | . | which extends that of L and satisfies { \t\ | x € A} = \L\. We will denote by v(.) the valuation associated to the absolute value, by V the subring {t e A | \x\ < 1} of A and by M the prime ideal {t € A \ \t\ < 1} of V. Then it is seen that M = MLV. Moreover, the absolute value || . || denned on A[x] obviously can be extended 37
38
Ultra-metric Ba.na.ch Algebra,
to yl{a;}. The valuation defined on A{x} by the Gauss norm || . ||, which is an absolute value extending that of A, will be denoted by v(.) again. Let W = {/ G A{X) | H/ll < 1} and let M = {/ G A{X} | ||/|| < 1}. Given / G W we will denote by a the residue class of a in ^ . Given / = Xi^Lo a " ^ " E A{x}, here we will denote by N(f) the greatest of the integers m such that |a m | = ||/||. Lemmas 7.1 and 7.2 are immediate: Lemma 7.1:
M = MLV, M = MLW,
Lemma 7.2
Le£ / G A{x) be such that ||/|| = 1. Then N(f) = deg(/).
Lemma 7.3
Let / ,
5
$ =
G ,4{x}. Then N(fg)
£[x].
= AT(/) + JV(5).
Proof: Suppose first that v(f) = w(g) = 0. Let q = AT(/), < = N(g). By Lemma 7.1, in -^[x], / is a polynomial of degree q, g~ is a polynomial of degree t, hence / v(P) andv(R) >v(P). Proof: We can clearly assume P / 0. Then, by multiplying P by a suitable constant A G L, we can also assume v(P) = 0. Since D is monic, the Euclidean division of P by D is clearly possible in V[x], and therefore Q is the quotient, R is the rest of this division, due to the fact that deg(R) < deg(D). So we have v(Q) > 0, v{R) > 0 because both Q, R belong t o V [a;]. D Theorem 7.5: Let A be complete with respect to \ . \. Let f G A{x} and let D G V[x] be monic. There exists a unique pair (g, R) G A{x} xA[x] such that f = Dg + R, and deg(i?) < deg(D). Moreover, we have v(g) > v(f) andv(R) >v(f). Proof: Since A is complete, then so is A{x} with respect to the norm || . ||. Consider the mapping , defined on A[x], which associates to each P G A[x] the quotient Q in the division by D. By Lemma 7.4 <j> is continuous with respect to the absolute value || . ||. Since A{x} is complete and since A[x] is dense in ^4{x}, then
Hensel Lemma
39
defined on A[x] which associates to each P G A[x] the rest R in the division of P by D. By Lemma 7.4 ip is continuous, therefore it can be continuously extended to a mapping i\) from A{x\ to ^4[x] such that v(ip(f)) > v(f) and deg(ip(f)) < deg(D). Thus, by putting g = (/) and i? = ?/>(/) we have proven the existence of the pair (g,R). Now, suppose we have another pair (h, S) e A{x} x A[x] such that f = Dh+ S, and deg(S) < deg(D). Then D(h - g) = R- S. Since D is monic, by Lemma 7.3 we have N(D(h - g)) > deg(D), while N(R - S) < deg(D) — 1, a contradiction. Hence h = g, S = R. □ Corollary 7.6: Let A be complete with respect to \ . \. Let f € A{x} and let D 6 V[x] be monic. If D divides f in A{x} then it divides f in A[x}. N o t a t i o n : Given a ring B and g, h € B[x], I(g, h) will denote the ideal generated by g and h in B[x\. Lemma 7.7: Let g,h G V[x] be monic, such that Klj, h) = ^[x] and let q € N be such that q < deg(g) + deg(/i). There exist S, W € L[x] satisfying v(Sg + Wh-
xq) > 0, v(S) > 0, v(W) > 0,
deg(5) < deg(ft), deg(W) < deg(g). Proof: Since T(lj,h) = -b\x\, there exists
(7.1)
Moreover, v(B) = 0. We now consider the Euclidean division of Bxq by h and Exq by g, respectively. We obtain Bxg = B^h + S and Exq = E0g + W. By Lemma 7.4 we have min(v(S),v(B0)) > v(Bxq) = 0. Next, by hypothesis we have deg(S) < deg(/i)
(7.2)
deg(W) < deg(g)
(7.3)
and
Let F = Bg + Eh-1. Then we have Bxq = (B0 + E0)gh + Sg + WhSince q < deg(
xq) < deg( 5 ) + deg(/i)
xq.
40
Ultrametric Banach
Algebra
and therefore Sg+Wh~xq is just the remainder of the Euclidean division of Fxq by gh. But then, by Lemma 7.4, we have v(Sg+Wh~xq) > v(Fxq) = v(F). And by (7.1) and by definition of F it is seen that v(F) > 0. This finishes proving that v(Sg + Wh — xq) > v(xq). □ Notation: Let g,h 6 V[x] be monic and satisfy I{g,h) — j^[x]- We will denote by E(g, h) the set of constants A € R + such that, for every polynomial Q £ L[x\ satisfying deg(Q) < deg(g) + deg(/i), there exist F,G € L[x) satisfying v(Fg + Gh - Q) > v(Q) + A, v(F) > v(Q), v(G) > v(Q), deg(F) < deg(ft), deg(G) < deg(g). Lemma 7.8: Let g,h e V\x], be monic and satisfy X(g,h) = j^[x], and let d = deg(g) + deg(h). Then E(g,h) is a non empty interval whose lower bound is 0. Moreover, given A € E(g,h), and monic polynomials
(0 < n < d - 1).
Now let Q = £ ^ o a « * n > let S = E ^ o a « S « > W = EtJo^Wn A = min 0 < n
min
and let
(v(an) + A„)
0
>
min
v(an) +
~~ 0
But trivially: v(S) > min 0 < n
min
A„ = v(Q) + A.
0
v(S) > v(Q),
max (deg(5 n )) < deg(h), deg(W) 0
So, A lies in E(g, h). Then it is obviously seen that E(g, h) is a non empty interval, and that its lower bound is 0. Now, let y e E{g, h) and let <j>, ip e V[x] be monic and satisfy v{g — (f>) > y, v{h — ip) > y. Since v(S) > v(Q), v(W) > v(Q), it is easy to see that
Hensel Lemma
41
v{S(g - <j>) + W{h -ip) > y + v{Q), and therefore v{Scj> + Wip - Q) > V + v(Q)- This shows that y € E(<j>,ip), and therefore E(g,h) C E(4>,ip). But similarly we have E(<j>, i/>) C E(g, h). □ Lemma 7.9: Let A be complete and let f(x) = J27Lo aix^ € A{x} be such that 11/11 = \aq\ = 1, with N(f) = q. Let g,h € V[x] be monic and satisfy Z(g,h) = ^\x]. Let A € E(g,h). There exist monic polynomials S, W € L[x] satisfying v(Sg + Wh-f)>\
+ v(f),
deg(S) < deg( ff ),
deg(W) < max(deg(/i),v(f),
v(W)>v(f).
Proof: By Theorem 7.5 we consider the Euclidean division of / by gh : f = £gh + Qx. Hence &eg{Q\) < deg(ff) + deg(/i). By Theorem 7.5 we have v(Qi) > v(f),
(7.4)
v(l) > v{f)
(7.5)
By Lemma 7.8 there exist Si,W\
G L[x] satisfying
v(Sl9 + Wlh-Q1)>
v{Qi) + X(g,h),
«(5i) > v(Qi), viWJ
> t/(Qi),
(7.6) (7.7) (7.8)
degfSO < deg(h),
(7.9)
deg(Wi) < deg( 5 ),
(7.10)
Now we put S = Si+eh, W = Wx. So we have Sg + Wh - f = Si+lgh+Wih-tgh-Qi and therefore by (7.6) we obtain v(Sg+Wh-f) > v(Qi) + A. Hence by (7.4) we obtain v(Sg + Wh-f)>v(f)+X.
(7.11)
By (7.4), (7.8) it is seen that v{W) > v(f).
(7.12)
Next, we have v(h) = 0, hence by (7.5) we see that v(£h) > « ( / ) .
(7.13)
42
Ultrametric Banach
Algebra
But by (7.7) we have v(Si)
> v{f) and therefore by (7.13) we obtain v{S) > v(f).
(7.14)
Finally, by definition we have deg(£) = deg(/) — deg(gh) and therefore deg{W) < max(deg(5 1 ), deg(lh)) < max(deg(/i), q - deg(ff)). Thanks to (7.10), (7.11), (7.12), (7.14), (7.15).
(7.15) □
Theorem 7.10 (Hensel Lemma): A is supposed to be complete. Let f € A{x} be such that ||/|| = 1 and such that f splits in -^ [x] in the form 777 with T{~f, 77) = ^ [ x ] . There exists a unique pair (g, h) G A[x) x A{x} such that g is monic and satisfies f = gh, 3 = 7, h = 77, deg(g) = deg(7). Proof: We can obviously take monic polynomials go, ho € V[x] such that ffo = 7, ho = T). We put v = v(f — goho), and take r € E(go, ho) satisfying T < v. We will construct sequences {gn)nen, {hn)neN in L[x] satisfying for all n > 0: {in) ("n) {iiin) K ) {vn) {vin)
v{f - gnhn) >{n + 1)T, v{gn - gn-i) > nr,v{hn - hn-i) > TIT, deg(/i„)j5 deg(/) - deg{g0), deg{gn) = deg{g0), lhl = J,hn = T], r G E{gn,hn). gn is monic.
First we put /1 = / — go^o- We notice that d e g ^ ) = deg(/). We now apply Lemma 7.9, to the case when {Q,g, h) = {f\,go, ho): there exist S\,Wi e A[x] satisfying deg(Wi) < deg(3o), deg(S1)<deg(/)-deg(5o), v{S,) > r, v{Wx) > r, v{Sig0 + W1h0-P1)>T
(7.16) (7.17) (7.18)
+ v{P1).
(7.19)
Next we put g\ = go + W\, h\ = h0 + S\. We check that {i\),{iii),{iiii),{ivi) are satisfied. Moreover, by (7.17) and (7.18), and by Lemma 7.8, r lies in E{g\,hi), hence vi is satisfied. Now we suppose we have already constructed pairs {gm,hm) satisfying {im), ("m), {Him), {ivm), {vm) for every m = 0, . . . , n . Then we put
Hensel Lemma
43
fn+i = f — 9nhn. We can apply Lemma 7.9 to the case when (Q,g, h) is equal to (fn+i>9n,hn). So, we can obtain Sn+i,Wn+i 6 A[x] satisfying v{Sn+1gn
+ Wn+ihn - fn+i)
deg(Wn+1)
>T + u(/„+i),
< deg( 5 n ),
(7.20) (7.21)
deg(S n + i) < max(deg(/i n ), deg(/ n + i) - deg(s„)),
(7.22)
v(Sn+1)
(7.23)
> v(fn+1),v(Wn+1)
> v(fn+1).
By (7.20), and by vn) we obtain v(Sn+1gn
+ Wn+lhn
- fn+1)
>(n + 2)T.
Now we put gn+i = g-n + Wn+i, hn+i — hn + Sn+i. / - gn+ihn+i = (fn+i - hnWn+i + (
_
We check that
- gnSn+i) -
= fn+i — hWn+\ — gn+iSn+i
(7.24)
Sn+iWn+i + (hn+\ — hn)Wn+i
gn)Sn+l + S n + i W n + l .
By (7.21) we notice that Gm+\ is monic, hence vin+{) is satisfied. By (iim) true for m < n, we notice that v(gn ~ gn+i) >{n + 1)T, v(hn - hn+i) > (n + l ) r
(7.25)
which gives (M n+ o), and by (7.23) and (7.24), we obtain (in+i) v(f — gn+ihn+i) > (n + 2)T. Relations ( m n + i ) , (ivn+i) are easily checked. By (7.25) and Lemma 7.8 Relation (vn+i) is also clear. Therefore the sequences (gn)nen, (MneN satisfying (?„), (« n ), (m„), (ivn), (vn) are now constructed. Since A is complete, the yl-module Fq[x] of the poly nomials of degree m < q is obviously complete with respect to the Gauss norm. Then by Relations iin) both sequences (gn)n
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Chapter 8
Infraconnected Sets
Infraconnected sets were introduced in order to provide a class of sets as wide as possible, aimed at playing the same role as connected sets do in complex analysis towards holomorphic functions. We will see that for many things, this class of sets is quite satisfactory. Many proofs of results given in this chapter can be found in [30, Chapter 2]. N o t a t i o n a n d definitions: Throughout this chapter the valuation of L is supposed to be dense. Let a e L and let r € K + . We will denote by d(a, r) the disk {x € L\ \x — a\ < r } , by d(a,r~) the disk {x € L\ \x — a\ < r} and we will denote by C(a,r) the set = {x € L | \x — a\ = r}. Given r\ and r^ such that 0 < r\ < T2 we will denote by Y(a,r\,T'i) the annulus {x S L\ r\ < \x - a\ < r2] and by A(a,r\,r2) the annulus {x € K\ ri < \x - a\ < r 2 } . We know that if b e d(a,r) then d(b,r) = d(a,r). In the same way if b € d(a,r~) then d(b,r~) = d(a,r~). Moreover given two disks T and T' such that T f l T ' ^ 0 then either T c f o r T ' c T. Of course the following three statements are equivalent: (i) d{a,r) = d(a,r-); (ii) C(o,r) = 0; (iii) ri\L\. Moreover the disks d(b,r~) included in C(a,r) (resp. in d(a,r)) are the disks d(b,r~) such that b € C(a,r) (resp. in d(a1r)). They are called the classes of C(a, r) (resp. of d(a,r)). Henceforth D will denote a subset of L. 45
46
Ultrametric Banach
Algebra
The closure of D is denoted by D and the interior of D is denoted by D. We put diam(D) = sup{|a; — y\ \x € D,y e D} and diam(D) is named the diameter of D. If -D is bounded of diameter R we denote by D the disk d(a, R) for any a G D. If Z) is not bounded we put D — L. Given a point a G £ we put £(a, D) = inf{|x — a\ \x € D}. Then 5(a, D) is named £/ie distance of a to D. Lemma 8.1: D\D admits a unique partition of the form ( T ^ g / , where each Ti is a disk of the form d(ai,r~) with r» = <5(a,, D). Definition:
Such disks d(a,i,r~) are called the holes of D.
Examples: (1) The holes of a disk d(a,r~) with r G | L \ are the classes of C(a,r). (2) The only one hole of L \ d(0,1") is d(0,1"). (3) The holes of L \ d(0,1) are the disks d(a, 1~) with a e d(0,1). Definition: A set D is said to be infraconnected if for every a E D, the mapping J a from D to K + defined by Ia{x) = \x — a\ has an image whose closure in R + is an interval. In other words, a set D is not infraconnected if and only if there exist a and b G D and an annulus T(a,r\,r2) with 0 < 71 < r 2 < |a - 6| such that T(a,n,r2) H D = 0. D is said to be strongly infraconnected if for every hole T — d{a, r~) with r € \ L \, there exists a sequence (xn)n€f^ in D such that \xn — a\ — \%n — xm\ — r whenever n ^ m. Lemma 8.2: If D is infraconnected of diameter /„(£)) = [0,ii] (resp. Ia(D) = [0, +oo[).
fi£l
(resp. +oo) i/ien
Lemma 8.3: Lei D 6e strongly infraconnected. Let a € D and let r G | L | 6e 5«c/i i/iai r < diam(D). There exists a sequence (xn)nefi in D such that \xn — a\ = \xn — xm\ = r whenever n ^ m. Corollary 8.4:
A strongly infraconnected set is infraconnected.
Lemma 8.5: Let D be infraconnected and let a belong to a hole T of diameter p. The closure of the set {\x — a\ \x G D} is an interval whose lower bound is p. Theorem 8.6: Let A and B be two infraconnected subsets of L such that A n B ^ 0. Then Au B is infraconnected.
Infraconnected
Sets
47
Notation: Let TZD be the relation defined in D as xTZoy if there exists an infraconnected subset of D containing both x, y. Corollary 8.7:
The relation TZD is an equivalence relation.
Definition: The equivalence classes with respect to TZD are called the infraconnected components. Examples: (1) d(0,1~) U d ( l , l ~ ) is infraconnected. Its holes are the disks d(a, 1~) with |a| = \a — 1| = 1. (2) Let r e}0,1[ and let D = d(0, l ~ ) U d ( l , r ) . Then D is not infracon nected, its infraconnected components are d(0,1") and d(l,r). The holes of D are the disks d(a, 1~) with |a| = |a — 1| = 1 and the disks d(a, \a — l\~) with r < \a — 1| < 1. Theorem 8.8: Let A and B be infraconnected subsets such that A = B. Then Al) B is infraconnected. Definition: such that
We will call an empty annulus of D an annulus
T{a,r\,r2)
(i) r\ = sup{|x — a\ \x G D, \x - a\ < r 2 } , (ii) r-2 = inf{ \x - a\ \x e D, \x - a\ > r i } . The set d(a,ri) n D will be denoted by J£>(r(a,ri,r2)) while the set (L \ d(a,r2~)) n D will be denoted by f£>(r(a,ri,r 2 )). When there is no risk of confusion about the set D we will just write J ( r ( a , r 1 , r 2 ) ) , (resp. £ ( r ( a , r i , r 2 ) ) ) instead of J D ( r ( a , r i , r 2 ) ) , (resp. £ o ( r ( a , r i , r 2 ) ) ) . R e m a r k s : (1) By definition, D is not infraconnected if and only if it admits an empty annulus. (2) If a set D admits an empty annulus r ( a , r i , r 2 ) , then {J(r(a,ri,r 2 )),£(r(a,7-i,r 2 ))} is a partition of D. Examples: Let r e]0,1[, let D = d(0,r)ud(l, l~) and let D' = d(0,r~)u d(l,r). Then r(0,r, 1) is an empty annulus of D and also of D'. In the same way r ( l , r , 1) is also an empty annulus of D'. Notation: Let O(D) be the set of empty annuli of D. Given Ai and A2 £ O(D), it is easily seen that ^(A!) c 2"(A2) is equivalent to £(Ai) D £(A 2 )- We will denote by < the relation defined on O(D) by Ai < A2 if I ( A i ) C J(A 2 ). It is easily seen that < is an order relation on O(D). We will denote by < the relation defined by Ai < A2 if Ai < A2 and Ai ^ A2.
48
Ultrametric Banach
Algebra
The following Lemmas 8.9 and 8.10 are easily seen. Lemma 8.9: Let Ai and A2 be two empty annuli of D. The following assertions are equivalent: (i) (it) (in) (iv)
Ai and A2 are not comparable with respect to the order <; I ( A i ) C 5(A 2 ); I ( A 2 ) C f (Ai); I ( A i ) n l ( A 2 ) = 0.
Lemma 8.10: Lei A s= £>(£>) and let x e J(A) (resp. x e £(A)). T/ie infraconnected component of x is included in 1(A) (resp. in £(A)). If A' e O(D) is such that A < A' then l(A') n £(A) ^ 0. Corollary 8.11: Let £ 6e an empty annulus of D. The family of the empty annuli A of D such that A > £ is totally ordered. We will also need Lemma 8.12 (Lemma 3.12 in [30]). The most classical example of an ultrametric complete algebraically closed field is the field C p (one can find a description in [30, Chapter 8]). Lemma 8.12: Let a € D, let p be the distance from a to D and let R be such that p < R < diam(D). For j = 1 , . . . , q let (Xj € d(a, R) and let r'^r'l € R+ be such that r'j < R < r'J. Then
f](%,r;,r;)nD)^. Sets having finitely many infraconnected components are characterized in [30]: Theorem 8.13: D has finitely many infraconnected components if and only if it has finitely many empty annuli. Moreover if so does D then one of the infraconnected components is
A0=
f|
f(E)
xex(D) while the others are of the form Ai = 2{Ai) f | ( p | £(£) j , \£
/
with Ai e O(D).
Infraconnected Sets
49
Definition: A bounded closed infraconnected subset A of L is said to be affinoid if it has finitely many holes, and if diam(A) and each diameter of hole lies in \L\. More generally, a set A is said to be affinoid if it is a finite union of infraconnected affinoid sets. Rrom Theorem 8.13, Lemma 8.14 is easily proven: Lemma 8.14: A finite union and a finite intersection of affinoid subsets of L is affinoid. Theorem 8.15: Let A be a commutative ultrametric normed K-algebra with unity. Suppose that A has an element u such that sp{u) admits an empty annulus F(a,ri,r2). Then there exist tpi, ip2 € Mult(A, \\ . ||) such that ipi(t) = r 1 ; ip2{i) = r2. Proof: By definition, r\ = sup{|A| | A € sp(u) |A| < r2}, and r2 = inf{|A| | A € sp(u) |A| > ri}. Consequently, both r\, r2 lie in the closure of {tp{u) | V € Mult(A, || . ||)}, which, by Lemma 6.3, is a closed subset of R. hence there exist Vii V"2 € Mult(A,\\ . ||) such that ipi(t) = n,
Mt) = r2-
□
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Chapter 9
Monotonous Filters
Monotonous filters are indispensable to understand the behaviour of ratio nal functions, and all kind of analytic functions. Circular filters which are closely to monotonous filters, will be introduced in the next chapter. Most of the proofs of results given in this chapter can be found in [30, Chapter 3]. Definitions a n d notation: In all this chapter L is a complete ultrametric field whose valuation is dense and D is a subset of L. According to classical definitions, we call base of a filter T a subset B of T such that every element of T contains an element of B. A filter T is said to be secant with D if the family of sets { B n D | B e T} is a filter. When a filter T is secant with a subset D of L, the filter {BnD | B £ J7} is called intersection of T with D, and denoted by T C\ D. A filter T on L will be said to be thinner than a filter Q if every element of Q belongs to J-. In such a case, Q will also said to be less thin than T. A sequence («n)n€N in L will be said to be thinner than a filter Q if so is the filter defined by the sets Aq — {un\n > q] (q € N). In such a case, Q will also said to be less thin than the sequence (w„)neNA sequence (un)ne^ in L will be said to be an increasing distances sequence (resp. a decreasing distances sequence) if the sequence \un+\ —un\ is strictly increasing (resp. decreasing) and has a limit (. £ W+. The sequence (wn)n€N will be said to be a monotonous distances sequence if it is either an increasing distances sequence or a decreasing distances sequence. A sequence (u„)„epsj in L will be said to be an equal distances sequence if \un — um\ = \um — uq\ whenever n, m,q G N such that n ^ m ^ q. 51
52
Ultrametric
Banach
Algebra
Let a G D and i? G 1R+ be such that T(a, r,R) D D ^ 0 whenever r G]0, i?[ (resp. T(a, R,r)(~) D ^ 0 whenever r > i?). We call an increasing (resp. a decreasing) filter of center a and diameter R, on D the filter T on D that admits for base the family of sets V(a, r,R) (1 D (resp. T(a, R, r) n £>). For every sequence (r„)„ e N such that r„ < r n + i (resp. rn > r n + i ) and lirrin-^oorn = R, it is seen that the sequence T(a,rn,R) D I? (resp. T(a, i£, r„) n D) is a base of T. Such a base will be called a canonical base. We call o decreasing filter with no center, of canonical base (£>„)n6N and diameter R > 0, on D a filter T on D that admits for base a sequence (Ai)n G N in the form Dn = d(an,rn) n £> with D n + i C D„ , r n + i < 7-„,lim n _ >00 r n = #, and DneNd(a™'r™) = ®Given an increasing (resp. a decreasing) filter J- on D of center a and diameter r we will denote by VD^) the set {x £ D\ \x — a\ > r} (resp. the set {x G D\ \x — a\ < r}) and by CD{F) the set {x G D\ \x — a\ < r} (resp. the set {x G D\ \x — a\ > r}). When there is no risk of confusion we will only write V{!F) instead of VD(^), and C{!F) instead of CD(F)Besides CD{3~) will be named the body of T and VD{^) will be named the beach of T. We call a monotonous filter on D a filter which is either an increasing filter or a decreasing filter (with or without a center). Given a monotonous filter T we will denote by diam{T) its diameter. The field L is said to be spherically complete if every decreasing filter on L has a center in L. The field C p for example is known not to be spherically complete [10]. However, every algebraically closed complete ultrametric field admits a spherically complete algebraically closed extension (see for example [30, Chapter 7]). Theorem 9.1: Let (un)neN be a bounded sequence in L. Either we may extract a convergent subsequence or we may extract a monotonous distances subsequence or we may extract an equal distances subsequence from the sequence (W„) 7I€ N-
Lemma 9.2: Let (an)n G N be an increasing distances (resp. a decreas ing distances) sequence in D. There exists a unique increasing (resp. decreasing) filter T on D such that the sequence (an)n G N is thinner than J-. Moreover, given a monotonous filter T on D, there exists a unique monotonous filter Q on L inducing T. Lemma 9.3: Let D be infraconnected. Let T be an increasing filter (resp. a decreasing filter) on L, of center a G D and diameter R < diam(D)
Monotonous
Filters
53
(resp. R < diam(D)) such that a does not belong to a hole of diameter p > R (resp. p > R). Then T is secant with D and induces on D an increasing filter (resp. a decreasing filter) of center a and diameter R, on D. Definition: Let T be an increasing (resp. a decreasing) filter of center a and diameter R on D. Then T is said to be pierced if for every r €]0, R[, (resp. r < R), T(a,r,R) (resp. T(a,R,r)) contains some hole Tm of D. A decreasing filter with no center T on D is said to be pierced if for every m G N, Dm \ Dm+i contains some hole Tm of D. R e m a r k s : The definition of a pierced filter with no center also applies to a decreasing filter with a center and then is equivalent to the one given just above for such a filter. If T is an increasing (resp. a decreasing) filter of center a, of diam eter R, then T is pierced if and only if there exists a sequence of holes (T„) n€ N of D such that S(a,Tn) < 5(a,Tn+i) (resp. S(a,Tn) > 8(a,Tn+1)), lim n—KXI
6(a,Tn) = R. Given a Cauchy filter T on D, of limit a in L, we will call a canonical base of T & sequence Dm in the form d(a, rm) n D with 0 < rm < rm+i and lim m —>oorm — 0- The filter J- is said to be pierced if for every m € N, Dm contains some hole of D. Let a G D. Let (Tm,i)i
= dm (1 < i < s(m), l<j<
s(m + 1)), dm > dm+1,
lim dm = R > 0, m—>oo
where the filter T of base Dm = d(am, dm) f l D i s a decreasing filter with no center. The sequence (T^^j)Ki-<s{m.),rTi€N is called a decreasing distances holes sequence that runs T. In each case, the sequence (dm)me^ is called monotony of the sequence (Tm,i)l<j<s(m+1),
m€N-
Summarizing these definitions, an increasing (resp. decreasing) dis tances holes sequence that runs an increasing (resp. decreasing) filter T will
54
Ultrametric Banach
Algebra
be just named an increasing (resp. decreasing) distances holes sequence and t h e filter T will be named the increasing (resp. decreasing) filter associated to the sequence (T m j j)i<j< s ( m ) i m € N- T h e diameter of T will be called the diameter of the sequence *{Fm,i)\
If a monotonous distances holes sequence has a piercing p > 0, it will be said to be well pierced. If a monotonous filter T is run by a well pierced monotonous holes sequence, T will be said t o be well pierced. In each case the sequence of circles C(a, dm) when JF has center a (resp. C(am+i, dm) when T has no center) will be said to run the filter T, and to carry the monotonous distances holes sequence (T , m i j)i< i < s ( m ) r n e p(. A monotonous distances holes sequences (Tm,i)i
Then
D e f i n i t i o n : Given a monotonous filter J-, we will call superior gauge of T the upper bound of all superior gauges of monotonous holes sequences running T.
Monotonous
Filters
55
From the definition, Lemma 9.6 is obvious again: Lemma 9.6: Let T be an increasing (resp. decreasing) filter on D of center a, of diameter r, of superior gauge s. For every e > 0, there exists a hole T of D included in T(a,r,r + e) (resp. T(a,r — e,r)) such that diam(T) > s — e. Let T be a decreasing filter on D with no center, of diameter r, of superior gauge s. For every e > 0, there exists a hole T of D included in a disk d(a, r + e) such that diam(T) > s — e. Notation: From now on, to the end of the chapter, 7 denotes the homographic function b + -^^ with a,b £ L. Proposition 9.7: Let a £ L, r > 0 and let a £ L be such that \a — a\ < r. Thenl(C(a,r)) = C(b,'-). Proof:
We may assume 6 = 0 and then the proof is immediate.
Corollary 9.8:
Let a £ L:ri,r2
□
€]0, +oo[ with \a — a\ < r\ < r-i- Then
l(r(a,r1,r2))=r(b,-,-).
V. r2
rxJ
Corollary 9.9: Let T be the increasing (resp. decreasing) filter of center a and diameter R> \a — a\, on L\{a}. Then ^(J7) is the decreasing (resp. increasing) filter of center b and diameter -^. Lemma 9.10:
Let a £ L be such that \a — a\ ^ r. Then 7(C(a,r))
= c(
7
(a),^-^).
Corollary 9.11: Let a £ L and r, r' e]0, +oo[ be such that 0 < r < r' < \a — a\. Then we have 7
7Ka,r))
(r(a,rIrO)=r(7(a),^>]^L5),
= d ( T ( Q ) , [ f l ^ | 2 ) , 7(d(a,r-))=d(7(a), ( ^ ^ p )
)■
Corollary 9.12: LetT be the increasing (resp. decreasing) filter of center a and diameter R on L\{a} with \a — a\ > R. Then j(J-) is an increasing (resp. a decreasing) filter of center 7(a), of diameter >a^a>i on L\ {&}.
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Chapter 10
Circular Filters
We are going to define circular filters, which roughly characterize the abso lute values on K{x). We shall construct a tree structure and a distance on the set of circular filters. Definitions a n d n o t a t i o n : Throughout the chapter, D will denote a subset of L. Let a € D, let p = 5(a, D) and let R e]0, +oo[ be such that p < R < diam(D). We call circular filter of center a and diameter R on L the filter T which admits as a generating system the family of sets F(a,r',r") with a € d(a,R),r' < R < r", i.e. T is the filter which admits for base the family of sets of the form r i i = i r ( a i > r i i r i ' ) ) with Q« € d(a, R),r'i
58
Ultrarnetric Banach
Algebra
or is of the form d{a,r") \ UQi=1d(ai,r'~), with r', r" G \L\, r' < r < r", and )tti — a,j) = r Vi ^ j . Lemma 10.1 is easily checked: Lemma 10.1: Let T, Q be two different circular filters. Then T is not secant with Q. Moreover, if QfT) £ 0, then Q(T) ^ Q{Q). Definitions and Notation: We call circular filter on D the filter induced on D by a circular filter on L secant with D. The set of circular filters on L secant with D will be denoted by <&{D). In the same way, the set of large circular filters on L secant with D will be denoted by $'(£)). Thus by definition, every circular filter on D is induced by a unique circular filter on L. Given a circular filter T on D induced by a circular filter Q on L, we will denote by Q{T) the set of its centers in L, i.e. Q{G)Given a circular filter J o n £ > , induced by a circular filter Q on L, we shall call T-affinoid any (/-affinoid. Lemma 10.2: Let T be a circular filter. Then T admits a base consisting of the family of all T-affinoids. If J- does not admit a countable base, it has a center and its diameter belongs to )L\. If T has no center and is secant with an affinoid subset E of L then E lies in J-. If J- has center a and diameter r, then an affinoid set E lies in T if and only if satisfy En{L\d(a,r)) ^ 0, End{b,r~) ^ 0 Vb e d(a,r). Proof: By definition, a circular filter with no center has a countable base, and of course so does a Cauchy circular filter. In both cases, it admits a base consisting of a family of diks which are jF-affinoid sets. Now, consider a circular filter of center a and diameter r. Therefore, F admits for base the family of annuli d(a, r + ~ \ (U? =1 d(ai, (r — ^)~) where the a* are centers of T satisfying |a* - af\=r. In particular, if r ^ \L\ we have q = 1 and we obtain a base of the form T(a,r — ^ , r + ~) which is countable. Now, suppose that T is secant with an affinoid subset E of L. Let (A n ) n€ N be a canonical base of J-. Since each An admits common points with E, each is included in E, and therefore it is included in E if and only if it contains no hole of E. But since T has no center, fl^Lo ^n = $' n e n c e there exists q eN such that An C E Vn > q, and therefore E e f. Now suppose that T has center a and diameter r. If E € J7, it obviously satisfies Ef\(L\d(a,r)) ^ 0, En d(b,r~) ^ 0 Vb e d(a,r). Now, suppose that E satisfies En(L\d(a,r)) ^ 0, End{b,r~) ± 0 Vb e d{a,r). Since E
Circular Filters
59
has finitely many holes, on one hand there exists s > r such that T(a, r, s) C E, and on the other hand, all classes of d(a, r) are included in E, except finitely many: d(bj,r~), 1 < j < n. And for each j = l , . . . , n , there exists Tj < r such that T(bj,rj,r) C E. Finally, E contains the set d(a, s) \ Wj=ld(bj,rj) which obviously lies in T. □ R e m a r k : By Lemma 10.2 it is easily seen that there exist circular filters without countable base if and only if the residue class field of L is not countable. The following Proposition 10.3 is just a translation of the definitions. Proposition 10.3: LetT be an increasing filter (resp. a decreasing filter) of center a and diameter r, on D. Then the circular filter of center a and diameter r on L is secant with D, and is the only circular filter on D less thin than T. Conversely let J7 be a circular filter of center a and diameter r on D, secant with d(a,r~) (resp. L \d(a,r)). Then the increasing filter (resp. decreasing filter) of center a and diameter r on L is secant with D and thinner than T. Proposition 10.4 is immediate and is Proposition 3.14 in [30]: Proposition 10.4: Let D be infraconnected, let a G D and let S be the closure of {\x — a\ \x G D} in R. For every r G S the circular filter T of center a and diameter r on L is secant with D. L e m m a 10.5: Let J7 be a circular filter secant with two disks d(a,r) and d(b, s). Then either d(a, r) C d(b, s) or d(b, s) C d(a, r). Proof: Let R = 5(d(a, r),d(b, s)). Suppose that none of the two inclusions is satisfied. Then d(a,r) H d(b,s) = 0. But since T is secant with d(a,r), then diam(J-) < r, and therefore T is not secant with C(a, R), in particular neither is it with d(b, s). □ Corollary 10.6: Let D be infraconnected, and let d(a,r~) be a hole of D. Then, the circular filter of center a and diameter r on L is secant with D. Proposition 10.7: Let (an)neN be a sequence in L that is either a monotonous distances sequence or a constant distances sequence. Then there exists a unique circular filter on L less thin than the sequence (an). Proof: By Proposition 3.15 in [30] we know that there exists a unique circular filter on L less thin than the sequence (an). By definition, this filter is secant with the set {an | n G N}, hence with D. □
60
Ultrametric Banach
Algebra
Corollary 10.8: Let (o n )„ e ^ be a bounded sequence in L, Then there exists a subsequence (ane )ten and a unique circular filter J- on L less thin than the subsequence (a n JtgNProof: Since the sequence (a n ) n gN is bounded, by Theorem 9.1 we can extract either a monotonous distances subsequence or a constant distances subsequence, or a converging subsequence. In all cases, once such a sub sequence is chosen, there exists a unique circular filter T on L less thin than the subsequence. □ Remark: If T is the circular filter of center a and diameter r, it is not secant with C(a,r) if and only if r ^ \L\. Definitions and notation: A circular filter T on L will be said to be (a, r)-approaching if it is secant with the circle C(a, r), or if it is the circular filter of center a and diameter r. A circular filter T on L will be said to be D-bordering if it is secant with both D and L\D. In the same way, a circular filter on D will be said to be D-bordering if it is induced by a .D-bordering circular filter on L. If D is bounded, T will be said to be D*-peripheral, or to be peripheral to D if Q{T) = D. Finally, a circular filter T will be said to be strictly D-bordering if either it is £>-peripheral, or it is peripheral to some hole of D. We will denote by £(.D) the set of ip? e Mult(K[x)) such that T is D-bordering, and by Eo(-D) the set of ipjr g Mult(K[x\) such that T is strictly .D-bordering. Proposition 10.9: Let D be closed and infraconnected and let T be a circular filter on L. If every element of T contains infinitely many holes of D, then T is D-bordering. Conversely, if J- is D-bordering but not strictly D-bordering, then every element of T contains infinitely many holes of D and then, either D admits a monotonous pierced filter thinner than J-', or J- has a center a and D admits an equal distances holes sequence {Tn)nH such that S(Tn,Tm) — S(Tn,a) = diam^) for all m^= n. Proof: Suppose that every element of T contains infinitely many holes of D. Then T is obviously secant with L\D. Let A e J7 be affinoid and let s = diam^). Suppose that A n D = 0. If D C L \ A, then every hole of D has empty intersection with A, a contradiction. But then, since D is infraconnected, there exists a hole T = d(a,r~) of A such that D C T. Since A is affinoid, r belongs to \L\, hence there exists another affinoid
Circular Filters
61
B € T, with a hole d(a,s~), with s > r, thereby £> n B = 0, and finally every hole of D has empty intersection with B, a contradiction again. Thus, T is also secant with D, and therefore T is D-bordering. Conversely, consider a D-bordering filter T which is not strictly Dbordering. Let s — diam(T). Let A e T be affinoid and let a e 7 l n ( L \ D ) . Since D is closed, a belongs to a hole T = d(a,r~) of D. Since ^ n D ^ O , >1 is not included in d(a,r~). We will show that there exists B e T such that B C A &nd T 11 B =
Then the set of
Proof: If D is infraconnected, it has finitely many holes. And if it is not infraconnected, it has finitely many infraconnected components D\,..., Dq, hence the set of its .D-bordering filters is the union of the sets of Dj-bordering filters, 1 < j < q. □
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Lemma 10.11: Let T be a circular filter on L and let a 6 L. There exists a unique r > 0 such that T is (a, r)-approaching. Moreover, for every s > diam^) there exists a unique disk E of the form d(b, s) which belongs to T. Proof: Let p = diam{T). If a is a center of T, then T is (a,p)approaching. Suppose now a is not a center of T. Then we can find a disk D £ T which does not contain a: let r = 5(a,D). Then T is secant with C(a, r). Let s > diam^F). If T has a center a, then T is secant with d(a,s). Now, suppose that T has no center. By definition of the circu lar filters with no center, there exists a disk d(b,r) £ T such that r < s. Then d(b, s) obviously belongs to T. Moreover, in both cases, any disk d(b,s) other than d(a,s) has empty intersection with d(a,s), hence T is not secant with d(b, s). □ Lemma 10.12: Let T be a filter admitting a base (Ai)i&j consisting of infraconnected affinoid sets. There exists at least one circular filter Q secant with T. Proof: Suppose first that f]ieJ Ai — 0. Then (Ai)i£j is a base of a cir cular filter with no center Q which actually is equal to T. Suppose now that fliej -^» 7^ $ a n d l e t a € fliej -^i- Let r = inf i € j diam(Ai). Then the circular filter of center a and diameter r is clearly secant with F, which ends the proof. □ Lemma 10.13: Let Q be a circular filter on K of center a and diameter r and let T be a filter secant with G, admitting a base {Ai)i
Circular Filters
63
case, since the Ai are infraconnected and affinoid, for each i € J and for every class D of C(a, t) except at most for finitely many, D is included in Ai. Consequently, T is secant with the circular filter Q' of center a and diameter t. Similarly, suppose now that there exists a hole T = d(b, p) of B such that T n At ± 0 Vi G J. We can find t e]/o,r[n|I| and then all classes of C(b, t) except at most finitely many are included in A, whenever i € J. Consequently, T is secant with the circular filter Q' of center b and diameter t, which ends the proof. □ Lemma 10.14: Let T be a circular filter secant with B, of diameter r and let B s T be d(a,r") \ U| = 1 d(aj,r') where the ai are centers ofT satisfying | a j - O j | = r\/i ^ j . Lett=\vam{r" — r,r — r'),lets"=\r,r + €[r\\L\, s' £ }r — e,r[n|L|. Let A — d(a,s") \ U? =1 d(oj, s'). Let T be a circular filter secant with A and let Q be a circular filter satisfying 5{T,Q) < e. Then A € T , Ac B and Q is secant with B. Proof: It is obvious that A 6 T, and we check that A C B because A c B, and each hole of B is included in a hole of A. Now, let us check that Q is secant with B. Indeed, if Q is not secant with A, then due to the hypothesis 5{T, Q) < e we can check that it is secant with a disk containing A, of diameter r", hence it is secant with B. Similarly, if Q is secant with a hole d(a,r'~) of A, due to the hypothesis 5{T,Q) < e, then Q is secant with r(a,r',r) C B. □
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Chapter 11
Tree Structure and Metric on Circular Filters
In this chapter, we will show that the set of circular filters is provided with a tree structure as defined in Chapter 2, and that the diameter here is an increasing function with values in M, defining distances associated to this structure, as shown in [9, Chapter 2]. The first remarks on that tree structures are due to E. Motzkin [41]. Finally, J.C. Yoccoz and J. RiveraLetellier considered a similar distance [42]. Definitions and notation: Given two circular filters T and Q, T is said to surround Q if either Q is secant with Q{T), or if T — Q. Similarly, a circular filter T is said to surround a monotonous filter Q if it surrounds the circular filter associated to Q. A monotonous filter T is said to surround a circular filter Q if its associated circular filter surrounds Q, and T is said to surround a monotonous filter Q if the circular filter associated to T surrounds the circular filter associated to Q. We will denote by -< the relation on the set of circular filters defined as T < Q if Q surrounds T and by -< the relation defined as T' -< Q if T
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Algebra
Proof: Indeed, the relation is reflexive by definition. Let T,Q,l-i be 3 circular filters. Suppose that T < G and Q < T. If T =£ G, then by definition Q(T) ^ 0, Q(G) ± 0, T is secant with Q(G) and G is secant with Q{T). Consequently, <2(.F) = Q(G) and therefore by Corollary 11.2, T =G- Finally, suppose that T < G and G -< H, with T ^ G ^ H. Then T is secant with Q(G), hence with Q(W) and therefore T -< H□ Theorem 11.4: Let D be infraconnected and let T € $(D). a minimal element in $(£>) if and only if:
Then T is
(1) it is punctual, or (2) it has no center, or (3) Q(F) HD = ®. Proof: On one hand it is easily seen that if T is punctual, or has no center, then it is minimal in $ ( £ ) , and therefore in $(D). Suppose now that Q(T) n D = 0. Suppose that G € $(D) satisfies Q ■< T. Then Q is secant with Q{T), and has a diameter strictly inferior to the diameter r of T, hence there exists a disk d(b, s) € G, with s < r. Consequently, d(b, s) C Q(^F)- Since d(b, s) C\ D = 0, we see that G is not secant with D. Now, let T S ${D) be minimal in $(D). Suppose that T is not punctual and has a center a, and let r = diam^). If Q(!F) H D ^ 0, then for any a £ Q{J~) H D, of course T surrounds the filter of neighborhoods of a, a contradiction to the hypothesis ".T7 minimal in $(£>)". Thus, 2(5") n£> = 0, which completes the proof. □ Lemma 11.5: Let T, G be two circular filters such that Q(Jr)r\Q(G) ^ 0Then T and G are comparable for <. Proof: Since Q{F) n Q(G) ^ 0, and since both Q(F), Q(G) are disks, we can suppose for instance Q(T) C Q{G)- Then T is secant with Q(<7), hence T
Tree Structure
and Metric on Circular Filters
67
Proof: Suppose that both T, Q surrounds H. If T = H, then Q just surrounds T. Now suppose that both J7, Q strictly surrounds "H. Then Ti. is secant with both Q{T) and Q(Q). But by Lemma 10.5 either Q(T) c Q{Q) or Q(Q) C Q(.F), hence by Lemma 11.5 T and Q are comparable. D By Proposition 10.4, we obtain Theorem 11.8: T h e o r e m 11.8: Let D be infraconnected, let T be a circular filter on L secant with D and letr £}diam{!F), diam(D)[. The unique circular filter of diameter r surrounding T is secant with D. Proof: By Lemma 10.11 there exists a unique disk d(a,r) T is secant with. By Proposition 10.4 the circular filter of center a and diameter r is secant with D, and on the other hand, by Theorem 11.6 this is the unique circular filter of diameter r surrounding T. □ Proposition 11.9: Let J-, Q be two circular filters on L which are not comparable for the relation -<. There exist disks F £ T and G G Q such that F n G = 0. Moreover, given F' e T, G' e Q such that F'nG' = 0, we have 5(F,G) = 5(F',G') > max(diam(f),diam{g)). Proof: Suppose first that both T, Q have no center. Then T (resp. Q) admits a canonical base T> (resp. £) consisting of a family of disks. But since the two filters are not secant, we can obviously find F G V and G 6 £ such that F n G = 0. Suppose now that F has centers and let d(a,r) = Q(-F)- If all disks G G Q contain d(a,r), then Q surrounds J7, hence we can find a disk G = d(b,u) G Q which does not contain d(a,r). Of course, if G C d(a,r), then T surrounds Q, hence Gf)d(a, r) = 0. Consequently, we have \a — b\ > r and u < \a — b\. Let s e]r, \a — b\[. Then the disk F = d(a, s) belongs to T and is such that F n G = 0. Since J 7 is a filter, we have F C\ F' ^ 0, hence either F C F', or F' C F. In the same way, G n G' =£ 0. hence either G C G', or G' C G. Without loss of generality, we can assume that F C F'. If G C G', then our claim is obvious. Suppose now that G' C G. If F' n G ^ 0, then either G C F', therefore G C G', a contradiction, or F' C G, hence F C G, a contradiction again. Consequently, we have F' n G = 0. Putting Z = 5(F',G) we have |a — 6| = I Va € F ' , b £ G, hence in particular, (5(F, G) = <5(.F', G') = J. Then we notice that I > diam{F) > diam(T), and I > diam(G) > diam(G). □
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Notation: Let T, Q be two circular filters which are not comparable for -<, we notice that, by Proposition 11.9, 6(F, G) does not depend on the choice of disks F € T and G e G, so we can put A(.F, G) = 5(F, G) with F, G disks such that F € .F, G € £, Ff)G = 0. Theorem 11.10: Let D be infraconnected and let T,G £ ®(D)- There exists sup(.F, G) S &(D) and it is the unique circular filter of diameter MJF, G) which surrounds both T, GProof: The claim is trivial when the two filters are comparable for <. So we assume they are not. Let I = \(J-, G)- By Theorem 11.6 there exists a unique circular filter S e $(L) of diameter / surrounding T. Then <S has centers. Let Q(S) = d(a,l). Then by Proposition 11.9 G contains disks included in d(a, I) and therefore is secant with d(a, I), hence S surrounds Q. We will check that <S is the smallest element of the set of filters on L surrounding both T and G- Indeed, let W be a circular filter surrounding T and G- Then both T and G are secant with Q{H). Let d{b, s) = Q(H). Consider disks F = d(a, p) £ T and G = d(fi, v) € G such that F n G = 0. Since F n d(b, s) ^ 0 and G n d(b, s) ^ 0, and since F n G = 0, it is easily seen that both F, G are included in d(b,s), hence s > I. But since both <S, TC surround JF, by Proposition 11.7 they are comparable for ■<. Then, since I < s, 7i surrounds <S, and therefore S is the smallest element of the set of filters on L surrounding both T and GNow, suppose that both .F, G are secant with D. Then d(a,p) n D ^ 0, and d(/9, i/) fl D ^= 0, hence S which is the circular filter of center a and diameter I — \a — /3\ is secant with D because D is infraconnected. D Theorem 11.11: If D is infraconnected, $(D) is a tree with respect to the order ^ and the mapping diam is strictly increasing from <&(£>) to R + . Proof: This is an obvious consequence of Theorems 11.3, 11.4, 11.10 and Proposition 11.7. D Notation: Let J7, G be two circular filters and let S = sup(J r ,5). We put 5{!F, G) = max(diam(5) — diam{!F), diam{S) — diam(G)) and 5'{T,G) = 2diam(S) — diam^) — diam{G)Remark:
Particularly, if J- < G, we have 8(F, G) = diam{G) -
diam{T).
Tree Structure
and Metric on Circular Filters
69
According to Chapter 2, we can state Corollary 11.12 Corollary 11.12: 5 is the supremum distance associated to the mapping diam defined on the tree $(K) and 5' is the whole distance associated to the mapping diam defined on the tree &(K). Lemma 11.13 is immediate and will be useful: L e m m a 11.13: Let T be a circular filter and let B e T be infraconnected and affinoid. Let Q\,...,Qn be the B-bordering filters and let A = mmi<j
first that it is a decreasing sequence. For each n € N we can find a center an $ d(an+\,rn+i). Thus, the sequence (a„) ng N is such that the sequence (|a n + i — fln|)n€N 1S strictly decreasing. Hence by Proposition 10.7 there exists a unique circular filter T less thin than the sequence (ara)neN of radius I. And then, we check that the sequence (.Fn)neN converges to Jwith respect to the metric topology. If the sequence is increasing, it is easily seen that the sequence converges to the circular filter of center a\ and diameter /. □ T h e o r e m 11.15 ([11]): topology.
$(L) is complete with respect to the metric
Proof: Let (^rn)nsN be a Cauchy sequence with respect to the metric topology. For every m,n e N, let <Sm,n denote sup(^ r m ,^ r „). Next, for each n £ N, let rn = diam{Tn), and sn — supm>ri{c?iam(<S„jm)}. We will show that the sequence d(an,sn)n€^ is decreasing with respect to inclusion.
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Algebra
Indeed, let m,t > n. Both <Snjt,<Snjm are secant with d(an,sn) and sur round Tn, hence are comparable with respect to <. Then, sup(Snj,Sntm) is secant with d(an,sn), and surrounds Tn, Tm, Tf, hence surround Sm^Consequently, Smj is secant with d(an, sn). Now, let us fix t > n. This true for all m >t, hence diam{Sm}t) < sn Vm > t, and therefore sm < sn. Thus, for every n € N, let Qn be the circular filter of center an and diameter sn. The sequence (Gn)n€N is then decreasing with respect to the order ■< and therefore, by Theorem 11.14, has a limit Q with respect to the metric topolo gies. Let 5 = lim sn. Since the sequence (Tnjnen is a Cauchy sequence, n—>oo
and since by definition of the distance 5, we have 8{J-n,Sn<m) < S^n^m), it is clear that lim sn — rn = 0. Consequently, lim S(!Fn,G) ~ 0 and n—*oo
n—>oo
therefore the Cauchy sequence has limit Q. □ Remark: In [42] another distance is defined on the set of large circular filters, as 2log(diam(sup(J7, Q)— log(diam(siamJr) —\og(diam(Q). This last distance defines the same topology on the set of large circular filters, but does not admit the same Cauchy sequences and does not expand the usual distance on the field L defined by the absolute value. As a consequence, the set of large circular filters appears to be complete for such a distance and therefore a sequence converging to a point a £ I in the usual sense, actually has no limit for this distance. Theorem 11.16: Let B be a totally ordered subset of $(K)Then B admits an infimum T with respect to the other <. Further, subset A of $(K) admits a supremum with respect to the order ■< if and only if it is bounded with respect to 5. Proof: Let B be a totally ordered subset of $(K) and let r = ini{diamQ\G € B}. We can see that all sequences (5ra)neN of B satisfying limn_,oo diam(Gn) = f are Cauchy sequences and admit the same limit T. Therefore T is clearly the infimum of B with respect to ■<. Now, let A be a subset of $(K). If it admits a supremum with respect to the order X, it is obviously bounded. Conversely, assume that A is bounded. We can clearly find a disk d(b, s) such that all elements of A is secant with d(b, s). Conse quently, the circular filter of center b and diameter s surrounds all elements of A. Now, by Proposition 11.7, the set A* of circular filters surrounding all elements of A is totally ordered, and therefore admits an infimum <S, and we have Q < S for every Q £ A. Consequently, S is the supremum of A. D
Chapter 12
Rational Functions and Algebras R(D)
N o t a t i o n : Throughout this chapter and in the following, K will denote an algebraically closed complete ultrametric field and D will be an infinite subset of K. Rational without poles in a set D of the field K are the only material we handle to define a kind of holomorphic functions in D. But first, we have to know perfectly the properties of rational functions, with regards to the ultrametric structure of K. Given a function / from D to K, | | / | | D denotes sup{|/(a;)| | x € D} € [0,+00]. And we denote by R(D) (resp. Rb(D)) the i-f-algebra of all rational functions h 6 K(x) with no pole in D (resp. the K-algebra of all rational functions h 6 K{x) with no pole in D which are bounded in D). Given P{x) — X)n=o a ™ x "
G
v(P,fi)=
K[x]> a n d M € K, we put inf
v(an)+niJ,,
0<j
we denote by iV+ (P, p) the greatest of the integers j such that v(aj) +j[i = info<j<9 v(an) + nfi, and by iV~ (P, /j.) the least of the integers j such that v(a,j) + jfj, = info<jT.(P) when \x — a\ approaches r but remains different from r. 71
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Ultrametric Banach
Algebra
Moreover, if P(x) = Y^j=oaj(x ~ a)"'> then <Pa,r(jP) = max \ctj\ri = ||P|| d ( o , r ). 0<j
Thus, <pair belongs to Mult(K[x\) and has continuation to K(x). given h € K(x), (fa^r lies in K if and only if so does r. Corollary 12.2:
Let P(x) = Y?j=o ajxJ
Moreover,
€ K[x). Then
< P , - l o g r ) = -log(^ 0 , r (P)). Lemma 12.3: Let h e K(x) and let r e R + . For every a e d(0,r) we have lim| x _ a |^ r i | x _ a |^o \h(x)\ = fa,r{h)- Let x € C(0,r). Ifh has no zeros (resp. no poles) in the class of x in d(0,r) then \h(x)\ >
Rational Functions
and Algebras R(D)
73
Corollary 12.8: Let h e K{x) \ {0}. / / d{0,r{) contains s zeros and t -poles of h, (taking multiplicities into account), and if r ( 0 , r i , r 2 ) contains neither zeros nor poles of h, then \h{x)\ = y~) <po,r2(h) = ( £ ? ) ' " V n ( > 0 , VzGlXO.n.ra). Theorem 12.9: Let h e K{x) \ {0}. If r e \K\, then vo,r(h) lies in \K\. Conversely, if h has s zeros and t poles in d(0,r), if s ^ t with
. But since p € \K\
and since s — t ^ 0, by the direct claim it is clear that \K\, and so does r.
p
<■■, also lies in D
N o t a t i o n : Henceforth, we denote by U the disk d(0,1) in K. Given a disk A = d(a,r) or A = d(a,r~), we put tpA = y>a,rTheorems 12.10 and 12.11 will be indispensable in Chapters 33, 34 and 35: Theorem 12.10: Let h € K(x) satisfy deg(/i) > 0. h^iU) = {x € K | \h{x)\ < 1} is affinoid.
Then the set
Proof: Let D = h~l(U) and let h = ^, with P, Q G K[x] relatively prime, and of course deg(P) > deg(Q). We first notice that D is bounded. Let T(a, /', I"), with a £ D, be an empty annulus of D. Suppose that h has no zeros and no poles in d(a, I'). There exists I > I' such that h admits no zeros and no poles in all d(a,l), hence by Lemma 12.2 \h(x)\ is constant in all d(a,l), hence \h(x)\ = \h(a)\ < 1 Vi e d(a,l), thereby d{a,l) C D, a contradiction. Thus, for each empty annulus A of D, 1(A) contains at least a zero or a pole of h. Consequently, D admits finitely many empty annuli and therefore by Theorem 8.13 D has finitely many infraconnected components. Let E be an infraconnected component of D, of diameter r, and let a € E. Let t (resp. s) be the number of poles (resp. zeros) of h inside E. Since E is infraconnected, the circular filter of center a and diameter
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Ultrametric Banach
Algebra
r is secant with E and thereby <pa,r(h) < 1. It is also secant with K \E, hence ipa,r(h) > 1, therefore ipa,r(h) = 1. On the other hand we have s — t > 0 because if s — t < 0, then by Lemma 12.7 there exists r' > r such that \h(x)\ = ipa^r(h) Vrr e r ( a , r , r ' ) , a contradiction to the hypothesis D C c!(a, r). Consequently, by Lemma 12.7 the equality (fa,r{h) = 1 implies re\K\. Similarly, consider now a hole T = d(b,p~) of E. Let I =
There exists h e R(D) such that
Proof: Suppose first D is infraconnected. Then D is of the form d(a, ro) \ \J]=id(bj,rJ), with \bi - bj\ > max(rj,r,), Vi ^ j , bj € d(a,r0) rj < ro Vj = 1 , . . . , n, Tj e |if | \/j = 0 , . . . , n. For each j = 0 , . . . , n, we can take Xj € K such that |Aj| = r,-. Let h(x) = ^ + YTj=\ ^b~- Outside d(b - j,r~), we have l^z^rl < li inside d(6 - j,rj), we have | j ^ - | > 1. Inside d(a,r0) we have | ^ ^ | < 1, outside d(a,r0) we have \^f\ > 1. Consequently, we check that \h(x)\ < 1 if and only if x G D, so we have constructed h when -D is infraconnected. Moreover, by construction, we check that lim^i^oo |/i(x)| = +oo, so deg(/i) > 0. We now consider the general case and denote by Di,... ,Dq the infraconnected components of D. For each i — 1 , . . . ,q, there exists hi € -R(-Di) such that \h(x)\ < 1 if and only if x € Di. We then put g = 5Z"=i ?T an<^ h = K We can easily check that \g(x)\ > 1 if and only if x € D. Moreover, we notice that for each i = 1 , . . . , n we have lim|T|_)(X) |/H(X)| = +oo, hence lim^i^oo \g(x)\ = 0, therefore lim| :E |_ 00 \h{x)\ = +00. So, deg(/i) > 0 and h satisfies our claim. □ According to Theorem 9.3 in [30] we have Theorem 12.12: Theorem 12.12: Rb(D) = R{D) if and only if D is closed and bounded. Moreover, if D is closed and bounded, || . ||D is a semi-multiplicative ultrametric norm of K-algebra on Rb(D).
Chapter 13
Simple Convergence on
Mult(K[x])
As we saw in Chapter 5, Mult(K[x}) is provided with the topology of simple convergence. On the other hand, according to Chapter 11, circular filters which characterize the elements of Mult{K[x\), are provided with a distance. Thus, Mult(K[x\) admits two topologies that we shall compare. Notation: We denote by V? the mapping \£ from (K) into Mult{K[x\) defined as ^(T) = ipF. Throughout the chapter, D will denote a subset of K and UD will denote the topology of uniform convergence on D. The characterization of all absolute values on K(x) is given in [32, 34 and 30]. According to Theorems 4.14 and 10.5 in [30], we have this theorem: Theorem 13.1 (Garandel-Guennebaud): The mapping \& from $ ( i Q into Mult(K[x\) defined as ^(.F) = tp^ is a bijection. Moreover, the restriction \J/' of 9 to $'(K) is a bijection from <&'(K) onto the set of multiplicative norms on K[x], and therefore onto the set of multiplica tive norms on K(x). Moreover, if T has center a and diameter r, then
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Algebra
In a field F which is complete but not algebraically closed, such as Q p , we have to consider the (algebraically closed) completion O of an alge braic closure. Then every element ip of Mult(Q.[x}) has a restriction to F[x] which obviously belongs to Mult(F[x}). And conversely, every ele ment of Mult(F[x\) admits extensions to MultQ[x}). Therefore, we have a surjection from the set of circular filters on £1 onto Mult(F[x\). But this surjection is not injective: if a and b are conjugated over F, the circular filters of centers a and b and of same diameter r define the same absolute values on F[x]. Consider a circular filter T of center a and diameter r £ \K\. The point of generic disk consists of considering a transcendental element tp over K, laying in the circle |x — a\ = r, such that \ty: — u\ — r Vu S K such that |u — a| < r [15]. Consequently we can check that ipj? defines an absolute value | . | on K(tp) satisfying PF(P) = \P(tjr). Theorem 13.2: Multa(K[x\) = Multm(K[x\) is dense in with respect to the topology of simple convergence.
Mult{K[x\)
Proof: It is well known that all maximal ideal of K[x\ have codimension 1, hence Multa(K[x}) = Multm(K[x\). Now, let ip? e Mult(K[x\). Let Pi,..., Pn e K[x) and let e > 0. For each j = 1 , . . . , n, there exists Ej € T such that \Pj(x) - <pr(Pj)\oo <e\/x e Ej. Let E = n^=1Ej. Then E lies in T and we have \Pj(x) — <pr{Pj)\oo < e Vx £ E, which shows that <pa belongs to the neighborhood V(
Simple Convergence on Mult(K[x])
77
we can find a rank s € N such that Tt, is secant with A. Therefore, for every e > 0 and for every / e R(D), we can find a rank S G N such that !&»(/) ~ ¥>.?"(/)|oo < e. hence the sequence (ts) converges to (fyr. Finally, suppose that we cannot find b € d(a,r) and a subsequence (0mJt€N) with V'mj = V^t i (t € N), such that for a certain fixed 6 € d(a, r ) , each j^j be (b,rt )-approaching with 6 and diameter rt ^ r. We first notice that there exists r' > r such that none of the ipj?t is secant with T(a, r, r'). In the same way, for every c s d(a, r), there exists p(c) €]0, r[ such that none of the (f>jrt is secant with F(a, p, r). Let (ct)teN be a sequence in d(a, r) such that \CJ — Cfcl = r Vj ^ /c. We can easily construct a decreasing sequence of afRnoid sets (At)teN of the form At =
d(a,r')\l[Jd(cj,p(cj)-)\.
For each i e N, let et = min(j,r' — r,r — p(ci),...,r — p(ct)). By hypothesis, for each t e N , there exists n ( e N such that \ipnt(x—ct) — cpjr(x —ct)\oo < et, hence, putting ipnt = ippt, we can see that Tt is secant with d{a,r') but is not secant with T(a,r,r') and is not secant with Ufj=1d(cj,r~). Conse quently, Tt is secant with a class of d(a,r) different from all d(cj,r) Vj = l,...,t. So, we can construct a sequence (bt)ten such that \bj — a\ = \bj —bk\=r\/j ^ k and such that Tt is secant with d(bt,r~). Now, let / e R(D). For each t 6 N, since Tt is secant with d(bt,r~), we can find at € d(bt,r~) such that \ipnt(f) ~ |/(at)||oo < £t- Then the sequence (at)teN is thinner than T, and therefore lim^oo \f(at)\ = ipp(f). But of course lim^oo \f(at)\ = limt-oo Vv^ (/), so the sequence (ipnt) converges to ip? again. D N o t a t i o n : In Chapter 11 the set of circular filters was provided with an order relation ■< that makes it a tree, and we have denoted by -< the strict order associated to <. On the other hand, we can apply to Mult(K[x]) the usual order < on functions with values in R as
Let tpjr,
Then ipj? < tpg if and only if
Proof: Let T,G € $(K). Suppose first that they satisfy T
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Algebra
included in d(b, s). Let P & K[x\. We have V>r(P) < \\P\ka,r)
< \\P\U(a,a) =
Now, let Q{x) = x — a. Then fAQ)
< \\Q\\d(a,r) = r and
hence fp(Q) <
< \\P\\F = r and
hence tpj?(P) <
Let T be a large circular filter on K. Then if o-nd only if T is secant with D.
Corollary 13.8: The mapping from <&(£>) to Mult(R(D)) which associates to each circular filter T secant with D the multiplicative semi-norm
Simple Convergence on Mult(K[xJ)
79
Theorem 13.9: Let
q),
and therefore
I \Hfj)\ - vAfj)\oo < e, Vx e £,- (l < i < 9), W> € Mult(R(E
0 £>), || . lUnc).
80
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Algebra
Now, by Lemma 10.2 F contains a ^"-affinoid E. Then Mult(R(E), || . || E ) is a neighborhood of ip?- in Mult(K(x)) and Mult(R(E n D), |j . WEC^D) is a neighborhood of
Mult(K[x\)
is locally compact and locally sequentially
Proof: Let ipj? £ Mult(K[x]) and let E be an affinoid set such that Mult(R(E),\\ . \\E) is a neighborhood of ipjr in Mult(K[x}). Then Mult(R(E), || . \\E) is a compact neighborhood of ip?. Moreover, it is a sequentially compact neighborhood. □ Notation: Given a set D C K, we will denote by £(-D) the set of ip? g Mult{K[x\) such that T is D-bordering, and by So(-D) the set of ip? e Mult(K[x}) such that T is strictly D-bordering.
Chapter 14
Topologies on
Mult(K[x])
In Chapter 11 we defined a metric S on $(K) and we showed in Chapter 12 that $(K) is complete with respect to this metric. In Chapter 14 we saw that $(K) is in bijection with Mult{K[x\) which is provided with the topol ogy of simple convergence. Henceforth, thanks to this one to one correspon dence, we will consider that both $(K) and Mult{K[x\) are provided with both topologies. Theorem 14.1: Let D be closed and bounded. Then the boundary of Mult(R(D), || . \\D) inside Mult(K[x]), with respect to the topology of sim ple convergence, is equal to S(D). Proof: Let T be £>-bordering. Let W be a neighborhood of (pr in Mult(K[x]). By Theorem 13.10 there exists an affinoid set E e T such that Mult(R(E), || . \\E)CW. Since En(K\D) ^
\ Mult(R(D),
||. | | D ) .
Conversely, let ip? € Mult(R(D),\\. \\D) belong to the closure of Mult(K[x])\Mult(R{D), ||. \\D) in Mult{K[x}). Let E e T be an affinoid set. Then Mult(R(E), || . \\E) is a neighborhood of
||. \\D)) ^ 0.
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Banach
Algebra
Moreover, W n {Mult(K[x]) \ Mult(R(D),
\\. \\D))
is open in Mult(K[x]) because so are W and Mult(K[x]) \ Mult(R(D), ||. \\D). Since Multa(K[x\) is dense inside Mult(K[x\), there exists ya e {Multa(K[x})
\ Mult(R(D),
||. || D )) n Mult(R(E),
\\ . \\E).
Consequently, a lies in E, which proves that JF is secant with K\D. There fore the boundary of Mult(R(D), \\ . \\D) inside Mult(K[x}) is included in E(.D), and finally these two sets are equal. □ R e m a r k : Since the topology of simple convergence is weaker than the 5-topology, the boundary of Mult(R(D),\\ . \\D) inside Mult(K[x]), with respect to the (5-topology is obviously included in the boundary of Mult(R(D), || . \\o) inside Mult(K[x]), with respect to the topology of simple convergence. The equality does not hold in the general case. Theorem 14.2: Let D be closed, bounded and infraconnected. Then Eo(r > ) is included in the boundary of Mult(R(D),\\ . ||o) inside Mult(K[x}), with respect to the 5-topology. Moreover, the boundary of Mult(R(D), || . \\p) inside Mult(K[x]), with respect to the 5-topology, is equal to E(D) if and only if for every tpg £ T,(D)\T.Q(D) , there exists either a monotonous distances holes sequence or an equal distances holes sequence which is thinner than Q, whose superior gauge is equal to its diameter. Proof: Let D = d{a,r), with a £ D. Let (fg G £(£>). Suppose first that Q is the peripheral of D, hence
Topologies on Mult(K[x])
83
topologies. But since D is infraconnected, for every s € [I, r], by Proposition 10.4 the circular filter of center b and diameter s is secant with D, and then, by Corollary 13.8, ipb,s belongs to Mult(R(D), \\ . \\D). Consequently, ipg lies in the closure of Mult(R(D), \\ . \\D) for both topologies and therefore belongs to the boundary of Mult(R(D), || . \\D) inside Mult(K[x\) with respect to the 5-topology. Now, let ipg e E(Z)) \ So(-D)- Let r = diam(Q). Suppose first that there exists a monotonous holes sequence, or an equal distances holes sequence, thinner than Q, whose superior gauge is equal to its diame ter. Let (Dn)ne^ be a base of Q consisting of afiinoid sets. Then of course limn-.oo diam(Dn) = r. Thus, there exists a monotonous distances holes sequence or an equal distances holes sequence, (Tn)n&^ such that limn^oo diam(Tn) = r, and Tn c Dn Vn 6 N. For each n € N, let Tn = d(bn, r~), let sn = diam(Dn) and A„ = S(ifg, r — A. Thus we have proven that for every (pT e Mult(K[x}) \ Mult(R(D),
\\ . \\D),
we have S(G,T) > min
min \r - s,|oo, \r - A|oo 1.
\o<j
)
84
Ultrametric
Banach
Algebra
This finishes proving that tpg does not belong to the boundary of Mult(R(D), || . ||D) inside Mult(K[x\) with respect to the 5-topology. □ T h e o r e m 1 4 . 3 : Let D be a closed bounded set. The topology of simple convergence is weaker than the 5-topology on Mult(R(D), || . \\£>). If at least one large circular filter on K is secant with D, then Mult(R(D), || . H^) is not compact for the 5-topology. However, on a totally ordered subset of Mult(R(D), || . \\D) the two topologies are equal. Proof: Let
Topologies on Mult(K[xJ) 85
of diameter s e]max(r 0 ,r — e),r[ such that T§ is secant with d(b,s) and we denote by E the annulus T(b,s,r + e). Now, in all cases, consider tps e S D (Mult(R(E n Z>), || . WEHD))- Thus, £ is secant with d(b,r + e), hence diam{Q) < r + e. If T -< Q then r < diam(G) < r + e, hence of course Q lies in B. Now suppose G < T, hence T§ < Q -< T, hence s < diam(G) < r, and therefore Q lies in B again. This proves that (Mult(R(E n D), || . HBOD)) n •? is included in JB, which obviously shows that the two topologies are equal on S. □ Corollary 14.4: The topology of simple convergence is weaker than the S-topology on Mult{K[x\). However, on a totally ordered subset of Mult(K[x\) the two topologies are equal. Later, we will need this Proposition: Proposition 14.5: Let (
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Ultrametric Banach
Algebra
(v) There exists no annulus E together with filters T, Q G T such that Jbe secant with 1(H), Q be secant with £(E), and none of circular filters H € T be secant with E. Proof: First, (iv) trivially implies (Hi) and (ii) trivially implies (i). Next, by Corollary 14.4 it is obviously seen that (iv) implies (ii) and that (m) implies (i). Consequently, (iv) implies (i). We can also easily check that (i) implies (v). Indeed, suppose that (v) is not true. Thus there exists an annulus E = T(a, r, s) together with F, Q e T such that T is secant with 1(E), Q is secant with £ (E), but none of the circular filters TL are secant with H whenever TL G T. Let E be the set of (pn, TL £ T such that TL is secant with d(a,r), and let F be the set of ip-j-c, TL € T such that TL is secant with K \ d(a,s~). Then E, F are two subsets of S closed in Mult(K\x\) for the topology of simple convergence, and make a partition of S, therefore S is not connected with respect to the topology of simple convergence. Consequently, (i) implies (v). Thus, it only remains us to prove that (v) implies (iv). The map ping diam defined on $(K) is strictly increasing by Corollary 11.2. Let T, Q G T be such Q surrounds T, let r = diam^), s = diam(Q). Let I € [r,s\. Let H be the unique circular filter of diameter I surrounding T. By Theorem 13.11, every neighborhood of
Let D be a closed bounded set. The following statements
(i) Mult(R(D), || . \\D) is connected with respect to the topology of simple convergence, (ii) Mult(R(D), || . \\D) is arcwise connected with respect to the topology of simple convergence, (Hi) Mult(R(D), || . \\D) is connected with respect to the S-topology, (iv) Mult(R(D), || . \\D) is arcwise connected with respect to the S-topology, (v) D is infraconnected.
Chapter 15
Spectral Properties and Gelfand Transforms
The chapter is first devoted to present several semi-multiplicative seminorms and compare them to the spectral semi-norm. We introduce prop erties (o), (p), (q), (r), (s) which will keep the same meaning throught the book. Most of results in domain chapter were stated in [23]. Next, we consider two kinds of Gelfand transform in an ultrametric if-algebra: one is similar to the definition in complex analysis, but does not show strong properties. The other is more sophisticated and shows certain specify prop erties linked to circular filters. Notation: Henceforth, K will denote an algebraically closed field com plete for a non trivial ultrametric absolute value. Throughout the chapter, (A, || . ||) will be a commutative normed if-algebra with unity. We can define several usual semi-multiplicative semi-norms: \\x\\.a =sup{
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Ultrametric Banach Algebra
Remark: r is not a semi-norm. Indeed, as we will see later, there might exist x e %{A) \ X(A, K) and x,y € A such that \ix) = 1 + 1 , x(u) = —t, with t ^ K, and max(r(x),T(y)) < 1. However we have x{x + y) = 1, hence r ( x + y) > 1. Notation: (o) (p) (q) (r)
In ^4 we will denote by (o), (p), (q), (r), (s) these properties:
||x|| sa = T(X) Vx e A, \\x\\sa = ||x|| si Vx e A, T(X) = \\x\\si Vx £ A, ||x|| sa = ||x|| 5m Vx e A,
(s) \\x\\Sm = \\x\\si Vx € A.
Thus, by definition, Property (p) implies Properties (o), (q), (r), and (s), and (r) implies (o). Proposition 15.2: Assume that A satisfies Property (p). Then A is semisimple if and only if \\ . ||Sj is a norm. Theorem 15.3: Let A be complete, let \\ . ||^ be its spectral semi-norm and let B be a commutative normed K-algebra with unity whose spectral semi-norm \\ . ||^ satisfies Property (q). Then every K-algebra homomorphism <j> from A to B satisfies W(x)\\fi<\\x\\?iVxzA. Proof: Suppose that for certain x € A we have ||(x)||^ > ||x||^ and let y =
Spectral Properties and Gelfand Transforms
89
in complex analysis, consisting of associating to each element / of A the mapping / f r o m X{A,K) to K denned as f(x) = x(f), (x € X(A,K)). The second, denoted by GM^ consists of associating to each element / of A the mapping /* from Mult(A, || . ||) to Mult{K[x\) defined as f*{(f>){P) = 4>(Pof). As in complex analysis, Propositions 15.6, 15.7 and 15.8 are immediate: Proposition 15.6: G^ is injective if and only if the intersection of all maximal ideals of codimension 1 is null. Proposition 15.7: X(A,K) being provided with the topology of simple convergence, for every f £ A, f belongs to C(X((A, K), K). Proposition 15.8: C(X((A, K), K) being provided with the norm of uni form convergence, GA satisfies \\f\\ = \\f\\Sa for every f £ A. Theorem 15.9: Assume that A satisfies Property (p). IfC(X(A,K),K) being provided with the norm of uniform convergence, then GA is an isometry if and only if \\x2\\ = \\x\\2, Va: £ A. Proof:
Suppose ||x 2 || = ||:r||2 is true for all x £ A, let x £ A and let p =
■UgiP Then we check that ]-^Jf = p(2") hence ||x|| ai = plim„-oo I M I " , and therefore p = 1. So, G^ is an isometry. The converse is trivial. □ Theorem 15.10: Assume that A satisfies Property (p). The following two properties are equivalent on A: (i) A is semi-simple and GA(A) is closed in (ii) A is uniform.
C(X(A,K),K),
Proof: First, suppose that A is uniform. Let x belong to the Jacobson radical of A. In particular, x belongs to the intersection of maximal ideals of codimension 1 of A. So, we have \\x\\sa = 0, so, by Property (p), \\x\\Si = 0, and therefore x = 0, hence A is semi-simple. Let h belong to the closure of GA(A) in C(X(A,K),K). Since C{X{A,K),K) is provided with the norm of uniform convergence, we just have to consider a sequence /„ of G^ converging to h. Thus, \\h-Jn\\ = sup{|x(/n)-ft(x)l Ix £ X{A,K)}. The sequence (fn)nen is a Cauchy sequence in A, with respect to ||a;||s» = 0, and therefore with respect to the norm of A because A is uniform. Let / = lim n _ 0 0 fn. Then we can check that / = limn^oo fn in C(X(A, K), K) and therefore h = f. Conversely, suppose that A is semi-simple and that G^(A) is closed in C(X(A, K),K). Since A is semi-simple and satisfies Property p), j| . || sa
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Ultrametric Banach
Algebra
is a norm equal to ||x|| S j- Consider a Cauchy sequence (/„) with respect to the norm || . | | S j . Then the sequence (/„) is a Cauchy sequence in C(X(A,K),K), and has a limit h which actually lies in GA(A) because GA{A) is closed in C(X(A,K),K). Hence, there exists / e A such that f = h. Consequently, / is the limit of the sequence (fn) with respect to the norm || . || si . Thus, A is a K-algebra complete for both || . \\si and || . ||. And since ||x||Sj < ||a:|| Vx £ A, by Banach's Theorem the two norms are equivalent. □ Now we will examine the mapping GM^. Theorem 15.11: Given f € A, the mapping f* from Mult(A, |] . ||) to Mult(K\x\) is continuous with respect to the topology of simple convergence. Proof: Let Q = f*{<j>). By Theorem 14.10 the family of sets Mult(H(E), || . \\E), where E is an infraconnected affinoid subset of K lying in Q, makes a base of neighborhoods of ipg. So, we take an arbitrary £/-affmoid B and will show that there exists a neighborhood W of <j> inside Mult(A, || . ||) whose image by / * is included in Mult{H{B), || . \\B). If Q has center b and diameter r > 0, we can take an arbitrary ^-affinoid B of the form d(b,r + t) \ (Uq,=1d(bj, (r — e - ) ) , and if Q has no center and but has diameter r, or is a Cauchy filter, we can take an arbitrary 5-affinoid B of the form d(b,r + e). Putting b = bo, we consider now the neighborhood of <£ W = {^ € Mult(A, || . ||) | Mf - bj) - 4>{f - bj)\ < e, (0 < j < q)}. Then we can check that f*(W) claim.
C Mult(H(B),
\\ . \\B), which proves the □
Lemma 15.12 is immediate: Lemma 15.12:
Let x e X{A,K)
and let f e A. Then f*(\x\)
=
fx(f)-
Proof: Let T be the circular filter such that ipyr = /*(|xl)- Then (fyr clearly belongs to Multm(K[x]) and T is the Cauchy filter of neighborhoods of xU) b e c a u s e Vr(P) = I^(X(/))I VP € K[x]. O Theorem 15.13: Assume that is uniform and that the intersection of maximal ideals of codimension 1 is null. Then GM^ is infective. Proof: Suppose / * = g* and / ^ g. let r = \\f — g\\. Since the inter section of maximal ideals of codimension 1, there exists (x € X(A,K) such that x(9 — / ) ¥" 0- By Lemma 15.12 we have /*(|xl) = fx(f) an<^
Spectral Properties and G elfand Transforms
91
5*(lxl) = ¥>x(g)- B u t s i n c e / * = 9*, we have /*(|xl) = 5*(lxD- Let T be the circular filter such that ipjr = /*(|x|) = 5*0x1)- Then
f,geA. \\f-g\\.
Notation: Given (f> € Mult(A, || . ||), we will denote by Z^ the mapping from A into Mult(K[x\) defined as Z^(f) = f*{<j)). Corollary 15.16: Let A be uniform. The family of functions Z^, <j> € Mult(A, || . ||) is uniformly equicontinuous with respect to the 5-topology on Mult(K[x}). Corollary 15.17: Let A be uniform. Each function Z$, € Mult (A, || . ||) is continuous with respect to the topology of simple con vergence on Mult(K[x\). Remark: Theorem 15.13 and its Corollaries apply to reduced affinoid if-algebras, as it is shown in Chapter 35.
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Chapter 16
Analytic Elements
Let us briefly recall that analytic elements where introduced by Marc Krasner in order to define a general notion of holomorphic functions in sets which are not just disks: indeed, when two disks have a non empty intersection, this which has the biggest radius contains the other. As a con sequence, it is hopeless to cover a set (which is not a disk) with a chained family of disks. But, according to Runge's Theorem, complex holomor phic functions may be viewed as uniform limits of rational functions. Marc Krasner adopted this point of view to defining holomorphic functions in a set D.
Definition and notation: In this chapter D is an infinite subset of K. We denote by H(D) the completion of R(D) for the topology UD of uniform convergence on D. The elements of H(D) are called the analytic elements on D [30,36]. By definition, the set H(D) is then provided with the topology of uni form convergence on D for which it is complete, and every / 6 H(D) defines a function on D which is the uniform limit (on D) of a sequence (^n)nGN in R(D). Thus, given another set D' containing D, the restriction to D of elements of H(D') enables us to consider that H(D') is included in H(D). Here, we will only consider bounded analytic elements, and we will denote by Hb(D) the set of the elements / e H(D) bounded on D. Then Hb{D) is clearly a K-veciox subspace of H(D) and is closed in H(D). Moreover, || . |£> is a /if-algebra norm on Rb(D), hence its continuation to Hb(D) is a /^-algebra norm that makes Hb{D) a Banach Af-algebra. If D 93
94
Ultrametric Banach
Algebra
is unbounded, we will denote by HQ(D) the set of the / G H(D) such that Hm\x\-*+oo,x€Df(x)
= 0.
According to Theorem 6.8 and Corollary 13.8, it is obviously seen that each element ip? of Mult(R(D), || . ||£>) has a unique continuation to Mult(H(D), || . \\D) and will be denoted by ipjr again. Particularly, when T is the circular filter of center a and diameter r, we will also denote by (pa,r the continuation of <par to Mult(H(D), || . \\D)- Thus we identify Mult(H(D),\\ . \\D) to Mult(R(D),\\ . ||D), and therefore all properties already proven in Mult(R(D), || . \\D) also hold in Mult(H(D), || . \\D)Theorem 16.1 is well known [30, Chapter 10]: Theorem 16.1: Let D be an infinite closed subset of K. Then Hb(D) is a Banach K-subalgebra of KD. The following three conditions are equivalent (i) (ii) (Hi) (iv)
Hb(D) = H{D) H(D) is topological K-vector space, (H(D), || . ||o) is a K-Banach algebra D is closed and bounded.
If these conditions are satisfied, then \\ . \\p is a semi-multiplicative
norm.
The following two lemmas are classical and easy. L e m m a 16.2:
R(D) is a full K-subalgebra of H(D).
L e m m a 16.3: Let f e H(D) be such that inf l 6 D|/(x)| > 0. Then / _ 1 belongs to H(D). Moreover if D is closed and bounded, an element g € h(D) belongs to H(D) if and only ifmix^r)\g{x)\ > 0. Let g, h G H(D) satisfy \g(x)\ = 1 for allx S D and \\h—l\\o > \\g—l\\DThen we have \\hg — l\\D = \\h — 1\\D Lemma 16.4: Let a e K\D and let f € R(D) be such that \f(a)\ > \\f\\oThen —^ belongs to the closure of K[f,x] in H{D). Proof: Let A be the closure of K[f,x] in H{D). Since A is a Banach i^-algebra, by Theorem 16.3 f(a) — f is invertible in A, hence of course in H(D). But by Lemma 16.2 its inverse g actually belongs to R(D). Now, since a is clearly a zero of f(a) — f, it is a pole of g. Consequently, by Lemma 1.8 —^— belongs to K{g,x\ which is included in A. □ Lemma 16.5: Let a G D\D and let f G H(D) be such that there exists n € N such that (x — ot)nf G H(D U {a}). There exists a unique q € N such that (x — a)q f G H(D U {a}) and such that the value of (x — a)qf at a is not zero.
Analytic Elements
95
Definitions and notation: In the hypothesis of Lemma 16.5, / is said to be meromorphic at a and to admit a as a pole of order q. Let D C K be infraconnected closed and bounded, and let T be a hole of D. Let / € H{D). Then / will be said to be meromorphic in T ii there exist finitely many points (aj)(i
SUch
that
H(B) is also equal to the set of the series f(x) = lim
\hn\r~n
= 0.
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Ultrametric Banach Algebra
By Theorem 14.1 and 14.4 in [30] and by Theorem 16.7 we have Theorem 16.8 T h e o r e m 16.8: Let a G D and r G R^ be such that d(a,r) C D. Let f G H(D). In d(a, r), f{x) is equal to a power series of the form ]CnLo a™ ( x — a)n such that linin-^oo |a«|'*n = 0. If f(a) = 0 and if f(x) is not identically zero in d(a,r), then there exists a unique integer q 6 N* such that an = 0 for every n < q and aq ^ 0, and a is an isolated zero of f in d(a,r). Moreover, there exists g £ H(D) such that f(x) = (x — a)qg{x). As a consequence of Theorems 16.7 and 16.8, we deduce Theorem 16.9: T h e o r e m 16.9: The characteristic of K is supposed different from 0. Let f G H(d(0,r~) and for every j = 0, ...,n let aj = * ./ '. Then f is equal to the power series Y^=Qan^n, and ||/||d(o,r—) = s u P n eNl a n|Moreover, for every q € N, / is of the form X)?=o aix^ + xU9> w^ 9 e H(M) and ||/|| d ( 0 ,r-) = max(|o 0 |, | a i | , . . . lon-ilr"" 1 , ||a;n5||d(o,r-))We must also recall Theorem 13.12 in [30]: T h e o r e m 16.10: If K has characteristic 0, then an element f G H(d(0, r)) has a derivative identically equal to 0 if and only if it is equal to a constant. If K has a characteristic p ^ 0, then an element f € H(d(0,r)) has a derivative identically equal to 0 if and only if there exists g £ H(d(0, r)) such that f(x) = (g(x))p. o
Definitions a n d notation: Let / € H(D), let a &D, let r > 0 be such that d(a,r) C D, and let f(x) = J2^=qbn(x ~ a)n whenever x G d(a,r), with bg(a) ^ 0, and q > 0. Then a is called a zero of multiplicity order q, or more simply, a zero of order q. In the same way, q will be named the multiplicity order of a. An element / G H(D) is said to be quasi-invertible if it factorizes in the o
form Pg with P G K\x] all zeros of which lie D, and g G H(D) invertible in H(D). T h e o r e m 16.11: Let D be closed and bounded and let f G H(D) be quasi-invertible. There exists h G R{D) such that \f{x)\ = \h{x)\, Vx G D. Proof: Since / is quasi-invertible, there exists a polynomial P whose zeros lie in D, and an invertible element g in H(D) such that / = Pg. Let m = infxg£> \g(x)|. Since g is invertible, we have m = ..±.. , hence m > 0, and then we can find / € R(D) such that \\g — 1\\D < m. Consequently, we
Analytic Elements
97
have |<7(a:)| = \l(x)\, Mx e D and we obtain h by putting h = PL Moreover, if r € \K\, then (po,r(h) € \K\. Conversely, if s ^ t and if <potT(h) € \K\, t h e n r € \K\. The following Theorem 16.12 is Theorem 15.1 in [30]. Theorem 16.12 and infraconnected holes (Tn)ra£N* °f /o 6 H(D) (resp. and
(M.Krasner): Let D be bounded (resp. unbounded) and let f £ Hb(D). There exists a unique sequence of D and a unique sequence (fn)nen in H(D) such that f0 e K), fn S H0(K \ Tn) (n > 0), lim^oo \\fn\\D = 0
oo
(i)
/ = £/„an
||/ n || D .
Moreover for every hole Tn — d(an, r~), we have («)
l l / » L = \\fn\\KXTn
= fan,r,Xfn)
< Van,rn{f)
< ll/L-
If D is bounded and if D = d(a, r) we have (Hi)
H A L = | | / 0 | | 5 = VaAfo)
< <Pa,r(f) <
ll/L-
x
If D is not bounded then |/ 0 | = lini|a;|_oo,a:€C \f( )\ < l l / L Let D' = D\ (U£°=1Tn). Then f belongs to H(D') (resp. Hb(D')) and its decomposition in H(D') is given again by (i) and then f satisfies
ll/IL = ll/LCorollary 16.13: Let D be closed and infraconnected. Let [Ti)iej be the family of holes of D. Let J be a subset of D and let L = I\J. Let E = DU (Ui€jTi) and let F = DU(Ui£LTi). Then we have H(D) = H0(E)@H(F), and for each g £ HQ(E), h e H(F), we have \\g + h\\D = m ax(||g||£;, | | / I | | F ) . T h e o r e m 16.14: Let D be closed and infraconnected. Then Eo(-D) is a minimal boundary for (H(D), \\ . \\D)Proof: By Theorem 16.12 it is a boundary for (H(D), \\ . ||D). Conversely, let B be a boundary included in Eo(D). Let D = d(a,r). Considering (x — a) we can see that a,r must belong to B because if
98
Ultrametric Banach Algebra
Proof: Let B be the closure of K[h,x] in H(D) and let T\,...,Tq be the holes of D containing at least one hole of D and let D' = D\ Uj =1 T,-. Clearly K[h, x] is included in H(D'). But since each hole of D' is a hole of D, and since D' = D, by Corollary 16.14 H(D') is a closed if-subalgebra of H(D). Consequently D = D'. □ The Mittag-Leffler Theorem suggests some new definitions. Definitions and notation: Let / € Hb(D). We consider the series J2™=ofn obtained in Theorem 16.12, whose sum is equal to / in H(D), with /o € H(D), fn e H(K\Tn)\{0} whereas the Tn are holes of D. Each Tn will be called a f-hole and /„ will be called the Mittag-Leffler term, of f associated to Tn, whereas /o will be called the principal term of / . For each /-hole T of. D, the Mittag-Leffler term of / associated to T will be denoted by fr whereas the principal term of / will be denoted by /o- The series Y^=ofn will D e called the Mittag-Leffler series of f on the infraconnected set D. More generally, let E be an infraconnected set and / e H(E). According to Theorem 9.5 [30], / is of the form g + h with g € R(K \(E\ E)), and h £ Hb(E), and such a decomposition is unique, with respect to an additive constant. For every hole T of E, we will denote by fT the Mittag-Leffler term of h associated to T, and fT will still be named the Mittag-Leffler term of f associated to T. Corollary 16.16: Let D be infraconnected. Let f € Hb{D) let (T„)nSK* be the sequence of the f-holes, with Tn = d(an,p~), let /o = f0, and fn = / T „ for ever~y n e N*. Let D = d(a, s) (resp. D = K). There exists q 6 N such that \\f\\D = \\fq\\D. Ifq>\ then \\f\\D = y>„„ r ,(/) = ¥>«„,-,(/,)■ If q = 0 and if D is bounded (resp. is not bounded) then \\f\\D = <pa,s(f) = faAfo) [resp. \\f\\D = |/o|). Further, given a hole T of D, if f belongs to E~b(D), and if g belongs to H0(K\T) and satisfies f -g € H(DuT), then fT is equal to g. Corollary 16.17: Let D be infraconnected. Let f € Hb{D). There exists a large circular filter T with center a S D secant with D such that if-pif) — Ij/ll D . If D is bounded, there exists a D-bordering filter T such that <pr(f) = \\f\\D.
Analytic Elements
Corollary 16.18: Let f £ H(d(Q,l )) and let (d(am,l family of the f-holes. Then f is of the form
99
))mgN* be the
oo \ ~ *
/ ■ \
n
n=0
i
\~~*
m,n€N- ^
a
n,m
""^
wii/i linin-too an = 0, linin-xx, |a„, m | = 0 whenever m € N* anrf lim ( sup |a„ j m |) = 0. Moreover f satisfies ll/IU(o,i-) =
sup
|a„, m |.
m,n£N*
Conversely, every function of the form (iv), with the am satisfying \am\ = \dj — am\ = 1 whenever m ^ j , belongs to H(d(0,l~)). The norm || . ||d(o,i-) *s multiplicative and equal to tfio,iTheorem 16.19 is proven in Chapter 15 of [30]: Theorem 16.19: Let r i , r 2 G R+ satisfy 0 < ri < r 2 . T/ien i?(A(0, r\,r2)) is equal to the set of Laurent series X^-^0™2-™ with lim-n-^-oo | a n | r " = linin-^oo la^lrj = 0 and we have -t-oo
y]anxn — oo
= max (sup|a„|r™, sup|a„|r2 ). A(0,n,r 2 )
V>0
n<0
'
About maximal ideals of codimension 1, we must also recall Theorem 16.2 in [30]: Notations: Let D 6 A. For every a € D we will denote by y{a) the ideal of the / e H(D) such that f(a) = 0. T h e o r e m 16.20: Let D be closed hounded infraconnected. The mapping 4> from D into the set of ideals of H(D) defined by 0(a) = y(a) is a bijection from D onto Maxa(H(D)). Finally, we can notice that analytic elements define uniformly continu ous functions, in a closed bounded set: T h e o r e m 16.21: Let D be closed and bounded and let f € H(D). f is uniformly continuous on D.
Then
100
Ultra-metric Banach Algebra
Proof: Let h € R(D). Since h has finitely many poles, we can find r 0 > 0 such that for all a £ D, d(a,ro) contains no poles of h. Let M = \\h\\DThus, by Corollary 16.18, for all a € D, we have sup fc>i
hW(a) k\
7-Q
< SUp fc>0
hW{a) k\
r g < |WU(a,ro) <
M.
Consequently, for every r €]0,ro], we have sup k>l
h,W(a) Jfe!
< sup fc>i
hW(a) k k r o(-) ) k\
<M-.
Thus, we can see that h(d(a,r)) C d(h(a),M^-), which proves that h is uniformly continuous on D. Then, so is every element oiH(D), as a uniform limit of uniformly continuous functions on D. □
Chapter 17
Holomorphic Properties on Infraconnected Sets
Throughout the chapter, D is a subset of K. Here we shall go back to the basic properties of analytic elements which are transmitted by rational functions. Proposition 17.1 is classical and given in [30] as Proposition 20.3: Proposition 17.1: Let a G D, let r G [S(a,D),diam(D)\, and let f € H(D). If
have t zeros in C(0,r).
Let q =
aq+t
Theorem 17.4: Let f = ^2n^>_00anxn € H(C(Q,r)) have no zero in C(0,r). Let t = J V + ( / , - l o g r ) . Then we have N+ (/, - logr) = N~(f, — logr) and v(f(x)) = v(f, — \ogr) = v(at) — tlogr whenever
ieC(0,r). Corollary 17.5: Let f £ H(T(0,r,s)) have no zero in F(0,r,s) and be equal to a Laurent series X^n^-cxs anXn ^x € r ( 0 , r , s ) . Assume that there 101
102
Ultrametric Banach Algebra
exists an integer m e Z* such that \am\pm > \an\pn Vn ^ m, V/J e]r, s[. / / m > 0, we /lave /(F(0, r,s)) = T(0, \am\rm, \am\sm), and ifm < 0, we have f(F(0,r,s)) =r(0,\am\sm,\am\rm). Theorem 17.6 will be useful when considering restricted power series: Theorem 17.6: Let f(x) = X^Lo a " x ™ € H{U)- Then f is invertible in H(U) if and only if |a.01 > su P«>o \an\- And f is irreducible in H{U) if and only if\a0\ < |oi|, |aj| > s u p n > 1 \an\. Proof: If |ao| > s u p n > 0 | a „ | , by Theorem 6.13 / is obviously invertible in H(U). Else, by Theorem 23.5 in [30], / admits at least one zero, and therefore is not invertible. Precisely, let q be the unique integer such that \aq\ = sup r a € N \an\. If q = 1, then by Theorems 23.5 and 23.6 in [30], / has exactly one zero a, and since it is quasi-invertible, it is of the form {x — a)g, with g invertible, hence / is irreducible. If q > 1, since / admits q zeros, / factorizes in the form n ^ i l 3 - ~~ aj)9ix)> whereas g e H(U) has no zeros, and therefore is invertible, and therefore is not irreducible. □ Definition: D will be said to be calibrated if every hole has a diameter that belongs to | K | and if diam(D) either belongs to | K \ or is infinite. Theorem 17.7 is proven in Chapter 30 of [30], (as Theorem 30.6): Theorem 17.7: Let D be closed and bounded. Then D is calibrated and strongly infraconnected if and only if for every f e H{D) there exists a € D such that | / ( a ) | = ||/|| D . As far as closed bounded sets are concerned, the composition of analytic elements is an analytic element (Corollary 11.2 in [30]): Proposition 17.8: Let D, D' be closed bounded subsets of K, and let f € H{D) be such that f(D) C D'. Let g € H(D'). Then go f belongs to H(D). Homomorphisms between A'-algebras H(D) are characterized in [30], (Proposition 11.6 and Theorem 11.7). Here, for convenience, we will restrict the statement to the case of closed bounded sets: Proposition 17.9: Let D, D', D" be closed bounded subsets of K and let 7 € H(D') satisfy j(D') C D. Let 0 7 be the mapping from H(D) into H(D') defined as 0 7 ( / ) = / ° 7- Then ^ 7 is a K-algebra homomorphism from H{D) into H(D') continuous with respect to the topology of uniform convergence on D for H(D) and on D' for H(D'). 1 / 7 6 H(D') and C € H(D") satisfy -y(D') = D and £(D") = D', then
Holomorphic
Properties on Infraconnected
Sets
103
If 7 is a bijection from D' onto D and if ^~l € H(D) then
104
Ultrametric Banach Algebra
As an immediate consequence, we have Corollary 17.16. Corollary 17.16: If D has infinitely many infraconnected components then H(D) is not noetherian. Henceforth, we will also use Theorem 17.17 which was proved in [30] as Theorem 21.12: Theorem 17.17:
Let f € H(D).
Then f(D) is infraconnected.
Chapter 18
T-Filters and T-Sequences
The behaviour of analytic elements is linked to the existence of certain pierced filters, called T-filters [20,30]. This has a strong implication on ultrametric spectral theory. Notation: In all of this chapter D denotes an infraconnected closed set of diameter S e ! + . Let a e D, let r e]S(a, D), S] n \K\ and let (Ti)ieI be the set of the holes of D included in C(a, r). We will denote by T(D, a, r) the set Ujg/Tj. For every q € N we will denote by S(D, a, r, q) the set of the monic polynomials P of degree q whose zeros lie in T(D, a, r). Let a e D. Let r e [6(a, D), diam(D)} n \K\ and let q € N. We put 1D(a,T,q)=i"
inf P&S(D,a,r,q)
Lemma 18.1: Let D = d(a,r
) \ d(a,p
), with 0 < p < r.
Then
7 D (°» »",«) = ( = ) • Proof: Indeed, for each monic polynomial P of degree q, having all its zeros in d(a, p ) we have
„-*■
In [30] we have shown this proposition, as Corollary 34.3. Proposition 18.2: Let a e D, let r e [5(a,D),diam(D)] n \K\ and let q € N. Let (Ti)iei be the set of the holes of D included in C(a, r) with 105
106
Ultrametric Banach
Ti = d(ai,ri
Algebra
) . Then we have
7 D ( a > r >
sup <
inf
3 ni
0>m ~
a
i
l
(h,
■ ■,i()€le,
^ t j
=q
l<m
Definitions: Let T be an increasing filter on D of center a € D and diameter 5. The filter !F will be called an increasing T- filter if there exists a sequence of circles (£ m )meN with E m = C{a,dm) (m € N) such that linim^oo dm = S and d m < d m + i together with a sequence of natural integers (? m ) m e N satisfying lim -yD(a,dm,qm) m —> o n
iV-o.
TT ■*- -■•
(18.1)
A decreasing filter .F will be called o decreasing T-filter if it admits a base (An)meN with D m = d(am, dm) nD\ ( n m e N d ( o m , rm)) together with a sequence of natural integers (qm)meN such that, putting S m = C(am, dm), the sequences ( a m ) m S N , (d m ) m 6 N, (gm)m€N satisfy m _—11
/d
\9j
lim 7 D ( a , d m , ? m ) TT ( - ^ I
0.
(18.2)
In particular, this definition holds when T is a decreasing filter of center a and diameter S, and then we can take am = a for every m G N. Another way to define T-filters consists of introducing T-sequences. We will call a weighted sequence a sequence (T mi i,g m) j)i<j< s ( TO ) )me N with (Trnti)i
T-Filters and T-Sequences
107
be called, respectively, the diameter, the monotony, the piercing of the weighted sequence {Tm,i,qm,i)i
lim m—KX>
n
d„ sup
PTU,J
l< i 7<s(m)
m —1
n
\Q>m,i
®m,j\
l
dn^q
\ (18.3)
= 0 /
(resp.
lim
n
sup l<7<s(m)
*m,j |
l
n!
n=l
(18.4)
7
)• Given a T-sequence (Tm^, qm,i)i
108
Ultrametric Banach
Algebra
a zero of order qm>i in Tm^ and has no other zero in K, satisfying further
J5.((<^.<«JII£IU_,U»») n ( £ r ) (reap. ^ ( ^ ( O J I I ^ l c ^ „ , „ , , „ „ )
-°
II ( £ ) * " ) = °)'
Lemma 18.4: Let J- be a monotonous filter on D. Then T is a T-filter if and only if there exists a T-sequence associated to T. Proposition 18.5: Let T be an increasing (resp. decreasing) T-filter of center a and diameter r. Let h(x) = j ^ , let D' = h(D) and T' = h(T). If \a — a\ < r, T' is a decreasing (resp increasing) T-filter of center a and diameter £. If \a — a\ > r, T' is an increasing (resp. decreasing) T-filter of center h(a) and diameter r ' " , 2 . Proposition 18.6: Let D admit a T-sequence (T,m)j,gTOii)i<j
T-Filters and T-Sequences
109
As an improvement of Theorem 36.10 in [30], we can state: Theorem 18.10: Let K be strongly valued. Let a e D be such that 5{a,D) < diam(D). We assume that for every r e]5(a,D), diam(D)[n\K\, each class of C(a,r), but maybe finitely many ones, contains at least one hole of D. Then for each r £}5(a,D),diam(D)] there exists on D an increasing idempotent T-sequence and a decreasing idempotent T'-sequence of center a and diameter r. Proof: The proof is the same as that of Theorem 36.10 in [30]. Let r € ]5(a,D),diam(D)}. Let J =]5(a, D),diarn(D)[n\K\, and let (rn)ne^, r ( 'n)n€N be sequences of J satisfying rn < r'n < rn+i, l i m ^ o o rn = r. By hypothesis, either the residue class field K of K is not countable, or \K\ is not. First we suppose that \K\ is not countable. Let n be fixed. By Lemma 18.9 there exists pn > 0 together with an infinite family of circles C(a,r), with rn
m
'rn\v~r'n+1
r'J
<
pn+1
1
•
(18-5)
n+ l
Thus, we have defined the sequences (pn)n£N, (<7n)n€N- Now, for each n £ N, we put tn = Y^h=i Qh- From above, we can clearly find qn circles C(a,dm)tT,<.Tn
diam(Tm)
n
,
^
—— < — whenever tn < m < tn+\. n. ^-^ drf„ n m
3
This shows that m-^oo diam{lm)
xx
dm
and therefore the weighted sequence (Tm, 1) is an idempotent T-sequence. Now we suppose that the residue class field K. is not countable. By Lemma 18.9, for each n € N, there exist pn > 0 together with an infinite family of classes of C(a,rn), such that each contains at least one hole of
110
Ultrametric
Banach
Algebra
diameter bigger than pn. In the same way as the previous case, for each n G N, we take qn £ N satisfying (18.5) again and then, we have defined sequences (p„)„ 6 N, (qn)nen- For each n e N, we put tn = YJh=\ 4fc- Then in C(a,rn), we can clearly find qn different classes such that each contains at least one hole of diameter bigger than pn. Let (Tm)tn<m
<j
By (18.6) it is easily checked that 7 D ( G , r n , q n ) < —.
(18.6) Therefore the
weighted sequence (Tm, l)meN is an idempotent T-sequence because by (18.5) we have
Symmetrically, we can prove the existence of a decreasing idempotent Tsequence of center a and diameter r: in the proof, we just have to replace (18.5) by
/r^y-rvH \rnJ
pn+i
<
1 n + 1' D
T-filters are used to describe the ability of certain analytic elements to vanish in a non trivial way. We have to recall this. Definition: Let T be a monotonous filter of diameter S, on an infraconnected set D C K, and let / € H(D). First, / is said to be vanishing along T if lim^r fix) = 0. When T is decreasing (resp. increasing) of center a, then / is said to be strictly vanishing along T if it is vanishing along J7 and if there exists S' > S (resp. 5 ' < S) such that for every r e]S,S'] (resp. r 6 [S',S[) we have (fa,r(I) T^O. When T is decreasing with no center in K, admitting a canonical base (Z?n)n6N with Dn — d(an,rn) n D, then / is said to be strictly vanishing along T if it is vanishing along J- and if there exists S' > S such that V3a„+i,r(/) 7^ 0 whenever r 6 [rn, S'}, whenever n G N. In Chapter 37 of [30] it is shown that given a monotonous filter T on an infraconnected set D, there exist elements H{D) strictly vanishing along T
T-Filters and T-Sequences
111
if and only if T is a T-filter. Moreover, by Theorem 37.2 in [30], we have this theorem: Theorem 18.11: Let T be aT-filter on D, let (T mj j, m g> i)i<j< s ( m ) iTne N be a T-sequence associated to T, and (<7m)m£N be the sequence of circles carrying this T-sequence. Let D' = ^ \ ( U i < j < s ( m ) j m g ^ r m i i ) , and for every (m,i), let amii G TTOjj. Let T' be the T-filter on D' associated to the Tsequence (Tm^,qm^). Let a G C(F'). There exists g € H(D') satisfying these properties: (i) g is meromorphic in Tm^, admits am 0, of center a and diameter r. Let D' = K\ (Ui
— e)) ( resp. x G D \
(«) Ue\\D, < ( p 3 + r 3 e , (Hi) bmii is a pole of (fic of order um>j < qmti and (j>e has no pole different from
This page is intentionally left blank
Chapter 19
Applications of T-Filters and T- Sequences
T-filters help us characterize quasi-invertible analytic elements and solve various other problems (Theorem 38.4 in [30]). In particular, they provide a characterization of quasi-invertible elements and overall they let certain analytic elements be different from zero in an area, and identically zero in another area. Notation: Throughout this chapter D denotes an infraconnected set of diameter S s M+. Theorem 19.1: Let D be open closed and bounded and let f € H(D). Then f is quasi-invertible if and only if it is not vanishing along a T-filter. Prom Theorem 19.1 here, and from Theorems 16.2, 38.9, and 42.5 in [30] we must recall the characterization of principal if-algebras H(D). Theorem 19.2: Let D be infraconnected, open closed and bounded. Then the following statements are equivalent: (i) (ii) (Hi) (iv) (v)
D has no T-filter; H(D) is noetherian; H(D) is a principal ideal ring; every ideal of H(D) is generated by a polynomial whose zeros lie in D; Every element of H(D) is quasi-invertible.
Moreover, when the conditions above are satisfied, every maximal ideal M. of H(D) has codimension 1, and is associated to a point a € D such that M = {f £ H(D) | f(a) = 0} and for every f e H(D), we have sp(f) = f{D). 113
114
Ultrametric
Banach
Algebra
Proof: According to Theorem 19.1 here and to Theorems 16.2, 38.9, and 42.5 in [30], the only new statement here is that for every / e H(D), we have sp(f) = f(D), and this comes from the fact that every maximal ideal M of H{D) has codimension 1, and is associated to a point a € D such that M = {/ € H{D) | / ( a ) = 0}. □ Notation: Let T be a circular filter or a monotonous filter on D. Hence forth, we will denote by y(T) the set of the / € H(D) such that lira.? f(x) = 0. Hence y(T) is equal to Kerfa^), and therefore is a closed prime ideal of H(D). If T is a monotonous filter on D, we will denote by yo(T) the closed ideal of the / € H(D) such that f(x) = 0 whenever x e V(T). L e m m a 19.3: Let admitting a partition T-filter, and assume circular filter. Then
D be a closed bounded infraconnected set of diameter s of the form (C(Ti))iei where each T is an increasing that D has no other T-filter. Let Q be the D-peripheral y{T) = y{Q), Vi € / .
Proof: Let i £ I be fixed, let a, be a center of T, and let r, = diam(Ti). Let / 6 y{Ti), so we have r» such that <Pa,i,r(f) > 0- Let u = inf{r ) <pai,r(f) > 0}. Then / is strictly vanishing along the decreasing filter of center a, and diameter u, so this filter is a T-filter, a contradiction to the hypothesis. Consequently, we have Va.Af) = 0> and therefore y{Ti) C y(G). Now, let / € y(G), so we have <pai,r(f) = 0. Suppose that <pai,ri(f) > 0. Since <pai,r{f) = 0, we can consider u = inf{r | <pai,r(f) = 0}. Then / is strictly vanishing along the increasing filter of center a; and diameter u, so this filter is a T-filter, a contradiction to the hypothesis. Consequently, we have ipaitn(f) = 0, and therefore y{Ti) = y{Q). □ Corollary 19.4: Let D be a closed bounded infraconnected, let a £ D and r > 0 such that D n d(a, r~) admit a partition of the form (C(Jri))jg/ where each Ti is an increasing T-filter, and assume that every other T-filter on Dnd(a,r~) surrounds all the Ti- Then, y{Ti) = y{Tj), Vi,j G i\ Definition: A partition (Ai)iei of an infraconnected set E is said to be T-optimal if for each i € I, there exists an increasing T-filter Ti of D which is not surrounded by any increasing T-filter different from Ti, such that C(Ti) = Ai.
Applications
of T-Filters and T-Sequences
115
If D admits a T-optimal partition (C(J-"i))jg/, this partition will be said to be T-specific if for every decreasing T-filter Q on D there exists h & I such that Q is secant with C(.7-/j). Recall Lemma 41.3 in [30]: Lemma 19.5:
If D admits a T-optimal partition, it is unique.
Definition: A circular filter T on D is said to be distinguished if D\Q(Jr) either is empty, or is of the form C{Q), with Q a decreasing T-filter, and if Q{F) either is empty or admits a T-specific partition. A distinguished circular filter is said to be regular if either Q(J-) = 0 or all elements of the T-specific partition of Q{F) have a diameter equal to diam(J-). Else, it will be said to be irregular. In the same way, a T-specific partition (Ai)i&r of an infraconnected set E will be said to be regular if diam(Ai) = diam(E) Vi € I. Else, it will be said to be irregular. According to the proof of Theorem 41.7 in [30] we can state Proposition 19.6: Proposition 19.6: If K is strongly valued, all T-specific partition of infraconnected subsets of K are regular. If K is weakly valued, there exists a bounded open closed infraconnected set E such that E = d(0,1), admitting d(Q, (5)") as a hole, with an irregular T-specific partition whose elements are the sets C(0,r) n d(a,r~), whenever \a\ = r, \ < r < 1. By Proposition 42.4, Corollaries 42.7, 42.8 and Theorems 42.9, 42.10 in [30], we have the following statements: Proposition 19.7: Let J- be a distinguished circular filter such that Q{J-, D) ^ 0. For every increasing T-filter Ti of the T-specific partition of Q{T, D) we have y{^) = y{F). Corollary 19.8: The restriction of the mapping y to the set of distin guished circular filters is a bijection from this set onto MaXoo(H(D)). By Theorem 16.20, we can state Corollary 19.9: Corollary 19.9: Let D be a closed bounded infraconnected set with no T-filter. Then the mapping
116
Ultrametric Banach
Algebra
a kernel of the form y(a), hence ip(x) = \a\ < ||X||,D. This shows that there exists no tp e Multm(H(D), || . \\D) such that ip(x) = \\X\\DAs a consequence of Proposition 19.6, we have Theorem 19.10 which is proven in [30] as Theorem 42.10. Theorem 19.10: Let K be strongly valued and let D be closed and bounded. The mapping 0 from Multm(H(D), \\ . ||) into Max(H(D)) defined as G(ip) = Ker(ip) is a bisection. Theorem 19.11: If K is weakly valued there exists a closed bounded infraconnected set E in K such that H(E) admits multiplicative seminorms ipi,ip2 £ Multm(H(E), || . \\E), with ipi ^ tp2 such that Ker(ipi) — Ker(ip2)The following Theorem 19.12 is a consequence of Theorem 38.10 in [30]: Theorem 19.12: Let D be closed and bounded and have finitely many infraconnected components Di,...,Dq. Then H(D) is isomorphic to the direct product H(D\) x . . . x H(Dq). In [7] (and [30]) we characterized continuous multiplicative semi-norms in a K-algebra H(D) which are absolute values: Theorem 19.13: Let T be a large circular filter on D. Then ip? is not an absolute value if and only if it satisfies one of these conditions: (i) There exists a T-filter Q on D such that T is secant with P(Q), (ii) T is T-filter. Corollary 19.14: All non punctual continuous multiplicative of H(D) are absolute values if and only if D has no T-filter.
seminorms
Definitions and notations: Let D be closed and bounded. Let incT(D) (resp. decT(D)) be the set of the increasing (resp. decreasing) T-filters on D. We will denote by < the relation defined on incT(D) (resp. decT(D)) by T\ < Ti \lC{Ti) C C(T{) and C{T?) ^ C{T\). In other words, denoting by T the circular filter less thin than a monotonous filter T, if T\ and JF2 are decreasing T-filters, then T\ < Ti is equivalent to say that T\ < .7-2, and if T\ and Ti are increasing T-filters, then T\ < T^ is equivalent to say that T
Applications
of T-Filters and T-Sequences
117
We will call an ascending chain of increasing (resp. decreasing) Tfilters a sequence of increasing (resp. decreasing ) T-filters (^rra)„gN such that Tn < Jrn+1 whenever n e N . Let (^"n)n6N be an ascending chain of increasing T-filters. For each n G N let rn = diam(Tn). Since the sequence (rn)n^ is decreasing, we put r = lim n ^oo r n , and then r will be named the diameter of the chain. We put A = n nSN C(.F„) and for each n e N, Dn = C(.F„) \ A. The sequence (A,)n€N is then a base of a filter T on D of diameter r. If r = 0, since D is closed, A is a point a of D, hence T is the filter of the neighbourhoods of a in D. If r > 0, T is a decreasing filter on T> of diameter r. In both cases .F will be called the returning filter of the ascending chain (FnJneNNow let (-F„)ngN be an ascending chain of decreasing T-filters and let a G V{Tn) for some n G N. The sequence (rn)ne^ is an increasing sequence of limit r G]0, +OO], and r will be named the diameter of the chain. Since D bounded, we notice that r < +oo, and then we will call the returning filter of the ascending chain (J-'n)n€N the increasing filter T of center a and diameter r ( it is seen that T is not depending on the point a G V(Tn), whenever n G N). Now, we can recall the characterisation of K-algebras H(D) without non zero divisors of zero which admit no continuous absolute value [7]. Theorem 19.15: Let H(D) be a K-Banach algebra without non zero divisors of zero. Then Mult(H(D), \\ . \\o) contains no norm if and only if D admits an ascending chain of T-filters (,Fn)n£N whose returning filter is either a T-filter or a Cauchy filter. Remark: By Theorem 19.15 we notice that if a X-Banach algebra H(D) has no non zero divisors of zero and has no continuous absolute value, then Max{H{D)) admits a partition of the form {F, G}, where F is a non empty closed subset for Jacobson's topology, and G is of the form U^=1Fn, where each Fn is a non empty closed subset for Jacobson's topology, such that Fn c Fn+i, Fn T^ Fn+i. Indeed, each set V{Fn) is a closed subset of Max(H(D)) which is identified with D.
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Chapter 20
Analytic Elements on Classic Partitions
Classic partitions let us generalize the notion of holes for a subset D of K, relatively to a disk containing D. So, we can generalize algebras of analytic elements, a notion introduced in [23]. Definitions: Given a set closed set S C K, we call a classic partition of S a partition of S of the form (d(bj, rj)) . The disks d(bj, rj) are called the holes of the partition. Let 0 be a classic partition of d(a,r). A closed infraconnected set E included in d(a, r), will be said to be a sub-O-set if every hole of E is a hole of 0 . Moreover, a sub-O-set E will be called a 0-set if E = d(a,r). A weighted sequence (Trnti,qrnii)i
120
Ultrametric Banach Algebra
Theorem 20.1 (N. Mainetti [29,37]): Let O = (d(aj:rj))jeJ be a classic partition of the annulus T(a,r',r"). There exists h £ J and a Ominorated annulus of the form T(ah,rfl,p) included in r(a,r',r"). Proof: Suppose the claim is false. Without loss of generality we can assume a = 0. For every i € J, we put Tj = d(ai,r~). Since our claim is false, for every e > 0, we can find an index i G J such that r, < e. Let us fix e > 0, and let h G J be such that r^ < min(e,r') and Let Sh £ ]rh> rh + e[- Then T(a^, r^, s^) is included in T(a, r', r") and is not included in any hole Tj. Consequently, T{ah,rh,Sh) admits a classic partition of the form Oh = (Tj)j e j h , with J^ C J. Since our claim is false, we can find another index k e Jh such that r\. < | . In this way, by induction we can construct a sequence (in)nN of J such that Tin+1 C T(oi rl , rin, r iti + -^) and r,Ti < -^ Vn 6 N. The sequence (a„) is a Cauchy sequence whose limit a obviously lies in T(0,r',r"). Hence a belongs to a certain hole T;. Therefore, when n is big enough then a iri lies in T/, a contradiction to the definition of O. □ Proposition 20.2: Let S C d(a, r) and let O be a classic of d(a,r) \ S admitting a O-minorated annulus T(b,r'r"). For }r',r"[, there exists an increasing idempotent T-sequence of O of and diameter p, together with a decreasing idempotent T-sequence same center and diameter.
partition all p € center b of O of
Theorem 18.10 applied to classic partitions enables us to state Proposition 20.3. Proposition 20.3: Let K be strongly valued. Let S be a bounded set included in a disk d(a,r), and let O be a classic partition of d(a:r) \ S. We assume that for every r e]8(a,S), diam(S){C\\K\, each class ofC(a,r), except maybe finitely many ones, contains at least one hole of O. Then for each r e}5(a,S),diam(S)] there exists in O an increasing idempo tent T-sequence and a decreasing idempotent T-sequence of center a and diameter r. Corollary 20.4: Let K be strongly valued. Let O = (d(bj,r~))j£j be a classic partition of the annulus T(a,r',r") and let s €]r',r"[. Then O admits an increasing idempotent T-sequence and a decreasing idempotent T-sequence of center a and diameter s. Corollary 20.5: Let K be strongly valued. Let S be a countable set, included in a disk d(a, r) and let O be a classic partition of d(a, r) \ S.
Analytic Elements
on Classic Partitions
121
For every I G]0,r[, O admits an increasing and a decreasing idempotent T-sequence of center a and diameter I. N o t a t i o n : Let V be a compact subset of Mult(K[x)) and let D = {a 6 K | ifa € T>}. The if-algebra i?(D) is provided with the semi-multiplicative norm || . ||p denned as ||/i||z> = sup{ip(h) \ ip € 2?}. Then we check that \\h\\v > \\h\\D Vh e R(D). Let £ b e a closed subset of the disk d(a, r) and let D be a closed subset of E. Let O = (d(bi,r~)i£j) be a classic partition of d(a,r) \ E. We will denote by $(£>, O) the set of circular filters T on K such that all elements B € T contain points of D or holes of O. In particular, if D = 0, we just put $ ( 0 ) = $ ( 0 , 0 ) . Lemma 20.6 is an obvious corollary of Theorem 13.10: L e m m a 20.6: * ( $ ( D , 0 ) ) is the closure of Mult(R(D),\\ {ipT \Te O) inMult{K{x\).
. \\D) n
N o t a t i o n : Let £ be a closed subset of the disk d(a,r) and let D be a closed subset of E. Let O = (d(bi,r~)iej) be a classic partition of d(a, r)\E. Let V = #($(£>,) we put || . | | 0 ) o = II • ||z>We will denote by H(D, O) the completion of K(x) for the norm || . \\D,O- I n particular, if D = 0, we denote by H(0) the completion of K(x) for the norm || . ||e>. If£> = 0, we just put \\h\\0 = sup{
Any i
L e m m a 20.8: Let D be a bounded closed infraconnected subset of K and let (Ti)iej be the family of holes of D. Let O = {Ti)i&j. Then \\ . \\D,O = II • I I D . Proof: Let 1* = d(bi,r~), (i £ J) be a hole of O, hence a hole of D. Since D is infraconnected, by Corollary 10.6 the circular filter Ti of center
122
Ultrametric Banach
Algebra
bi and diameter r^ is secant with D, therefore ^p^{f) < so the claim is obvious.
||/||L>
V/ £ R(D), □
Lemma 20.9: Let O be a classic partition of a disk d(a,r), and let D be a sub-O-set. Then we have ||/I||D < ll^llo for every h £ R(D). Moreover, if D is a O-set, then \\h\\r> = \\h\\o for every h € R(D). Proof: Let h G R(D). By Theorem 16.14 there exists a Z)-bordering filter T such that ||/I||D =
Chapter 21
Holomorphic Properties on Partitions
Throughout this chapter D is a closed bounded subset of the disk d(a,r). Given a classic partition, we have a Mittag-Leffler Theorem which gen eralizes the well known Theorem 16.12 for analytic elements on an infraconnected set [29,37]. Next, properties of T-filters also apply to analytic elements on partitions. T h e o r e m 21.1: Let O be a classic partition of d(a,r) \ D. Let f € H(D,0). There exists a unique sequence of holes (T n ) ne j,j- of O and a unique sequence (fn)nen in H(D, O) such that /o G H(d(a, r)),fn € Ho(K\ Tn) (n > 0), lim^oo /„ = 0 satisfying oo
W
f = Y1 / "
and
= SUp
\\fn\\D,0-
Moreover for every hole Tn = d(an,r~),
we have
n
(«) ("0
\\f\\D,0
neN
||/n||D,0 = ll/n|| KNT „ = Va n ,r„(/n) < Vo„,r„(/) < II/IID.O| | / o l k o = | | / o l l 5 = P « , r ( / o ) < VaAf) < UWD.O-
Let D' = d(a,r)\(U^L1Tn). Then f e H(D') and its decomposition in H(D') is given again by (i) and satisfies | | / | k = ||/lke>Proof: Suppose first that / lies in R(D). Without loss of generality, we may denote by T\,..., Tq the holes of O that contain poles of / . Let E = d(a,r) \ U' = 1 Tj. Then E is a C-set. Since E is infraconnected, we can apply Theorem 16.12 that gives all statements above. Now, consider / € H(D, O) and let (hm)m€ji be a sequence in R(D) converging to / in H(D, O). The set of holes of O containing at least one pole of one term hm of this sequence is obviously countable and may be defined as a sequence 123
124
Ultrametric Banach
Algebra
(Tn)neN*- Thus, each hm has a unique decomposition h = X]^Lo^m>™> with hmio € H(d(0,r)), and /i m ,„ € H0(K\Tn) Vn G N*. So, similarly to the classical proof of the Mittag-LefHer Theorem for infraconnected sets, for any m, q G N, we have \\hm - hq\\Dto = supnm(\\hm>n - /I«,,„||D,O) and then, for each fixed n G N*, (resp. n = 0) the sequence (/im,n)meN converges in Ho(K\Tn) (resp. in H(d(a,r))) to a limit fn. We check that the series XmLo ^n converges to / in H(D, O), and satisfies all statements above. □ Corollary 21.2: Let {Ti)iei be the family of holes of D. Let J be a subset of I and let L = I\J. Let E = Du (UieJTi) and let F = K \ U ieL 7V Then we have H{D) = H{E) © H0(F), and for each g € H0{E), h e H{F), we have \\g + h\\D = max(||2|| B , \\h\\F). Definitions and notation: Let / G H(D,0). We consider the series Y^Lofn obtained in Theorem 21.1, whose sum is equal to / in H(D,0), with / 0 € H(D),fn e H{K \ Tn) \ {0} and with Tn holes of O. Each Tn will be called a f-hole and /„ will be called the Mittag-Leffler term of f associated to Tn, whereas /o will be called the principal term of / . For each /-hole T of O, the Mittag-Leffler term of / associated to T will be denoted by fr whereas the principal term of / will be denoted by To. The series E^=o/™ ( w i t h fn = ~h\t Vn € N*) will be called the Mittag-Leffler series of f on (D, O). Similarly to what was done with sets Mult(H(D), || . ||o), here we have a Garandel-Guennebaud's Theorem for algebras H(D,0). Theorem 21.3: Let E be a closed set such that D C E C d(a,r), and let O be a classic partition of d(a,r) \ E. For every T G $(D,0), the multi plicative semi-norm ip?- defined on R(D) extends by continuity to an ele ment D,O
Holomorphic Properties on Partitions
125
R(D) consisting of the h € R(D) of the form ^2i€l 5 3 ^ , w**n Ia finite sub set of J (resp. Y,i£L ^ b j + (x-'btf)- T h e completion of R'(D,0, (bi)i€J) (resp. R"(D,0,(bi)i£j)) is a Banach i^-subvector space of H(D,G) that we will denote by H'(D, O, (bi)ieJ) (resp. H"(D, O, (bi)i€J)). By Theorem 21.1, Proposition 21.4 is immediate: Proposition 21.4: Let O = (d(bi,r~)iej) be a classic partition of d(a,r) \D. Then H'(D,0,(bi)i&J) (resp. H"(D,0,(bi)i€J)) is equal to the set off e H(D, O) of the form £ i € / ^ , [resp. £ i 6 / ^ - + j ^ ) with I a countable subset of J, and lim^ -~ = 0, whereas H is the filter of complements of finite subsets of I. Proposition 21.5 will be useful in the following chapters. Proposition. 21.5: Let O = (d(bi,r~)i€j) be a classic partition of d(a, r) \ D, and suppose that O admits an increasing (resp. a decreasing) idempotent T-sequence (Tn) of center b and diameter I. Let s €]0, l[ (resp. s e]l, r[) and let e e]0,1[. There exists f € H'(D, O, (bi)i^j) satisfying (i) | 0 ( / - l)|oo < e W> e Mult(H(D, 0),\\ . || D ,o) such that ip(x -b)<s and ip(f) = 0 VV> € Mult(H(D,0), || . ||D,e>) such that ip(x -b)>l, (ii) 0 ( / ) + 0 VV e Mult(H(D, O), || . ||D,O) \ Multa(H(D, O), \\ . \\D,o) such that s < 4>(f) < I. (resp. (i) VK/ - 1) < e V^ e Mult(H(D,0), \\ . || D ,o) such that 4>(x - b) > s and tp(f) = 0 Vip e Mult(H(D, O), || . \\D<0) such that ip(x - b) < I (ii) iP(f) / O V ^ e Mult(H(D, O), || . \\'D,o) \ Multa(H(D, O), \\ . \\D,o) such that I < tp(f) < s.) Proof: We set E = K \ (U%LQTn). Since the sequence (T n ,l) n £ N is an idempotent T-sequence of E, of center b and diameter I, by Theorem 18.11, there exists / € H(E), strictly vanishing along the increasing (resp. decreasing) T-filter of center b and diameter I, meromorphic on each hole Tn and admitting each bn as a pole of order at most one, and having no other pole, satisfying further | / ( x ) | = 1 for alia; € DDd(b,s), f(x) ^ O f o r a l l x e D\d(b,l~) (resp. | / ( x ) - l | < e for all x € D\d(b,s~), f(x) = 0 for all x € D n d(b, I)), and for every circular filter Q of diameter u
126
Ultrametric Banach
Algebra
Corollary 21.6: Let O be a classic partition ofd(a, r) \ D, and suppose that O admits a O-minorated annulus T(b,r',r"). Let I, p',p" £]r',r"[ satisfy p' < I < p". There exist f,gG H{D, O) satisfying (i) ,/,(/) = 1 W e Mult(H(D,0), || . || D i 0 ) such that tp(x - b) < p' and tP(f) = 0 VV> € Mult(H(D, O), \\ . H^o) such that i>(x - & ) > / , (u) ^(g) = l Vz/» e Mult(H(D,0), || . || D i 0 ) such that ip{x -b)> p" and ip(g) = 0 V ^ £ Mult(H(D, O), \\ . \\D,o) such that xp(x - b) < I. Lemma 21.7: Let K he strongly valued, let O be a classic partition of d(a,r) and let f,Qe $ ( 0 ) , T ^ Q. There exists f G H{0) such that
Proof: Suppose first that T and Q are not comparable with respect to <. Then, by Proposition 11.9 we can find disks d(b, I) G J- and d(c,m) € Q such that d(b, I) n d(c,m) = 0. Let t = \b ~ c\. Then by Proposition 11.9 again, we have t > max(/,m), hence, denoting by O' the set of holes of O included in T(b, l,t), this annulus admits O' as a classic partition. Suppose now that T and Q are comparable with respect to <. So, we can suppose that Q surrounds T. Let p = diam(J-), t = diam(g) and let I €}p,t[. By Lemma 10.11, there exists a unique disk d(b,l) which belongs to T', and then denoting by O' the set of holes of O included in T(b,l,t), this annulus admits O' as a classic partition. Now, in both situations, we take s &]l,t[. By Corollary 20.4. O admits an increasing idempotent T-sequence and a decreasing idempotent T-sequence, both of center b and diameter s. Therefore, the conclusion comes from Proposition 21.5. □ Theorem 21.8: Let K be strongly valued and let O be a classic partition ofd(a,r). ThenMultm(H(0),\\ . ||o) = Mult(H(0),\\ . ||0). Proof: L e t ^ e Mult(H(0,\\ . \\a) and let M be a maximal ideal of H{0) containing Ker(tp). Then ip is of the form tpj:, with T € <&(£)). On the other hand, by Theorem 6.15 there exists
Chapter 22
Shilov Boundary for Algebras H(D, O)
In Chapter 6 we proved that, given a commutative ultrametric Banach /C-algebra with unity A, a Shilov boundary does exist for (A, ]| . || s i ). How ever, this existence is abstract and doesn't let us determine its nature. When we consider a Banach K-algebra H(D), we are able to describe very precisely its Shilov boundary by using the characterization of multiplicative semi-norms by circular filters. Definitions and notation: In all this chapter D is a closed and bounded subset of the disk d(a,r) and O = (d(bitr~).iej) is a classic par tition of d(a, r) \ D. A circular filter T will be said to be strictly O-bordering if it is the peripheral of a hole of O or of d(a,r). A circular filter O will be said to be O-bordering if either it is strictly O-bordering or all elements B G T contain holes of O. We will denote by E(O) the set of
128
Ultrametric Banach
Algebra
of T it is secant with a disk d(c, s) c d(b, p~) with s < p, and of course it has elements B of diameter I &]s, p[ that contain no hole of O, a contradiction. So, T is not secant with any hole of O. Now, let hi,..., hn € R{D, O), let e > 0. Since T admits a base of affinoids sets, we can find an affinoid set E £ T such that | \hj(x)\ - iprihj)^
<e\/xeE,Vj
=
l,...,n.
Then there exists a hole T of O included in E. Consequently, \fT(hj) -frih^loo
< e Vj = l , . . . , n .
This shows that E0(O) is dense in £(£>).
D
T h e o r e m 22.2: Let O be a classic partition ofd(a,r) \ D. Then S(C) is equal to the boundary of Mult(H(D, || . \\D,O) inside Mult(K[x\). Proof: The proof roughly will follow the same way as this of Theorem 14.1. Let O = (d(bi,r^)iej). Since S 0 (O) is dense inside T,(0), and since the boundary of Mult{H(D, || . ||o,) inside Mult(K[x}) is obviously closed, it is sufficient to show that So() is included in this boundary. Let fj: e So()If T is the peripheral of d(a, r), it is obvious that ipyr belongs to the closure of the subset of Mult(K[x\) \ Mult{H(D, O), \\ . \\D,O) consisting of all (pg such that Q is secant with K \ d(a, s) for some s > r. Consequently, we are led to assume that T is the peripheral of certain hole d(bj,rj) of O, and then symmetrically, it is also obvious that
O), ||. \\Dt0)) ^ 0.
Moreover, W n (Mult(K[x\) \ Mult(H(D,0),\\. \\D,o)) is open in Mult{K[x\) because so are W and Mult{K{x\) \ Mult(H(D,0), ||. \\D,O)Since Multa{K[x\) is dense inside Mult{K[x\), there exists <pa € (Multa(K[x\)
\ Mult(H(D,
O), ||. \\D,a)) n Mult(H(E),
|| . \\E).
Consequently, a lies in E \ D. But since E c d(a,r), a must belong to Y, which proves that T is secant with Y. Therefore the boundary of
Shilov Boundary for Algebras H(D, O)
Mult(H(D, O), ||. || D i o) inside Mult(K[x\) these two sets are equal.
129
is included in £(£>), and finally □
Theorem 22.3: Let O = (d(bi, r~)i€j) be a classic partition ofd(a, Then £(£>) is the Shilov boundary for (H(D,0), \\ . | | D j 0 ) .
r)\D.
Proof: First, we notice that S(C) is closed inside Mult(K[x}) because it is the boundary of Mult(H(D,0), \\ . ||o,e>)- By Theorem 21.1 it is easily seen that E 0 (O) is a Shilov set for H(D, O). Now, we will show that S(C) is the smallest closed Shilov set. Suppose it is not the smallest. Then, there exists another closed Shilov set S which does not contain E(O). Since S is closed, and since So(^) is dense in E(C) there exists i> e S 0 (C) \ S. Let Q be the strictly 0-bordering filter such that ip = (fig. Thus ip is of the form
h(x)
II?=i(*(x-b)
9+1
Then
(K. Boussaf):
T,(D) is the Shilov boundary for
Proof: By Theorem 14.1 E(D) is the boundary of Mult(H[D), \\ . \\D) inside Mult(K[x\), hence it is closed inside Mult(K[x]). Suppose first that D is infraconnected. By considering the natural par tition O of D \ D, we have S(C) = £(£>), so we can apply Theorem 22.3 showing that £(£>) is the Shilov boundary of H{D). Now, we shall consider the general case.
130
Ultrametric Banach Algebra
First, we check that £(£>) is a Shilov set. Let h G R(D). If there exists an infraconnected component E of D such that ||/i||r> = | | ^ | | B , then as we have seen, there exists ipjr G E(I?) such that
Shilov Boundary for Algebras H(D, O)
131
Suppose first that H is the peripheral of an infraconnected component B of D. There exists an annulus T(b, z', z") included in K \ D such that s < z' < z" < tfsi~s". Let c G T(6, z', z"), let I = \c\ and
nut*-*)
h(x) =
C)« +
(X -
1
Then we have ip-n(h) = ^ x - Now, let ipjr e S be such that T is secant with d(b, s). Since T is not secant with E, we check that s'q And now, let
sq
Thus if H is the peripheral of an infraconnected component B of D, we are led to a contradiction, and this shows that H is just the peripheral of a hole T = d(b,s~) of an infraconnected component B of D. Since B is an infraconnected component of D, there exists an annulus T(c, z', z") included in K \ D such that s' < z' < z" < s with c € d(b,s~). So, T = d(c,s~). Let a e T(c,z',z"), let I = \a - c\ and let
h{x)=
(x-a).+i
'
Then we have tpnih) = j . Now, let y2;r e 5. If J-" is secant with since J 7 is not secant with E, we check that
C(c,s),
s' <pr(h) <-» <
And if T is secant with K \ d(c, s), it must be secant with K \ and then we check that fAh)<
^77 < ~
d(c,s"~),
=fn{h)-
Finally, suppose that T is secant with d(c,s~). Then one of the Cj lies in d(c, s) and we can assume c = C\. Therefore T must be secant with
132
Ultra-metric Banach
d(ci,s'
Algebra
), and we check that
with respect
Proof: Theorem 22.6 is just a corollary of Corollary 22.5 when D is infraconnected. Now, suppose that D has finitely many infraconnected compo nents £>i, ...,£>,. Since by Theorem 19.12 H(D) = H(DX) x . . . x H(Dq), and the theorem holds for each algebra H(Dj), so we can easily see that the Shilov boundary of H(D) is equal to the union of Shilov boundaries of all algebras H(Dj), 1 < j < q, hence the conclusion follows. □
Shilov Boundary for Algebras H(D, O)
133
More precisely, we can state Corollary 22.6: Corollary 22.6: Let D be closed bounded having finitely many Dbordering filters. The following four sets are equal:
(0 £(#), (ii) The boundary of Mult(H(D), || . \\JJ) inside Mult(K[x\) with respect to the topology of simple convergence, (Hi) The boundary of Mult(H(D),\\ . \\D) inside Mult(K[x\) with respect to the metric topology, (iv) The Shilov boundary for (H(D), || . \\D)Proof: If D has finitely many D-bordering filters, then it has finitely many infraconnected components. □ R e m a r k : In particular, Theorem 22.6 applies to H(D) when D is an affinoid set.
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Chapter 23
Holomorphic Functional Calculus
Holomorphic functional calculus is well known and helpful in complex Banach algebra. We can also define it, from the spectrum of an element x, by considering first the rational functions without poles in sp(x). Now, con sidering an ultrametric Banach K-algebra A, thanks to specific ultrametric properties given in Lemma 23.1 bellow, this calculus has continuation from a Banach algebra of the form H(sp(x),0) to A, where O is a natural partition defined by the norm of A. This calculus will let us solve sev eral problems in spectral theory and about idempotents. The holomorphic functional calculus was first defined in [23]. Notation: In this chapter, A is a commutative Banach if-algebra with unity whose norm || . || is ultrametric. Recall that given <j> € Mult(K[x}), such that
the unique circular filter Q on K
Lemma 23.1: Let x £ A be invertible and let b € K be such that \b\ < M—=T]f- Then x — b is invertible and satisfies \\(x — 6) _ 1 || = | | x - 1 | | . Proof:
By hypothesis, we have |b| < ,, \«
, hence
lim IKfaT1)! =0. n—»oo
Consequently, the series X^Lo(z)™ *s converging in A. Then we check that
0 - b ) Er=o( 6 " a: " n_1 ) = L N o w s i n c e \\hx~l\\ < i . w e Hx-1!! > \\bnx-n-l\\ Vn e N*.
have
But since || . || is ultrametric, we have || X^o^™ 1 "™ - 1 !! ~ ll 2 - -1 !!' therefore ||(x - 6) _ 1 || = ||x _ 1 ||. 135
anc
^ D
136
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Algebra
Corollary 23.2: Let x G A and for each b € K \ sp{x), let p(b) = iks-M-'ll" Then p(a) = p(b) whenever a, b G K \ sp(x) such that \a — b\ < )i/j;_M-i[|- Letr > ||a:||Sj. Thend(0,r)\sp(x) admits a classic par _ with tition of the form (d(6,/?(6) ))6ed(o,r)W(x); d{b,p{b)~) = d(a,p(a)~) whenever a, b G K \ sp{x) such that \a — b\ < IKS-M-III ■ Definitions and notation: Let x £ A, let a £ K and let r G [||x — a\\si, \\x\\], with r > \\x — a\\Si if A is not uniform. We will call x-normal partition of center a and diameter r a classic partition O = (d(bi:r~)i£j) of d(a, r) \ sp(x) satisfying n ^ ^ - i n = n Vi G J. In the same way, we will call x-spectral partition of center a and diam eter r a classic partition O = (d(bi,r~)iej) of d(a,r) \ sp(x) satisfying 1 = r» Vi e J. A x-normal (resp. x-spectral) partition will be said to be centered if it has center a e sp(x). A x-spectral partition of center a will be said to be wide if it has a diameter r > \\x — a|| S j. We will call strict x-spectral partition the unique centered x-spectral partition of diameter ||x|| s j. Then, if A is uniform, the strict x-spectral partition is a x-normal partition too. Let r = ||x — a\\ and let O = (d(bi,r~)i£j) be a centered x-normal partition of d(a,r)\sp(x). Then x G A will be said to have bounded normal ratio if the family { iil^lfe.wil | « € J } is bounded. The if-algebra A will be said to have bounded normal ratio if every element of A has bounded normal ratio. T h e o r e m 23.3: Let t e A. There exists a unique homomorphism Qt from R(sp(t)) into A such that &t(P) = P{t) for all P € K[X]. Moreover, Qt is injective if and only if t has a null minimal polynomial. For every h € R(sp(t)), we have sp(h(t)) = h(sp(t)) and sa{h(t) = h{sa(t)). Proof: Let D = sp(t). We may obviously define 0 { from K[x] to A as 6 t ( P ) = P(t). Now let Q € K[X\ have its zeros in K \ D. Then Q(t) is invertible in A, so we may extend to R(D) the definition of Qt, as 9 t ( g ) = P(t) Q(t)-\ for all rational function g e R(D) (with (P, Q) = 1). The uniqueness to Qt is then obvious. Next, Ker(Qt) is an ideal of R{D) which is obviously generated by a polynomial G. Then G = 0 if and only if t has a null minimal polynomial. Now, let h = £ € R{sp(t)) (with (P,Q) = 1). Let A e sp{t)), and let x € X(A) be such that x(M*)) = A- T h e n x(M*)) = hW, h e n c e h(sp(t)) C sp(h(t)). Conversely, let ^ G sp(h(t)), let x € «V(i4) be such
Holomorphic Functional
Calculus
137
that x(/i(i)) = M, and let a = \(t). Then, we have x(P(t)) - Wt(Q(*)) = 0, hence a is a zero of the polynomial P{X) — fj,Q(X). Therefore a lies in K because K is algebraically closed. But then, as t — a belongs to the kernel of x, at lies in sp(t). Hence we have sp(h(t)) = h(sp(t)). Finally, sa(h(t)) = ix(h(t)) | X € X(A, K)} = {h(X(t)) | X € X(A, K)} = h(sa(t)). D Remark: When the homomorphism Qt defined in Theorem 23.3 is injective, the K-subalgebra B = Qt{R(sp(t))) is isomorphic to R(sp(t)), and in fact is the full subalgebra generated by t in A. So, in such a case, we will consider R(sp(t)) as a Af-subalgebra of A. Definition and notation: Let t G A. In this chapter and in the sequel we will denote by © t the canonical homomorphism from R(sp(t)) into A denned in Theorem 23.3, and we will call it the canonical homomorphism associated to t. Meanwhile, when it is injective, we will currently confound a rational function h G R(sp(t)) with its image 0t(/i). Given ip G Mult(A, \\ . ||) we will put ipt = ip ° ©t- Thus, we have
Vt(P)=VW))VPetf[z]. We will put (7/i(t) = {
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Ultrametric Banach
Algebra
For each j = 1 , . . . , n, since by Lemma 23.4 T is secant with K \ d(bj,r~), we have (fjr(hj) < <£b-,r{hj). Similarly, as T is secant with d(a,ro), we have
\{t-a)k)
<
0
1
*(^-)
\pJ
\ t — a D,o/
(t-a)
D,0
Now, let h e R(D), let T1:..., Tq be the /i-holes of O and let YH=o 9n be the Mittag-Lefner series of h on (D, O), with gn = HT„ Vj = 1 , . . . , q. Then we have ||©t(^,j)|j < ||/I||D,O for each j = 1 , . . . q. Finally, we check that there exists a constant L > 1 such that p n | | < n Lr . Indeed, if A is uniform, we just take L = 1. And if A is not uniform, then we have r > \\t\\Si, hence linin-xx, -LJi = 0. Thus we have shown the existence of L in all cases. Consequently, in H(D,0), we check that ||©t(5j')ll < -k||<7jlli>,c? for each j = 0,...,q. But by Theorem 21.1, we have ||/I||D,O = maxo<j
□
Corollary 23.7: Let t € A, let O be a t-normal partition, and let ip G Mult{A,\\ . ||). Thenipt belongs to Mult(R(sp(t),0),\\ . \\D,o)Definitions a n d notation: Let x € A, let a € sp(x), let 5 = ||x — a||5;. An annulus T(a,r',r") C d(b,s) will be said to be x-clear if there exist
Holomorphic
Functional
Calculus
139
tp', V" e Mult(A, || . ||) such that ip'{x - a) = r', tp"{x - a) = r", and V>(x - a) £}r',r"[ Vip £ Mult(A, || . ||). Remark: If r ( a , r ' , r " ) is a x-clear annulus, in particular we have T(a, r', r") n SJJ(X) = 0. But the converse is not true. Theorem 23.8: Let t £ A. If there exists a t-clear annulus, then both Mult(A, || . ||) and a(t) are not connected. Moreover, there exists a t-clear annulus if and only if a(t) is not connected. Proof: Suppose first that there exists a t-clear annulus T(a, r, s). Let F = {4> £ Mult(A, || . ||) | 4>{x-a) < r} and let G = { £ Mult(A, || . ||) |
||/|U<max(^,— J < - .
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Algebra
Now, by definition of O, it is clear that d(b,r~) belongs to O, therefore : II/IID.O > V6,r(/) = ^ this shows that the two norms are not equal when there exists a rr-clear annulus. Conversely, we now suppose that there exists no x-clear annulus. Let h G R(D). By Theorem 23.5 we have HM*)IU < \\h\\D,o-
(23.1)
Let T = d(b,r~) be a hole of O containing a pole of h and let T be the circular filter of center b and diameter r. We will show that there exists ip G Mult(A, || . ||) such that ipx =
Holomorphic
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Calculus
141
holes of O. The restriction of Qt to (R'(sp(t),(bi)iej), \\ . \\sp(t),o) *5 a continuous K-linear mapping and has continuation to H'(s(t), (bi)i^j), O). Proof: Let D = sp{t). Let / € (R'(sp(t), O, (bi)i€J). Let / 0 + £ ~ = 1 JTn be the Mittag-Leffler series of / on O. For each n e N*, let bn G TnThen fr is of the form \ . Since t has bounded normal ratio, there exists a constant M such that | | / T J | D , O < M\\Qt(fTn)\\ Vn G N*. Finally, let /o = E L o 8 " 1 " 1 - B y hypothesis, we have lim™-^ \am\rm = 0. If ||t|| 8i = ||*||, then r = ||i||, hence ||0 t (7o)|| < \\7O\\D,ONow, if ||t||Sj < ||£||, then r > ||i|| S i, hence there exists a constant TV such that ||t m || < Nrm Vm G N, and therefore, | | e t ( / 0 ) | | < N\\J0\\DtoThus, putting Q = max(M,7V), we have | | 9 t ( / ) | | < Q\\f\\D,o V / ' G (i2'(sp(x),(6i) i € j). D By Lemma 10.11, we have Lemma 23.12. Lemma 23.12: Let t £ A have a null minimal polynomial. Let a G K, let 4> £ Mult(A,\\ . ||), let <j>t be the restriction of <j> to R{sp(t)), and let r = 4>(t — a). Then ^~1(4>t) is (a,r)-approaching. Proposition 23.13: Let t G A be such that the mapping Qt is injective. Letae K\sp(t), and let r = l l f t - a ) - 1 ! ! " 1 . There exists 9 G Mult(A,\\ . ||) whose restriction to R(sp(t)) has a circular filter (a, r)-approaching. Proof: We consider R(sp(t)) as a if-subalgebra of A. Let <j> G Mult(A, || . ||). If ^ _ 1 ( ^ t ) is secant with a disk d(a,p) for some p G]0,r[, then clearly we have
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Chapter 24
Uniform K-Banach Algebras and Properties (s) and (q)
Notation: Throughout this chapter, A is a commutative ultrametric Banach if-algebra with unity, whose norm is || . j | . Given t € A we will denote by O t the canonical morphism associated to t. The question whether semi-norms || . ||Sj and || . || s m are equal is an old one. Actually, when K is weakly valued, there exist commutative ultramet ric Banach K-algebras with unity where the semi-norm || . || si is strictly superior to the semi-norm || . \\sm for certain elements. But if we assume that |[ . || si is a norm and that A is complete for this norm, then we can prove the equality [29, 37]. In Corollary 24.13 we find again a Theorem due to B. Guennebaud [34], stating that the completion of a field, with respect to a semi-multiplicative norm, admitting at least two continuous absolute values, has non zero divisors of zero. In particular, this shows why Corollary 4.3.4 in [2] is not true. L e m m a 24.1: Let t e A have bounded normal ratio and let O = (d(ai,r^)iej) be a wide t-spectral partition. The restrictions of Qt to R'(sp(t),0,(a,i)i£j) and R"(sp(t),0,(a,i)i£j) are continuous. Proof: The restriction of Qt to R'(sp(t), O, (a;) i e j) is obviously contin uous. Next, we check that the family of ~C TTTar^-fc> -)—211 ■ I * € J } is bounded because on one hand ||(x — bj)~2\\si = (||(x — bj)~l\\Si)2, and on the other handlKz-^ni^lK*-^)-1!!)2. □ Lemma 24.2: Let t e A. Assume that Ker{Qt) ^ {0}. Then \\t\\si = Pllsm- Moreover, if A^K, then A has non zero divisors of zero. 143
144
Ultra-metric Banach
Algebra
Proof: Let D = sp(t) and let B = Qt(R(D)). Then Ker(Bt) is an ideal of R(D) generated by a monic polynomial G(x) = r i i = i ( x ~~ a *)- Since G(t) = 0, for every ip £ Mult(A, ||. ||) we have ip(G(t)) = 0, hence there exists an integer l{ip) £ {l,...,q} such that ip{t — a\^) = 0, hence tp(t) = |fli(V>)|- Then t — a;(^,) lies in Ker(ip) and therefore belongs to a maximal ideal M. of A. But by Theorem 6.15, there exists 9$ £ Multm(A, || . ||) such that Ker{9^) = M.. Hence we have 0^(t) = |aj(^,)| = ip(t). Thus we have shown that ip(t) < ||t|| sm . But this is true for all ip £ Mult(A, \\ . ||). Consequently, by Theorem 6.19 we have \\t\\Si = ||i|| s m , which proves the first statement. Next, we notice that Ker(Ot) admits a generator G(x) £ K[x) whose zeros lie in D. If deg(G) = 1, then t lies in K (considered as a if-subalgebra of A), and obviously we have t/j(t) = \t\ Vip £ Mult(A, \\ . ||), and therefore, sup{|A| | A e D} = \\t\\3i. And if deg(G) > 1, then Ker(Qt) is not prime, hence A contains non zero divisors of zero, so the second statement is trivial. □ Theorem 24.3 ([29, 37]): \\t\\ai = \\t\\sm.
Let t £ A have bounded normal ratio. Then
Proof: Let D = sp(t). We put r = ||i|| S j, r' = \\t\\sm and we suppose r' < r. If ||t|| = ||t||Sj we put u = r, and if ||£|| > \\t\\si we take u e]r, ||£||[. Let B = Qt(R(D)). By Lemma 24.1 we can assume that Ker(Qt) = {0}. Hence B is isomorphic to R(D). Let O be a wide i-spectral partition of diameter u, and for convenience, we put || . ||t = || . \\D,O- Let (d(aj,rj)), (i £ J) be the family of holes of O and let B = Qt(R'(D, O, (a,)ieJ)). By Theorem 23.6 the restriction Q't of Q t to R'(D, (aj)j 6 j)) is contin uous once R'(D,0,(ai)i£j)) is provided with the norm || . || t . Therefore, Q't has continuation to a continuous K-vector space homomorphism from H'(D, O, (a,i)i£j) into the closure B of B in A. We will still denote it by Q't. Let s' £}r', r[. The annulus T(0, s', r), admits a partition by a subfamily S of holes of O. Hence by Theorem 20.1, T(0, s', r), contains a O-minorated annulus T(b,p,v). Of course, we may choose v as close to p as we want. Particularly, if \b\ > p, we take v £]p, \b\\. Next, we take A £}p, v[. Clearly b does not lie in D. Let r^ = \UX_^)-\H . ■ By Lemma 23.12 there exists 6 £ Mult (A, || . ||) such that ,i~1(6) is (6, r^-approaching. In fact, by definition, we have r^ < p, hence C(b, r{,) is included in d(b, p), hence ^~1(6) is secant with d(b,p). On the other hand, there certainly exists (f> £ Mult (A, || . ||) such that ^ , _ 1 (0) is (6, r)-approaching. Then by Propositions 21.5 there
Uniform K-Banach
exist / , g G H'(D, O, (a,i)iej)
Algebras and Properties (s) and (q)
145
satisfying:
(i) V(/) = 1 for all V € Mult(H(D, (ii) V(/) = 0 for all i/> G Mult(H{D, (iii) V(ff) = 1 for all ip G Mult{H(D,
O), || . || D j 0 ) such that V(< -b)>v, 0),\\ . | | D ' 0 ) such that V(* - &) < A, O), \\ . \\D,o) such that ip(t -b) < p,
and (iv) i)(g) = 0 for all V G Mult(H(D,
0),\\ . || D ,o) such that V(* - &) > A.
We put F = ©{(/), and G = ©((gr). Though 0 t is not supposed to be continuous on all H(D,0,{a,i)i£j), it is continuous on H'(D,0,(ai)i&j). Given 0 e Mult(A, || . ||), we will denote by t the K-vector space seminorm denned on R{D) + H'(D, O), \\ . \\D,oSuppose first |6| < p. It is seen that for every ip € Multm(A, \\ . ||), we have ip(t) < r', hence ipt(x) < r''■ Consequently, ipt(g) = 1, and therefore ip(G) = 1, hence G is invertible in A. On the other hand, ^~l({<j))t) is (0, r ) approaching, hence (4>)t(g) — 0, and therefore ()(G) = 0, a contradiction to the conclusion G is invertible in A. Finally, suppose |6| > p. in the same way we have d(0, r') C K \ d(b, A), and therefore, ip(F) = 1 for all ip G Multm(A, \\ . ||), and ip(F) = 0 for all tp such that ^~l{%pt) is secant with d(b,X). So, F is invertible. But ^~1(6t) is secant with d(b,X) and therefore, 6(F) = 0, a contradiction to the property "F invertible". □ Corollary 24.4: Property (s).
If A has bounded normal ratio, then A
Corollary 24.5:
If A is uniform, then A satisfies Property (s).
satisfies
Corollary 24.6: If A has bounded normal ratio and has no maximal ideals of infinite codimension, then A satisfies Property (p). Corollary 24.7: / / A is uniform and has no maximal ideals of infinite codimension, then A satisfies Property (p). Corollary 24.8:
If A is uniform then its Jacobson radical is null.
P r o p o s i t i o n 24.9: Let t G A have bounded normal ratio and assume that there exist ip', ip" G Mult(A, || . ||) and r,r', r" such that ip'(t)
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a subfamily <S of holes of O. Hence by Theorem 20.1, 17(0, r , r ' ) contains a 0-minorated annulus V(b, p, v). Moreover, we notice that there exists at least one element <j> of Mult(A, \\ . ||) such that ^ _ 1 ( ^ t ) is secant with d(b, p). Indeed, if t — b is not invertible, this is obvious. Now, suppose that t — b is invertible. Since F(b, p, v) admits a partition by holes of the i-spectral partition, according to Corollary 23.2 we have ||(* - &) _ 1 || - 1 < P a n d °y a n d Lemma 23.12 there exists <j> € Mult(A, || . ||) such that ^ _ 1 ( 0 t ) is secant with d(b,p). Then by Corollary 21.6 there exist / , g e H'(D,0,(ai)ieJ) satisfying fg = 0 and ip't(f) = t(g) = l. By Theorem 23.6 we know that Qt is continuous on R"(D, O, {ai)iej). And by Proposition 21.4 fg belongs to H"(0,(ai)i^j). Consequently, e t (/)e t (i(t) and r2 = ip2(t), we have T(0,ri,r2) n sp(t) = 0. Hence by Proposition 24.8 A has non zero divisors of zero, a contradiction. □ Corollary 24.11: Let A be uniform and have no non zero divisors of zero, and be such that sp(t) =fc 0 Vi e A. Then A satisfies Property (q). Corollary 24.12: Let B be a commutative uniform ultrametric Banach K-algebra without non zero divisors of zero, with unity, such that sps{t) ^ 0 Vt € B and let <j> be a K-algebra homomorphism from A to B. Let || . ||^ (resp. || . Wft) be the spectral norm of A (resp. B). Then satisfies
Mxnf^Mxn^vxeA. Thanks to Proposition 24.9, Guennebaud's Theorem [34] on the comple tion of a field which is a normed /iT-algebra having more than one continuous absolute value, appears as a simple corollary: Theorem 24.13: Let F be a field extension of K provided with a semimultiplicative norm and admitting two different continuous absolute values. Then the completion of F has non zero divisors of zero. Proof: Let A be the completion of F, with respect to its norm. It is a uniform Banach K-algebra such that Mult'(A, || . ||) contains at least
Uniform K-Banach Algebras and Properties (s) and (q)
147
the expansions tpi, ip2 of two different continuous absolute values defined on F. Let t £ F be such that ipi(t) < ip2{t)- As an element of F, the spectrum of t is empty, hence it is also empty as an element of A. Putting n = ipi(t), r e Mult(A, || . ||) such that * _1 (^>t) is secant with d(a, s'). Let r &]s',s"[. Suppose first T(a,s',s") admits 0 as a center. Since r(0, s',s") is a O- minorated annulus, we can find an increasing distances holes sequence (T^)„eN of O of center 0 and diameter r such that the weighted sequence (Tn, l) n gN makes a T-sequence. The T-filter of center 0 and diameter r will be denoted by T. Let D = d(Q,r") \ [jneNTn. Since D is clearly a C-set, H(D) is a .ftf-subalgebra of H(sp(t),0,(ai)i€j). Now, we denote by D' the set
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Ultra-metric Banach Algebra
D U (K \ d(0,r). Hence Hb{D') is a Banach if-subalgebra of H{D). Let R'(D') = Rb(D') n R'{{sp(t),0,(ai)ieJ) and ff'(D') = Hb(D') n H'((sp(t), O, (aj)jgj). Particularly, by Theorem 23.11 the canonical homomorphism Qt is denned and continuous on H'(D'). Then, by Theorem 18.12, there exists / e H(D), meromorphic in each hole Tn, admit ting in Tn an as unique pole, an having order at most 1, such that |/(c)| = 1 Vc € d(0,s'), /(c) = O V c e f l \ d(O.r-). Thus, we see that / belongs to H'(D'). Let F = 0 t ( / ) . Since r" > r, there exists V € Mult (A, || . ||) such that ip(t) > r, hence tp(Qt{f)) = 0, hence F belongs to a maximal ideal containing Ker(ip), and therefore 0 belongs to sp(F). On the other hand for every c 6 sp(x), we have |/(c) | = 1, hence sp(F)nC(0,1) ^ 0. Actually, we will show that sp(F) C C(0, l)u{0}, i.e. F — I is invertible for alH G K such that 0 < |Z| < 1. Let I £ if be such that 0 < |/| < 1 and suppose that F — I is not invertible in A. Let C, € X(A) be such that {(F-l) = 0. Let M = Ker(C,), and let Q, = -fa. Let ^ G Mult(A, || . ||) be such that Ker() = M. Then defines an element <j>t of Mult (R(D)) which of course is of the form tpg, with Q a circular filter on K. Moreover, in Rb(D'), the norm || . \\D' obviously satisfies ||/I||D' > l|ft||sp(t),0) hence the restriction of
Chapter 25
Properties (o) and (q) in Uniform Banach K-Algebras
We shall show that in a uniform Banach Af-algebra, Property (o) implies Property (q). Conversely, we shall see that Property (q) doesn't imply Property (o). Notation: Throughout this chapter, A is a commutative ultrametric Banach _ft"-algebra with unity. Several results come from [23]. Lemma 25.1: Let x £ A be such that T(X) < \\x\\. There exists <j> G Mult{A, || . ||), a number v > 0 an affinoid set F which belongs to the filter T = * _ 1 ( 0 X ) , such that diaru^'1^)) > v, for all ip e Mult(A, || . ||) such that *""1(V'i) e $ ( F ) . Proof: We assume Lemma is not true. Let r = T(X) and let 5 = ||x||, let m e]r, s[, let O be the strict x-spectral partition, and let (fro e Mult(A, || . ||) be such that ik~1({(j>o)x) is secant with T(0,m,s). Let To = <S>~1(((fro)x) and let So = diam(To). Suppose we have already constructed a finite sequence (
false, there of the form such that ^ g , and
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Algebra
5(J r n + i,^ r „) < ^ p j . This means that for every ip € Mult(A, || . ||) such
that 5{^'l{tpx),Fn)
< ^pj, we have diam^'1^))
> ^ .
Let F be
the unique disk of diameter ^pj such that Tn € $ ( F ) . Then for all Q £ $ ( F ) we have 5{T,G) < ^ p \ , hence diam(tf -1 (V' :B )) > ^ for all V> £ Mult(A, || . ||) such that S P - 1 ^ ) € $ ( F ) , which proves the conclusion of the Lemma. Thus, since we have supposed that the conclusion is false, we can find
s < — — , and 5{Tn+1,Tn) n+2
Is < ——. n+1
And this is true for all n £ N, hence the sequence ((^ n )i)n€N i s a Cauchy sequence with respect to the ^-topology. Consequently, its limit in Mult{K[x\) is of the form <pa, with a £ F(0, r,s). On the other hand, since Mult(A, || . ||) is compact for the topology of simple convergence, the sequence ((pn)nen admits a point of adherence 8 with respect to the topology of simple convergence. Consequently, 0X is a point of adherence of the sequence ((ipn)x)neN- Next, since Mult(K[x}) is sequentially com pact, we can extract from the sequence (ipn)x a subsequence converging to 8X with respect to the topology of simple convergence. Thus, without loss of generality, we can assume that the sequence (ipn)x converges to 9X with respect to the topology of simple convergence in Mult(K[x}). But since the sequence (ipn)x converges to tpa for the (5-topology, so much the more does it converge to <pa for the topology of simple convergence, hence ipa = 9X. Thus, 9a is punctual, a contradiction to the fact that a £ sp(x). Consequently, our hypothesis the conclusion is false is wrong. □ T h e o r e m 25.2: infective.
If A does not satisfy Property (q), then GMyi is not
Proof: Suppose that A does not satisfy Property (q) and let x £ A be such that T(X) < ||x||. Consider now the strict x-spectral partition O of d(Q, s)(x). Let r = T{X) and let s = ||x||. By Lemma 25.1 there exists
Properties (o) and (q) in Uniform Banach K-Algebras
there exists g G H(sp(x),0)
151
such that >fg(g) = 1 V£ € <&(d(a, 1)) and
Property
(o) and
Proof: Suppose A does not satisfy Property (p) hence it does not satisfy Property (q). Let t € A be such that r(t) < \\t\\. Since Maxa(A) =£ 0 we have sp(t) ^ 0. Taking a € sp(t), we notice that r(t — a) = r(t), and \\t — a\\ = \\t\\. So, without loss of generality, we can assume 0 6 sp(t). Let s = ||i||, let r = r(£), and let O be the strict t-spectral partition. There exists a 0-minorated annulus T(b,r',r") included in r ( 0 , r , s ) . Suppose 0 is (resp. is not) a center of F(b,r',r"). We can find in T(0,r',r") (resp. in T(6,r',r")) an increasing (resp. a decreasing) idempotent T-sequence (T„)raefsj of center 0 (resp. of center b) and diameter p £]r',r"{ whose holes are holes of O. Consider the O-set D admit ting each Tn as a hole, and no other holes. We notice that D admits a unique T-filter F run by the T-sequence (Tn)„epj- By Theorem 18.11 there
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Ultrametric Banach
Algebra
exists / € H(D) such that
Corollary 25.5: Let B be a uniform normed commutative K-algebra with unity satisfying Property (o) and let <j> be a K-algebra hornornorphism from A to B. Let || . ||^ be the norm of B and let || . ||A be the spectral semi-norm of A. Then <$> is continuous and satisfies ||0(x)||^ < ||z||^ Vx € A.
Properties (o) and (q) in Uniform Banach K-Algebras
153
Definitions and notation: Let E be a commutative ultrametric normed K-algebra with unity. Let n & N and let || . ||o be the norm of E. The .E-algebra of polynomials in n variables E[X\,... ,Xn] is provided with the Gauss norm j|| . ||| defined as = ii^in
sup
Hdii
i„||o-
il,...,in
By Proposition 6.18 the norm is a if-algebra norm and if the norm of E is multiplicative, this norm ] | . 111 is known to be a multiplicative norm of .ftf-algebra. Further, by Proposition 6.18, if E is complete, the completion of E[Xi,... ,Xn] with respect to this norm, denoted by E{X\,... ,Xn), con sists of the set of power series in n variables J ^ i a,ilt...,%nX\l ... X^ such that limj1+...+jTi_MX) ailt...tin = 0. The elements of such an algebra are called the restricted power series in n variables. Henceforth, in this chapter we will denote by M the disk d(0,1-) and by A the K-algebra H(M){Y}. We will denote by x the identical mapping on M, we fix r e]0,1[, we put t = 1 — xY and by S the multiplicative subset generated in A by the t — a, a G d(0,r~), and B will be the if-algebra Finally, we will denote by T(r) the set of ifi € Mult(A, || . ||) such that ip(X) > r. Lemma 25.6:
Let P € E[Y] and let Q{Y) = P ( l - Y).
Then \\\P\\\ =
IIIQIIIProof: Let P(Y) = Y^=oaiYJ a n d l e t Q(Y) = E " = o 6 j y 7 " ■ T h e n e a c h bj is of the form (-1)-7' Yl2=j ) a k - Hence |6j| < |||P||| and thereby |||Q||| < |||P|||. But on the other hand P(Y) = Q(l - Y), hence |||P||| < |||<3||1, consequently the equality holds. □ Lemma 25.7:
Let ip e T(r) and let h = £- e B, with f e A, g & S.
Then 0 ^ ip(S). The mapping ip defined in B by ip(-) = ^ W belongs to Mult(B). Further, the mapping <j> from B to [0,+co] defined as 4>{h) = su Pver(r) V'CO belongs to SM(B). Proof: First, we will show that cj>{h) < +oo. We can write g in the form n " = i ( i - « j ) 9 i , Oj € d(0,r~), q, € N*. Weset q = £ " = 1 qj. Sinceip{t) > r,
154
Ultrametric Banach
Algebra
clearly ip(t — a,j) = ij)(t), and hence we have ip(g) > rq. Consequently,
^(L)<m<mt \9J~
ri
~
ri
and therefore
B C H(M){t}{\}r
C B.
Lemma 25.10: For every a € C(0,1), (a — xY)A is a maximal ideal of infinite codimension of A. Moreover, we have sa(xY) = M, sp{xY) = U. Proof: Let N = sa(xY) and let D = sp(xY). Since |||xy||| = 1, sp(xY) is obviously included in U. On the other hand D is clearly included in M because given a € M, the homorphism Xa,i from A to K defined as
(
OO
YLfoY3 3=0
(X)
\
) /
=
$Z^'(a)
satisfies
Xa,i(xY) = a.
j=0
Consequently, we have M CN
CDCU.
(25.1)
Properties (o) and (q) in Uniform Banach K-Algebras
155
Let a G U \ M and consider the ideal / generated by a — xY. Clearly / n H{M) = {0}. Consequently the canonical surjection 0 from A onto the algebra A' = j induces an injection from H{M) to A'. We will show that j is just the completion of the field of fractions F of H{M). Since the norm of A is multiplicative, / is obviously closed in A, hence A' is provided with the quotient norm, which makes it a Banach A"-algebra. Let a € M. In A', we have
*(*)=(«(£
and
<®
<
1,
1. Consequently, the 'GO igi \)^+i converges in K for every a G M, which proves that
and similarly ||#(a;)|| < 1, hence \\9(x) series
2^,oo , j=0
x
9(b — x) is invertible in A' and admits for inverse Y^jLo rg(x))"+1 • T n u s ! A! contains a field isomorphic to F and we will confound it with F, due to the fact that the restriction of 9 to H(M) may be considered as the identity. In the field F, it is clearly seen that the relation \\(6(x — 6)) _ 1 || = = ||(0(x - b))|| _ 1 holds for all 6 ^ M because x - b is invertible in H(M), and that III . Ill is an absolute value. Consider now b e M. Indeed,
(9(b-x)-l)
n+1
= ^T j=0
(9{x)Y+ i
3=0
which proves that \\{9{b - x)~l)\\ <\. So, actually, ||(0(6 - x)- x )|| = 1, and of course \\9(b — x)|| = 1 because |||6 — x\\\ = 1. Thus the norm of A' induces on F the continuation of the absolute value of H(M). Since 9(H(M)[Y}) C F, then F is dense in A', hence A' is the completion of F with respect to its absolute value. Therefore, / is a maximal ideal of infinite codimension. Thus, given a e U \ M, then a belongs to D\N, hence U \M = D\N, and consequently, by (25.1), L = M, and D = U. □ Theorem 25.11 ([37]): B is a commutative ultrametric uniform Banach K-algebra without non zero divisors of zero, satisfying Property (q) but not Property (o). Moreover, it is equal to the set of Laurent series X^-^ anYn, with an 6 W, l i m n ^ + 0 0 an = 0, lim„ l&nll7"™ = 0, and we have +00
J>« yn
max ( sup||a n ||,sup||a n ||r ^n>0
n<0
156
Ultra-metric Banach Algebra
Proof: First, we will show that B does not satisfy Property o). Let t = 1 - xY. By Lemma 25.10 we have SPA{XY) = U, SO,A{XY) = M. Consequently, we can see that sa^(i) = d(l, 1~). Let x 6 X(A,K). For all g € S, we notice that x{9) ¥" 0- Thus, x has continuation to a K-algebra, homomorphism x fr°m B to K defined as x ( f ) = f?{y> ^ / G ^> 9 & S. Since |x| clearly lies in T(r), x is continuous with respect to the norm || . || of B, therefore x has continuation to a JiT-algebra homomorphism x from B to K. Consequently, SCLA(XY) C sag(xY). But of course, the inverse inequality is true because A C B, so we have sag(xY) = M, thereby 1 -
B
X
sa
=1.
(25.2)
\X(t)\ > r
(25.3)
Now, let x S - ^ ( ^ satisfy
and let A = x(A) be provided with an absolute value extending this of K. By (25.3) we have x(
sup jo, |A| A G SPB (jj } =-• Thus, by (25.2) we can see that B does not satisfy Property (o). Now, we will show that B has no non zero divisors of zero, and first we will prove that B is the if-vector space of the power series / of the form E^Lo a«
fe-l a xJYk
/i(x)y = Y, 3 j=0
xY h
+ ( ) 9 = Yl(aNxYY(Yk~J">+ j=0
(xY)k9-
Properties (o) and (q) in Uniform Banach K-Algebras 157
So, we can write fc-i
hYk = ^ Oj(l - t)jYk-j i=o
+ (1 - tfg.
(25.4)
Now consider the polynomial P(Z) = Yl^Zo a.iYk~iZi £ K[Y]0[Z\ and let Q{Z) = P{\ - Z). By Lemma 25.6 we have |||Q||| = |||P||| sa^<j
sup
116^)11=
0<3
sup
K-|.
(25.5)
0<j
Consequently, we obtain fc-i
fe-i
J2h(Y)ZJ
k
J2aj(i-tyY -i
= 1110111 = 111^111=
3=0
SUp
0<j
|Oj|
And finally, ||Q(y)||=max(|a0|,...,|a*-i|). j
On the other hand we have \\\g{\ - Z) \\\ || 1 — t\\ = 1, in A we obtain \\9(l-ty\\
(25.6)
= \\\gYi\\\ = \\g\\. But since
= \\gl
By (25.4) we deduce hY* = Q(t) + (1 -t)*g, g e H(M), And by (25.5), (25.6), (25.7) we obtain ||/i||=max(||Q(t)||,||(l-i)^||).
(25.7) Q(Z) e tf[yio[Z].
(25.8)
G A Now, consider / = Y^ZfjYJ - F o r e a c h 3 e N> h y i i s o f t h e form Qj(t) +gjSj{t), with Q3\Z) e #[*1o[.Z], <& € H ( M ) , 5j(Z) € ff[Z], satisfying further
\\\Qj\\\ < ll/ill
(25-9)
llldilll < ll/ill-
(25-10)
WlQiW+gjSjlW^Wfil
(25.11)
and
Consequently, we have
158
Ultrametric Banach
Algebra
By (25.9), the series Y^jLoQj(Z) is clearly converging in K{Y}o{Z} to a limit Y^T=o^iO'r)Z:i'■ ^ n t n e s a m e way, by (25.10) the series Y^jLo9j^j{^) converges in H{M){Z] to a limit Y^jLoV-i^■ Moreover, by (25.9) and (25.10), we can see that sup j € N ||Aj|| < ||/|| and sup j e N H^H < ||/||. On the other hand we have ||/|| < max(sup^ >0 HAjl^sup^x) ||//j||), thereby H/ll =max(sup||A ; ,-||,sup||/ij||). j>0
j>0
Putting a,j = Xj + fij, we have ||aj|| = max(||Aj||, ||//j||). This shows A to be the set of series G — Yo°° a«^™ w'ith coefficients a,j G A, satisfying lim n _ + O Q an = 0, \\G\\ = supjjo n ||. n>0
Now, we will show that B is the set Q. of series G = ]C-oo a "^"' w i t n coefficients a,j G W, satisfying limn_>+00 a n = 0, limn^-oo ||a n ||r n = 0, -, it is seen that Q C B.
and ||G|| = s u p n > 0 ||o n ||. Since
Given
a G d(0,r ), we notice that j ^ = Yn^o t^rr belongs to Q. So, by Lemma 25.9 B is included in fi, and therefore Q is dense in B. Since the norm of H(M) is an absolute value, on H(M)[Z] consider the two absolute values V'o, ^l defined by
we can
( " \ sup ||oj||, ipi j y^cijZi I = sup j|a.j11r-' 0<j
^ ) = ^ ^ = ^ ( i - 0 = Ki = o,i). Then given £ j L 0 b*Yk
H M
( )W]>
bkYk)
we have
< sup ||6fc|| (i = 0,1). J
0
Consequently, tp'(f) < \\f\\ V/ G H(M)[Y], hence each ip'{ has contin uation to an absolute value ip'/ defined on A, and actually belongs to
Properties (o) and (q) in Uniform Banach K-Algebras
159
Mult(A, || . ||) (i = 0,1). On the other hand, we notice that each ip" belongs to T(r), and therefore has extension tjj'/' to B and B (i = 0, 1). In particular we have $ " ( / ) < | | G | | V G e B ( i = 0,l). a tn
Now, let I = Y^£> n ||G|| > MG)
€ fi
-
B
(25.12)
y (25.12), we can check that
= sup \\an\\ and ||G|| > i(G) = sup ||o n ||r n . n>0
n>0 n
Thus we have ||G|| = max(sup n > 0 ||a n ||,sup n < 0 ||a n ||r ). Consequently, fi is complete for the norm || . || and therefore is equal to B. Let F be the field of fractions of H{M) and let F be its comple tion with respect to the absolute value || . ||M of H(M) extended to F. Then, F is provided with an absolute value that we still denote by II • || M- We can consider F + K{Y}o as a JiT-vector space provided with the norm || . || defined as it follows: given g £ F, h € i f j l ^ o , we set |<7 + h\\ = max(||gf||M) \\h\\). Let V be the iC-vector space consisting of all series ]C-c» a n * n an € F + K{Y}o){{t}}r. It is provided with the norm defined by +oo
Va„t" 11
-oo
= max(sup||a fc ||,sup||a fc ||r fc ). fc>0
fc<0
Thus, B is a X-subvector space of V which actually is included in because so is K{Y}, due to the fact that Y = - ^ and -
F{{t}}r
= 1 . Since both
the multiplication of B and its norm are induced by those of F{{t}}r, B is a K-subalgebra of .F{{t}} r , and therefore B is an integral domain. But the norm of B is semi-multiplicative, and therefore B is uniform, hence by Corollary 24.11 B satisfies Property (q). □
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Chapter 26
Properties (o) and (q) and Strongly Valued Fields
In Chapter 25 we saw that in a commutative uniform Banach .ftT-algebra, Property (o) implies Property (q). Here we examine whether this remains true when the algebra is not supposed to be uniform. Actually, this is depending on the ground field [23]: if it is strongly valued, then the property is always true. If it is weakly valued, counter-examples show that the property sometimes doesn't hold. N o t a t i o n : In the Chapter, A is a commutative ultrametric Banach K-algebra with unity. Theorem 26.1: K is supposed weakly valued. There exist a non semisimple commutative ultrametric Banach K-algebra without non zero divi sors of zero, with unity, all maximal ideals of which have codimension 1, such that || . ||Sj is a norm. Proof: For convenience, we put I — | . For each r £ [0,1], we denote by Tr the circular filter of center 0 and diameter r. By Proposition 19.6 we can construct an open closed bounded infraconnected set E C d(0, l~)\d(0, l~) such that E — d(0,1), admitting an irregular T-specific partition whose elements are the sets d(a,r~~) n E, whenever |a| = r, I < r <1. Now, let (rra)„£M be a strictly increasing sequence of [/, l[n|AT| such that limn^ooT-n = 1 and | X^=o logr„|oo = +oo, and for each n G N, let an e C(0,r n ) and let Tn = d{an,r~). Let E' = E \ (U n e N T n ) and D = r(0,1,2) U E'. Let T = y(Fi). By Theorem 18.8 the weighted sequence (Tn, l)n€N is an idempotent T-sequence of center 0 and diameter 1. And the subset E' of D admits a partition of the form {CE'(Gi,r)> i & I,r €\l,l\f\\K\) 161
162
Ultrametric Banach
Algebra
where the (Gi,r)i£i are T-friters of diameter r secant with C(0,r), and where the only T-filter on E' which is not one of the Gi,r, is the increasing filter of center 0 and diameter 1. Hence for every increasing filter Q of D secant with any circle C(0,r), by Corollary 19.4 we have y(G) = y{Tr). In particular, since for all r,r' £]l, 1[, Tr T'T are secant with VD(G), we check that y{Tr) = y{Tr'). So, we will just denote by J this closed prime ideal y(Tr) which actually does not depend on r £]l, 1[. Since D has no decreasing T-filter, as we just saw, each element / of J satisfies f(a) = 0 Vo £ T(0,1,2), and therefore J C T. Moreover, since D has no decreasing T-filter, T\ n D is not a distinguished circular filter, therefore D actually has no distinguished circular filter. Consequently every maximal ideal of H{D) has codimension 1. For every a £ D, let Ma be the maximal ideal of H(D) consisting of the / £ H(D) such that f(a) = 0 and let Xa G X(H(D), K) be defined as Xa(f) = /(«)• Then each maximal ideal of H{D) is of the form Ma. Let A = L ' and let 0 be the canonical surjection from H{D) onto A. For each a £ T(0,1,2), 9{Ma) is a maximal ideal of codimension 1, because Ma D J. But for each a € d(0,1) C\D, J is not included in Ma, hence 6(Ma) is not a maximal ideal of A. Consequently, by Lemma 1.13 all maximal ideals of A have codimension 1 and are of the form 9(Ma) with a e T(0,1, 2). But since D admits an increasing T-sequence of center 0 and diameter 1, there exists in H(D) an element / € T satisfying g(a) = 0 Va £ F(0,1,2), and y>o,r(ff) > 0 Vr e]l, 1[. Let h = 6(g). Then /i ^ 0, but since ^(a) = 0 Va € T(0,1, 2), we see that || . ||). Therefore we have ||/i|| s m = 0 < ||/i|| si . 1(h) = 0 V 7 G Multm(A, By construction, A has no non zero divisors of zero. Now, suppose that an element y of A is such that \\y\\Si = 0. Let / £ H(D) be such that $(/) = V- By definition, for each r e]i, 1[, <^o,r is of the form ^ r o ^, with ipr £ Mult(A, || . ||), therefore y>o,r(/) = 0, hence / belongs to ^7, thereby y = 0, hence || . ||Sj is a norm. □ Corollary 26.2: If K is weakly valued, there exist commutative ultrametric Banach K-algebra with unity, without non zero divisors of zero, all maximal ideals of which have codimension 1, satisfying property (o) but neither Property (q) nor Property (s). Theorem 26.3:
Let K be strongly valued. Then A satisfies Property (s).
Proof: Suppose Property (s) is not satisfied. So, there exists t £ A such that \\t\\si > j|t|| s m , and therefore, there exists ipa £ Mult(A, || . ||) such that ipo(t) > \\t\\sm- Let r = ||£|| sm and u = ipo(t). Thus, sp(t) is included in d(0,r). Let O be a i-normal partition. By Corollary 20.5 there exists
Properties (o) and (q) and Strongly Valued Fields
163
an increasing idempotent T-sequence (T n )„ 6 N of center 0 and diameter u with all Tn included in the annulus r(0, r, u). Therefore, by Proposition 21.5 there exists / e H(D, O) such that i/>(f) = 1 \/tp e Mult{H{D,
|| . ||), we have
Thus, <j>
7
is of the form ipp and .T is secant with d(0,r), so we have (/) = 1, and therefore (F) = 1, a contradiction. D Proposition 26.4: Assume K is strongly valued. Suppose there exists t G A, Vi, ^2 € Mult(A, || . ||) SMC/I thati>i{t) < tp2(t) andr(0,tpi(t),i/j2(t))n sp(t) = 0. 77ien /I admits non zero divisors of zero and A contains elements whose spectrum is not infraconnected. Moreover, if ipi(t) = r(t), then A does not satisfy Property (o) and the intersection of all maximal ideals of codimension 1 is not null. Proof: Let D = spA{t). We put r% = 4>i{t), r2 = ip2(t)- Let s € lIKIIsii \\t\\[- Let O be a t-normal partition of diameter s. Let r £\r\,r2\. By Corollary 20.5 there exist an increasing T-sequence and a decreasing T-sequence of center 0 and diameter r, and therefore by Proposition 21.5 there exist f,g& H(D, O) such that: ip(f) = 1 VV> € Mult(H(D,0), ^ t ( / ) = 0 V^ e Mult(H(D, ^ t ( / ) ± 0^
|| .
||D,O)
such that ipt(t) < ru
and
O), || . || D ) 0 ) such that VtW > r,
(26.1)
e Afuii(A || . ||) \ MuZt0(i4, || . ||) such that n < i/>t(f) < r, (26.2)
i/,t(g) = 1 W e Mult(H(D,0), ^ t ( 5 ) = 0 V ^ e Mult(H(D,
|| .
||L.,O)
such that V(i) > J"2, and
O), || . Ho.o) such that ip(t) < r.
(26.3)
Thus fg = 0. Since O is i-normal partition, by Theorem 23.6 we have the canonical continuous morphism Qt from H(D, O) into A. Now, Qt is defined on H{D, O). Let F = Qt(f), G = &t(g)- Let Tj be the circular filter of (tpj)t (j = 1,2). Then T\ is secant with d(0, n ) , hence (ipi)t(f) = 1, and therefore tpi(F) = 1. In the same way, .7-2 is secant with d(0, s) \ d(0,r~), hence (^2)4(5) = 1. a n d therefore ip2(G) = 1. Thus, both F, G are different from 0 though FG — 0 and therefore A contains non zero divisors of 0.
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Ultrametric Banach
Algebra
Now, we will show that SPA(F) is not infraconnected. Consider I S d ( 0 , l - ) \ {0}, and suppose that F - I is not invertible in A. Let C € X{A) be such that ({F-l) = 0. Let M = Ker((). Let (/> € Mult(A, || . ||) be such that Ker{4>) = .M. Then <> / defines an element
Properties (o) and (q) and Strongly Valued Fields
165
an empty annulus T(a, r\, r2) and then by Lemma 8.15 there exist ipi, ip2 G Mult(A, || . ||) such that ipx(t) = rx, ip2(t) = r2. D Theorem 24.7: Assume K is strongly valued and let A be such that MaXa(A) ^ 0. If A satisfies one of the following conditions bellow, then A satisfies Property (q): (i) (ii) (Hi) (iv)
Property (o), A has no non zero divisors of zero, sp(x) is infraconnected for every x £ A, the intersection of all maximal ideals of codimension 1 is null.
Proof: We assume that A does not satisfies Property q), and will show that A does not satisfy the three properties at the bottom of Theorem 26.7. Let t e A be such that r(t) < \\t\\si. Let si = r(t), let s2 = \\t\\SiBoth si, s2 lie in the closure of {
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Chapter 27
Multbijective Banach if-Algebras
In Theorem 6.15 we saw that the mapping from Multm(A,\\. ||) into Max(A) associating to each 0 e Multm(A,\\. ||) its kernel, is surjective. The natural question coming next, is whether the mapping is also injective. The answer was given in [23]. Definition and notation: In all this chapter, A is a commutative ultrametric Banach X-algebra with unity. A is said to be multbijective if for every maximal ideal A4 there exists only one ip e Multm{A, ||. ||) such that Ker(tp) = M. R e m a r k : As recalled in Chapter 5, in C it is well known that every maximal ideal of a C-Banach algebra is the kernel of one and only one multiplicative semi-norm, and that every multiplicative semi-norm has a kernel which is a maximal ideal. A partial answer to the question above was first given in [21] and is recalled in Theorem 42.10 in [30]: Theorem 27.1: If K is weakly valued there exists a closed bounded infraconnected set E in K such that H(E) admits multiplicative semi-norms ip\,ip2 € Multm(H(E), || . | | E ) , with ipi ± ip2 such that Ker(ipi) = Ker(i>2). Proof: The proof is based upon the same construction as in Theorem 26.1. Here we just take again the set E constructed at the beginning of the proof of Theorem 26.1. Let I = \. For each r G [0,1], we denote by TT the circular filter of center 0 and diameter r. By Proposition 19.6 we can construct an 167
168
Ultra-metric Banach
Algebra
open closed bounded infraconnected set E C d(0,1~) \ d(0, l~) such that E = d(0,1), admitting an irregular T-specific partition (C{Qitr),i € I,r £ }l,l[C\\K\) and we denote by T the circular filter of center 0 and diameter 1. By Lemma 19.3 we have
y(Qi,r) = y{9j,a) = y{T), Vi,j e 7,Vr,s e]Z, i[n\K\. Let M be this ideal y(T) which actually does not depend on i € 7, and r e]Z, l[n|AT|. By Corollary 19.8 At is a maximal ideal of 77(D) of infinite codimension. And for every r e]Z, l[n|7C|, we have Ker(ip0
□
When K is strongly valued, we have a different answer: Theorem 27.2: Let K be strongly valued, and let F be a field extension of K which is an ultrametric normed K-algebra admitting two different continuous absolute values. Then the completion of F admits non zero divisors of zero. Proof: Let F be the completion of F. Let ip\, ip2 be two different contin uous absolute values have continuation to F. We can find x € F such that tpi(x) < -02(x). We put r\ = ij)i{x), r 2 = i>2{x)- Since x does not belong to K, we have sp(x) = 0. Consequently, by Theorem 26.4 F has non zero divisors of zero. □ Remark: In Theorem 24.12 we saw that a commutative Banach Tf-algebra with unity which is the completion of a field extension F of K provided with a semi-multiplicative norm, has non zero divisors of zero. We then made no additional hypothesis on the ground field. Here we sup pose K strongly valued, but we no longer assume that the norm is the spectral norm of F. As a Corollary, we obtain Theorem 27.3: Theorem 27.3:
If K is strongly valued, then A is multbijective.
Proof: Suppose K is strongly valued. Let M b e a maximal ideal of A, and suppose there exist ipi, i>% € Multm(A,\\ . ||) such that Ker(ipi) = Ker(ip2) = M. Let F = -j^. Thus, the field F admits two different absolute values | . \j quotients of ipj respectively, (j = 1,2) which by construction, are continuous for the norm of F quotient of the norm of A. Since F is a
Multbijective
Banach K-Algebras
169
iiT-Banach algebra for its quotient norm, by Theorem 27.2 we are led to a contradiction. □ Corollary 27.4: There exist non multbijective commutative ultrametric Banach K-algebras with unity if and only if K is weakly valued. Theorem 27.5: Let K be strongly valued, let ip € Mult(A, \\ . ||) and sup pose that the set of maximal ideals containing Ker(ip) is countable. Then i> belongs to Multm(A,\\ . ||). Proof: Let J = Ker{t^) and let B = ^ and let 9 be the canonical surjection from A onto B. Then Max(B) = 9(Max(A)) is countable. Suppose that J ^ Max(A). Let M. be a maximal ideal of A containing J and let M! = 9(M). Then B admits an absolute value ip' such that ip = ip'od. Moreover, there exists (a unique) cj> € Multm(A, || . ||) such that Ker(
170
Ultrametric Banach
Algebra
of problem. We notice that our counter-example is an algebra of analytic elements H{D). Hence in particular, Multm(H(D),\\ . \\D) is dense in Mult(H(D), || . \\D). Consequently, density does not imply multbijectivity. However, we are not able to construct a noetherian non multbijective ultrametric Banach if-algebra. So, we can ask whether noetherian ultrametric Banach X-algebras are multbijective, no mater what the field.
Chapter 28
Pseudo-Density of Multm{A,
|| . ||)
Thanks to Property (s), we can ask whether in a commutative uni form Banach JsT-algebra with unity A, Multm(A, \\ . ||) is dense inside Mult(A, || . ||). Actually, we can only show a weaker property: pseudodensity. Notation: Throughout the chapter A is a commutative ultrametric Banach K-algebra with unity. Definition: A subset S of Mult(A, || . ||) will be said to be pseudo-dense in Mult(A, || . ||) if for every ip € Mult (A, \\ . ||) and if for every e > 0, there exists a neighborhood of the form V(ip, / , e) / € A, such that V{ip, / , e) n 5 ^ 0 [29,37]. Theorem 28.1 ([29]): Multm(A, \\ . ||) is pseudo-dense in Mult (A, || . ||) if and only if A satisfies Property (s). Proof: Suppose first that Multm(A, \\ . ||) is pseudo-dense in Mult(A,\\ . ||). Let x £ A and let r = \\x\\sl. Take e > 0. By The orem 6.19 there exists ip e Mult (A, || . ||) such that ip(x) = r. But since Multm(A,\\ . ||) is pseudo-dense in Mult(A,\\ . ||), we can find <j> £ Multm(A, || . ||) such that <j>{x) > ip(x) - e, hence finally
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Ultrametric Banach
Algebra
Let r = ip(x). If r = 0, then d(0, A) n sp(x) = 0, hence x is invertible, a contradiction to the hypothesis ip(x) = 0. So, we can assume r > 0. Then sp(x) n T(0,r ~ X,r + A) = 0. Let v e]0, ^[, a,!> e if such that r-i^<|a|
(28.1)
Suppose first that 0(x) > r + X. Then 6{x — a) = 9{x — b)= 9{x), hence 9(u) = 1. Consequently,
^ ( l t )-o( u )
=
M - i > r + Ar - y - i r
=
^zil. r
But since ;/ < | , we obtain iP(u) - 9(u) > ^ .
(28.2)
Suppose now that 9{x) < r — X. Then 9{x — a) = \a\ and 9(x — b) = \b\, hence 9{u) = ^ j £ l . By (28.1) we obtain
But since v < j , we have
(, + A-,)(l- 7 ^A_)>^ + A - f A - 2 ^ > ^ ^ > A . Vr
[r — uyj
rz
2r (28.3) Thus, in both cases, by (28.1), (28.2) and (28.3) we have proven that "^{u)9{u) > ^ . Consequently, we see that 9(u)<\\u\\si-~,
V0 e Multm(A,\\
r
. ||),
a contradiction to the hypothesis on Property (s). Corollary 28.2: Let A have bounded normal Multm(A, || . ||) is pseudo-dense in Mult(A, || . ||).
□ ratio.
Then
Proof: Indeed, since A has bounded normal ratio, by Theorem 24.3 A satisfies Property (s). O
Pseudo-Density
of MulUn(A,
\\ . ||)
Corollary 28.3: Let K be strongly valued. Then Multm(A, pseudo-dense in Mult(A, \\ . ||).
173
\\ . ||) is
Proof: Indeed, since K is strongly valued, by Theorem 26.3 A satisfies Property (s). D Lemma 28.4: Let
Let ip e V((j>, hi,...,
hq, | ) and let k € { 1 , . . . , q}. We have
l0(A)-tf(A)loo < |^(//c)-^fc)|oo + | ^ / = ) - V ' ( ^ ) | o o + |V'(^)-^(/fc)loo.
But \4>(fk) -
^ f> the conclusion is clear.
Theorem 28.5: Let A have bounded normal ratio and contain a dense Luroth K-subalgebra B. Then Multm(A, \\ . ||) is dense in Mult(A, \\ . ||). Proof: By definition, B is of the form K[h,x], with h € K(x). Let ip e Mult{A,\\ . ||), let fi,...,fq € A and let e be > 0. We can find hi,..., hq € B such that \\fj — hj\\ < | Vj = 1 , . . . ,q, and then by Lemma 28.3 we have
V (V,hi,...,h q , | ) C V(ip,h,...,
fg,e).
For every <j> e Mult(A, \\ . ||), we denote by c/)x its restriction to B. Let T be the circular filter of ipx. By Theorem 13.9 there exists V € T such that | \hj(a)\ - 0(fy)|oo < | Va e y Vj = 1 , . . . q. Now, by Corollary 28.2 Multm(A, \\ . ||) is pseudo-dense in Mult (A, \\ . ||), hence there exists 6 £ Multm(A, || . ||) such that the circular filter Q of 9X
□
174
Ultrametric Banach
Algebra
is secant with V. Then, since 9(hj) = lim^-0 \hj(a)\, it is clear that | ^ ) - V ( ^ ) | o c < ^ Vj = Consequently, 9 belongs to V(^, hi,..., V(iP,fi,...,fq,e).
l,-..,q.
hq, | ) and therefore it belongs to □
Chapter 29
Polnorm on Algebras and Algebraic Extensions
This chapter is designed to help us prove that a reduced affinoid algebra is complete for its spectral norm. In particular, this will be useful in charac teristic p ^ 0. We will roughly follow the same way as in [3]. Definition and notation: A field E is said to be perfect if every algebraic extension of E is separable. Given a field E of characteristic p / 0, we denote by E* the extension of E containing all pth-xoots of elements of E. Let L be a subfield of K. B is a commutative L-algebras with unity, without non zero divisors of zero and A is a L-subalgebra of B containing the unity of B, provided with a L-algebra semi-multiplicative semi-norm ipLet P(X) = J29j=0 aixi € A\x\ b e m o n i c - W e P u t S(P,ip) = max
(tp(aj))1'^.
0<j
When there is no risk of confusion on the L-algebra semi-multiplicative semi-norm ijj of A, we will only write S(P) instead of S(P,ip). In particu lar, when the monic polynomial P belongs to L[X\, then S(P) will denote max0<j
176
Ultrametric Banach
Algebra
Lemma 29.2: Let B be provided with a L-algebra semi-multiplicative semi-norm tp. Let P(X) £ A[X] and let b e B satisfy P(b) = 0. Then S(P,tP)>tP(b). Proof: Let P(X) =■ Y,]=o ajxi a n d suppose ip(b) >S(P). Thus, we have ip(aj) < *p(b)q-j, hence tp{ajV) < tp{aj)tp(by < tp(b)jtp(by-^ = ip(b)i whenever j = 0,...,q — 1. Consequently, S(]T^~Q a.jXi) < tp(bg). But by Lemma 5.3 ip is ultrametric. And since P(b) = 0 we have i K £ £ o *i&) = ip(bq), a contradiction. D Lemma 29.3 is almost classical and comes from the behaviour of poly nomials in an algebraically closed complete ultrametric field recalled in Lemmas 13.6 and 13.7. Lemma 29.3: Let P(X) = J2'j=oaJX'' e K[x] be its zeros in K. Then maxi<j< g \ctj\ = S(P).
be iconic, let c*i,
...,aq
Proof: Let {|ai|,..., \aq\) = {n,...,rh}, with rj < rj+i (1 < j < h). For each j = l,...,h, let m,j be the number of zeros of P inside C(0,rj), let tj = N+(P,-log(rj)), and Sj = N~(P, - l o g fa)). On one hand, by Corollary 17.3 we have rrij = tj — Sj (1 < j < h) and logfa) = jpi-(log(|a S j | - log(|ot3.)). In particular logfa) = —-log(|a s J) = max log(|Qj|). mix
*S3
But since N-(P, - logfa)) = sfc and JV+(P, - logfa)) = q, by definition we have rqh =
\ash\{rhy-h>\a3\{rhy\<j
hence — - log(|aj|) < l o g ( K J ) < rh = — — log(|a s J) Vj = 0 , . . . , q - 1. Consequently max log(|aj|) = 1<J<9
log(K,|) >
therefore S(P) = max 0 <j< g \atj\.
:log(KI) \/j = 0 , . . . ,q - 1, q-j
O
Polnorm on Algebras and Algebraic Extensions
177
Proposition 29.4: Let A be integrally closed such that B is finite over A. Let T be a set of L-algebra homomorphisms from A into K such that, for each t € A, there exists ^t^T satisfying |£t(t)l = sup{|x(i)| | X € T } . Let W be the set of L-algebra homomorphisms from B to K whose restrictions to A belong to T . On B we put \\x\\ = sup{|x(aO| I X e W } . Lety € B and let P = irr(y,A). Then S(P, || . ||) = ||y||. Moreover there exists ( € W such that \C(y)\ = \\y\\. Proof: By Lemma 5.4, the function || . || denned on B is a .L-algebra semi-multiplicative semi-norm. Therefore, by Lemma 29.1 we have s(P,||.||)>||y||. We will prove the opposite inequality. Let P(X) = Y^j=o aj^ € ^ [ ^ 1 and ft e { 0 , . . . , g-1} be such that S(P, || . ||£) = , "v / ||a / l || j 4 . By hypothesis e there exists £ € T such that \£{ah)\ = \\ah\\. Let f(X) = Y?j=ot(ai)xi L\X\. Then, of course, S(/, | . |) = S(P, || . ||) because |£( a j )| < \\aj\\ Vj = 0 , . . . ,q. Let a\,... ,aq be the zeros of / in K. By Lemma 29.3 we have maxj^xg |OJ| = S(/, | . |). Let (3 be a zero of / in K such that |/3| = S(/, | . |). Let
V=o
/
j=o
and let 6 be the canonical surjection from A[X] onto A[y}. Since P ( X ) J 4 [ X ] C Ker(cp),
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Ultra-metric Banach
Algebra
closed. Now let x € F, let P = irr(x, L) and let ati,..., aq be the zeros of P in fl. By Lemma 29.3 we have maxi<.,
Polnorm on Algebras and Algebraic Extensions
179
Theorem 29.10: Let L have characteristic p > 0. If LP is L-productal with respect to the L-polnorm, then so is the algebraic closure of L. Proof: Let L°° = \J^=1 L'p*, and let Q, be an algebraic closure of L con taining L°°. Then by Theorem 29.1 L°° is perfect, and by Proposition 29.9 the L-polnorm is an absolute value that continues this of L. By The orem 29.7, since L°° is perfect, fi is L°°- productal with respect to the L°°-polnorm. But by Proposition 29.8 the L°°-polnorm is equal to the L-polnorm on f2. Consequently Q, is L°°-productal with respect to the Lpolnorm which is an absolute value on L°°. Hence by Lemma 4.4, Cl is L-productal with respect to the L-polnorm. D
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Chapter 30
Definition of Affinoid Algebras
Affinoid algebras were introduced by John Tate in [49] who called them algebras topologically of finite type. As this last name suggests, such an algebra is the completion of an algebra of finite type for a certain norm. Definitions and notation: In this chapter, L is a complete ultrametric field, A, B are commutative ultrametric Banach L-algebras with unity. Recall that UL = d(0,1), ML = d(0,1~) and the residue class field of L is C. Here we put V = {x € A | ||x|| < 1}. Let (E, || . ||) be a normed L-vector space which also is a ^-module. The norm of E will be called a A-quasi-algebra-norm if it satisfies ||a/|| < llall 11/11 ^ a e A Vf e E- If E i s provided with such a A-quasi-algebranorm, (E, || . ||) will be called a A-quasi-algebra-normed A-module. A {//,-submodule E\ of A will be said to define the topology of E if it is a bounded neighborhood of zero i.e. if there exist r, s e]0, +oo[ with r < s such that {x € E\ \\x\\ < r } c £ , c { i e E\ \\x\\ < s}. We will denote by Eo the C^-submodule of the x £ E (resp. x & A) such that ||x|| < 1. We will recall definitions given in [49] about algebras topologically of finite type, now mainly called affinoid algebras. Proposition 30.1: Let E be a normed L-vector space which also is a Amodule. Let t 6 ML- Let E\ be a Ao-submodule defining the topology of E. Assume that there exists e\,..., eq € E\ such that E\ = 2 ? = i ^o e i + ^ i • Let
182
Ultrametric Banach Algebra
Proof: Let y € E\. We can write y in the form S£H=\ /i,»e» + tyii with /i,i € AQ and y\ € Ei, and in the same way, we can write j/i = ]£i=i /2,ie» + tj/2, with /2,j € ^0; and 1/2 € £1. By induction, it clearly appears that for every m € N*, we obtain y = Yli=i(I2'jLi fm,itm~1)el + tmym. And since all terms fj^ lie in A$, and all yj lie in £ 1 , each series X]?li/m.i*" 1 - 1 converges in the Banach algebra A to an element / j . Then, since each set tmEi is closed, y - Y^Li fiei l i e s i n tmEi f o r a11 m 6 N*- Therefore, y = Y11=ifiei- Consequently
Putting then c = maxi
>c\\(j>{ai,...,aq)\\.
Thus 4> is continuous, hence Ker{4>) is closed and therefore E is provided with the quotient norm || . ||' of ||| . |||. Then, E is complete for the norm || . ||'. On the other hand, given / 6 Aq, we have \\Hf)\\'=
inf xGner(tp)
Ml/+ slH > c
inf
||0(/ + x)||.
x€.Ker{
But since (/ + x) = <£(/), we check that \\(f)\\' > c\\<j>(f)\\. Thus E is complete for both norms || . |j and || . ||' and therefore by Banach's Theorem (Theorem 3.4), the two norms are equivalent. By definition, <j> is open with respect to the norm || . ||' on E. Consequently Y21=i ^e% i-s a ^4o-submodule E\ that defines the topology of E. Since F is dense in E, each e* is of the form fi + thi, with hi £ E\. Therefore E\ is included in ^ ? = i A)/i + tE\, and then by Proposition 30.1, we have E = Ylt=i Aofi C F. D Corollary 30.3: Let A be noetherian and let F be a complete A-quasialgebra-normed finite A-module. Then every A-submodule of F is closed. Proof: Indeed, since A is Noetherian, and since F is finite, F is Noethe rian, hence any ^4-submodule E has a finite closure in F, hence by Propo sition 30.2 is closed. Corollary 30.4:
Let A be noetherian. Every ideal of A is closed.
Definition
of Affinoid A Igebras
183
Proof: Indeed, the closure of an ideal I is of finite type, and therefore, / is closed. □ Definitions and notation: Let us recall the notation given in Chapter 8: Let E be a commutative ultrametric normed L-algebra with identity. Let n G M and let || . ||o be the norm of E. The U-algebra of polynomials in n variables E[Xi,... ,Xn] is provided with the Gauss norm |] . || defined as IIEi 1 „i„ a ii,-,»n^'i 1 ••■ x n"ll =sup i l i ... i i n ||a n ,..., I J|o- By Proposition 6.18 this norm is a L-algebra norm. In particular, if the norm of E is multi plicative, so is the norm || . ||. We denote by E{X\,... ,Xn) the set of power series in n variables E n „ i „ o.ili...!inX\1 . ..X%> such that lim il+ ... + j„_ >00 a^,...,^ = 0. The ele ments of such an algebra are called the restricted power series in n vari ables, with coefficients in E. Hence by definition E[Xi,..., Xn] is dense in E{X\,..., Xn). By Proposition 6.18, if E is complete, E{X\,..., Xn) is just the completion of E[X\,..., Xn}. More generally, consider a subring F of E, and elements tj (1 < j < q) of E{Yu...,Yn) such that ||ij|| < 1 Vj = l,...,q. Consider the canonical mapping
184
Ultrametric Banach
Algebra
(2) Let f(X) = EZoaixi G UL{X} = (Ti) 0 andJ e t q = iV+(/,0). Consider the residue class £-algebra A = M^ /j,0-, . In ^4 we denote by / the residue class of f(X). Then / is the polynomial YH=oUiXl(3) Since the Gauss norm on Tn is multiplicative, it is obviously equal to the associated semi-norm || . \\Si. Lemma 30.5: Let L = K, let / i , . . . , fm e T±. There exists a 6 UK such tkat\fj{a)\ = \\fj\\Vj = l,...,m. Proof: Indeed, for each j — 1 , . . . , m the equality |/j(a;)| = | | / j | | holds in all classes of UK except in finitely many ones. Consequently, all equalities \fj{x)\ = | | / j | | hold in all classes of Ui except in finitely many. □ Proposition 30.6: Let A, B be L-affinoid algebras such that B is Aaffinoid. Suppose that MBQB is finite over MAA . Then B is finite over A. Proof: By hypothesis, A is the quotient of certain L-algebra A{T\,... ,Tr}, and therefore is of the form A{t\,... ,tr}. Let A € ML, A ^ 0. For each t S B0, here we denote by i the residue class of t in ^ - . Thus, we have B0
_ A0 .-
-.
Now, by hypothesis for each i = 1 , . . . , r, there exists a polynomial Pi(T) = T" -hU-iT^-1 + ... + / 0 , with ^ e A0, and P^U) e MLB0. Let PZ(U) = Xigi, with Xi € Mi, and gi € B0- Now, we can choose k £ N such that £■* € XAo, and thus Pi(ti)li lies in AS 0 , hence Pi{U)li = 0. Therefore, ^ is finite over JJ~, and finally, by Proposition 30.1, B is finite over A. D Lemma 30.7: Let A be a L-affinoid algebra which is a quotient of a topologically pure extension L{X\,... ,Xn} of L by an ideal. Let 9 be the canonical surjection, and for each i = 1 , . . . ,n, we put U =
Definition
of Affinoid Algebras
185
from L{Y\,... ,YS} into a A defined as 4>{Yj) = yj, (1 < j < s) is an isomorphism from L{Yi,... ,YS} onto a closed subalgebra of A. Proof: Let F(Yi, ...,YS) € L{YX,..., Ys} be such that F(yx, ...,ys) € Ai. Suppose that F(Yi,...,Ys) $ U{Yi,...,Ys}. Since ||F|| lies in \L\, there exists X £ ML such that ||AF|| = 1. Let P = XF. Then the residue class P of P in £\Xi,...,Xs] is different from 0, but satisfies P(y~i, ■ ■ ■ ,ys) = 0 in ^ A , a contradiction. Consequently, F(Yi,...,Ys) belongs to UL{Y\, ..., Ys}, and therefore <j> is open. Hence, <j> is a bicontinuous isomorphism from L{Yi,..., Ys} onto a closed L-subalgebra of A □ Proposition 30.9: A is a L-affinoid algebra if and only if it is a finite extension of a topologically pure extension of L. Particularly, if A is of the form L{X1,...,Xn}[yi,...,yq] withyx,...yq integral over L{Xi,... ,Xn} and maxi<j< 9 \\yj\\ < I, then A is isomorphic to a quotient of L{X\,...
,Xn,Y\,..
■ ,Yq}.
Proof: First suppose that A is a finite extension of a topologically pure extension of L, say A = L{X\,... ,Xn}[yi, ...,yg]. Without loss of general ity, we may clearly assume that max(|j/i|,..., \yq\) < 1. Then the canonical homomorphism <j> from L[X\,..., Xn, Yi,...,Yq] into A defined by
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Chapter 31
Algebraic P r o p e r t i e s of Affinoid Algebras
We shall recall the proofs of many classical algebraic properties of ii'-affinoid algebras such as noetherianness, finite codimension of maximal ideals and other easy spectral properties [49]. Notation: As usual, K is an algebraically closed complete ultrametric field whose valuation is not trivial. We denote by U the disk d(0,1) and by Un the unit ball Kn : {(xi,...xn) | maxi<j<„ \Xj\ < 1}. We denote by M the disk d(0,1~), and by K. the residue class field of K. Throughout the chapter, A will be a K-affmoid algebra. Theorem 31.1: A is Noetherian, every ideal is closed, and each maximal ideal of A has codimension 1. Proof: We will prove by induction that topologically pure extensions are noetherian. This is trivial for K. Suppose this has been proven for K{X\,... yXs} whenever s < r, and consider B = K{X\,... ,Xr}. Let f e B, / ^ 0 and let B' = -fg. In B the Gauss norm satisfies ||/|| e \K\, so we can find A e K such that ||A/|| = 1 and therefore, without loss of generality, we can assume ||/|| = 1. Since the Gauss norm is multiplicative, now the mapping
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Ultrametric Banach
Algebra
Since ||/|| = 1, we have / ^ 0. However, we notice that f(x\,... ,xr) = 0. Consequently, { x i , . . . ,xr} is not algebraically free over K. Hence, by The orem 1.16 we can find T i , . . . , T S in -^r , with s < r, such that TI,...,TS are algebraically free over K, and such that ^ p - is finite over K.[T\, ... ,TS]. Therefore, taking t\,... ,ts in B' such that £$ = T» (1 < i < s), by Propo sition 30.6 K{t\,... ,ts} is a topologically pure extension of K and B' is finite over K{t\,...,is}. But by induction hypothesis K{t\,..., ts} is noetherian, and so is B' which is finite over K{t\,..., ts}. This is true for every / G B: and therefore by Proposition 1.5 B also is noetherian. Consequently, by Corollary 30.3 every ideal is closed. Finally, consider a maximal ideal M of B. Then, -^ is a field F and a AT-affinoid algebra. Thus, F is finite over a topologically pure extension K{Xi,... ,Xq}, and then by Proposition 1.4 K{Xi,..., Xq} is a field. But such a ring is never a field except when q = 0. Hence, F is a finite field extension of K, and then, since K is algebraically closed, we have F = K. O Corollary 31.2: The quotient of a topologically pure extension by an ideal is a K-affinoid algebra. Corollary 31.3: Let A be a K-affinoid algebra and let B be a A-affinoid algebra. Then B is a K-affinoid algebra. Corollary 31.4: Let B = Tn \y\,..., yg] be finite over a topologically pure extension Tn. Then B is isomorphic to a K-affinoid algebra of the form T„{y^...,y,} where j is an ideal 0fTn{Yi,...,Yg} such that TnDl = {0}. Corollary 31.5:
In A there are finitely many minimal prime ideals.
Corollary 31.6: Let A admit an idempotent u. Then uA is a K-affinoid algebra admitting u for unity. Theorem 31.7: For all a € Un and f € Tn, we put Xa(f) = / ( a ) . The mapping § from Un into X{Tn) defined as
Algebraic Properties of Affinoid Algebras
Proposition 31.9:
189
The following three properties are equivalent:
(i) ix(/)|
(Hi) ll/IU < 1. Proof: Let us show that (i) implies (ii). By Corollary 31.3 ^4{y} is a KafRnoid algebra. Next, by definition, we have ||V|| = 1. Let x £ X(A{Y}). Then \x(fY)\ < 1, hence x(l - fY) / 0. Consequently, by Theorem 31.1 1 — fY does not belong to any maximal ideal of ^4{^}, which proves (ii). Let us show that (ii) implies (Hi). Since 1—fY is invertible, its inverse is a series Y^^Lo 9nYn, with gn € A and lim n ^oo gn = 0. Since (1 - fY)g = 1, we check that g0 = 1, and gn+\ = fgn, V n g N . Consequently, gn = fn, and limn^oo fn = 0, hence by Theorem 6.9, ||/|| S j < 1. Finally, we check that (iii) implies (i): by Theorem 6.19 we have | x ( / ) | <
WfhiTheorem 31.10:
D Every K-affinoid algebra owns Property (p).
Proof: Suppose a A'-affinoid algebra A does not own Property p), and let f € A be such that ||/|| s o < \\f\\si. Taking X e K such that ||/|| s a < 1^1 < ll/llsi w e have \\j\\sa < 1 < ||f ||si, hence Condition (i) is satisfied but Condition (iii) is not, a contradiction. □ Lemma 31.11: Let I be an ideal of Tn, and let A = ^f. Let 8 be the canonical surfection from Tn onto A. Let ip £ Mult(Tn,\\ . ||). There exists (j) e Mult(A, || . ||) such that ip = <j> o 6 if and only ifZ C Ker(ip). Proof: Let || . || be the Gauss norm on Tn, and let || . ||9 be the quotient X-algebra norm of A. Let / € Tn. Suppose that I C Ker(tp). For every t € I we have i/j(f) = ip(f + t) = \\f + t\\, and therefore, ip(f) < Il^(/)ll9- Consequently, <j> o 8 belongs to Mult(A,\\ . ||). Conversely, if ip is of the form
For all x G Tn there exists x S X(Tn)
such that
Proof: Since the Gauss norm on Tn satisfies || • || = || ■ \\Si, we are reduced to show that for every / G Tn there exists x € X(Tn) such that lx(/)l = 11/11- We will proceed by induction. When n = 1, by Theorem 17.7, there exists a e U such that \f(a)\ = \\f\\. Consequently, the mapping
190
Ultrametric Banach
Algebra
X* e X{Ti) defined as Xa(g) = 9(a) (g G T{) satisfies \xM)\ = ll/IUNow, suppose we have already proven the following property: (Qn): For every rc < N, whenever h = l , . . . , m . Now by the induction hypothesis we can find x £ ^(^g) such that
xi
n
^^
0
9k h
II 0
> •
l
And since both x> II • || are multiplicative, and obviously satisfy |x(h(X) = YlT=oX(9k>h)Xk which obviously lies in T\. Since the prop erty holds for n =
Let f
€
Tn.
Then sp(f)
is the disk
d(f((0)),
ll/-/((o))ll). Proof: Without loss of generality, we can assume that /((0)) = 0. Let r = \\f\\. Of course sp(f) C d(0,r). Then / is of the form E ( j ) e N „ bU)X^>\ and there exists (A;) G N" such that ||b(fc)|| = ||/||- Since /((0)) = 0, at least one of the coordinates of (fc) is not 0. For convenience we can assume that (k) is of the form (fci,..., kn), with kn > 0. Now, we can write Tn in the form T n _ 1 { y } , and put / = ~Y^=oanYn■ Let I = kn. So, we have ll/H = || ai ||. Let a G d(0,r). Since l > 0, we have ||/ - a\\ = \\f\\ = ||a{||. By Theorem 31.12 there exists x £ X(Tn-{) such that |x(a;)l = llaHI-
Algebraic Properties of Affinoid Algebras
191
Let P(Y) = j:ZoX(an)Yn-a € K{Y}. Clearly ||P|| = | x ( a , ) | = r, hence by Theorem 17.2 P has its zeros in d(0,r). Now, by Corollary 31.8 there exists x € X(Tn) such that x(/i) = x(h) V/i G T n _i, and x ( ^ ) = a. Then X(f) = a. D Theorem 31.14:
Let f £ Tn.
Then f is invertible in Tn if and only if
l/((o))l>ll/-/((o))||. Proof: If |/((0))| > ||/ - /((0))||, / is invertible by Theorem 6.13. Now suppose |/((0))| < ||/ - /((0))||. Since sp(f) = d(/(0), ||/ - /(0)||), it is seen that 0 belongs to sp(f), hence / is not invertible. □
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Chapter 32
Jacobson Radical of Affinoid Algebras
Notation:
Throughout the Chapter, A is a K-affmoid algebra.
In A we will show that Multm(A, || . ||) is dense in Mult(A, ]| . ||). It will help us show that the nilradical of A is equal to the Jacobson radical, and more generally that A is a Jacobson ring. P r o p o s i t i o n 32.1: Let A = Tn[x\,... xq] be a K-affinoid algebra without non zero divisors of zero. Let <j> € Mult'(Tn, \\ . ||). There exists <j> € Mult'(A, || . ||) whose restriction to Tn is <j>. Proof: Let F be the completion of the field of fractions of Tn with respect to (f>. Then x\,..., Xq are algebraic over F. Let A' = F[xi,... ,Xq 1. Then A is isomorphic to a subring of A' through an isomorphism which induces the identity on T„. By Theorem 3.8 there exists a unique absolute value (j) extending that of F to A' and then, considering A as included in A', 4> defines on A an absolute value cf> extending <$>. Let m = maxj<j< 9 4>(xi), let A G K be such that |A| < ^ , and let y, = Ax; (1 < i < q). Let 9 be the canonical surjection from Tn{X\,..., Xq} onto A inducing the identity on Tn and such that 9{Xi) = y% (\ < i < q). Since 4>(yi) < 1, and since belongs to Mult(Tn, || . ||),
(32.1)
Now, let t € A. By definition ||i|| = inf{||y|| | 6{Y) =J}. So, given Y e Tn{X1,..., Xq} such that 9{Y) = t, we have 0(t) =
194
Ultra-metric Banach Algebra
considering such a Y we obtain 4>{t) < inf{||y|| | 6(Y) = t} = \\t\\, and therefore
Jacobson Radical of Affinoid A Igebras
195
family of compact subsets {Multm(B, \\ . \\F), F £ T(ip)} is a base of a filter because given F, G £ T(tp), (Multm(B, \\ . \\F))n(Multm(B, || . || G )) contains Multm(B,\\ . | | F u G ) . Let Y = ClFeT^) Multm(B, || . | | F ) . Then Y is not empty. For every x £ J3, we put i/(x) = sup{y>(x)| <^ £ Y}. Then we have v(x) < inf{||x|| F | F £ T(0)} Vx £ 5 . But inf{||x|| F | F e T(ip)} = tp{x) Vx £ B. Consequently, we have u(x) < ip(x) \/x £ B.
(32.2)
Now, we fix x £ B such that v(x) < 1. Since Y" is compact, we can easily construct in Mult(B) an open neighborhood W of Y such that ip(x) < 1 V{x) < 1. Thus v{x) < 1 implies i/>(x) < 1. Consequently, since, by Lemma 6.1, ip and v are two semimultiplicative semi-norms, we can conclude that ip(x) < v{x). So, by (32.2) the two semi-norms are equal. As a consequence, ip lies in Y. Then, for every F £ T(ip), ip lies in Multm(B, \\ . \\F). Since Multm(B, \\ . \\F) is a subset of Multm(B), this finishes proving that Multm(B) is dense in Mult(B). Now, since A is a quotient of a topologically pure extension Tn = K{Xi,..., Xn} by an ideal J, we have a canonical homomorphism 6 from T n onto A. Putting Xj = 0(Xj), 1 < j < n, the ii'-subalgebra of finite type B =K\ dense in A. Putting F = {xi,... xn}, the norm of A is equivalent to || . || F . We now take
The nilradical of A is equal to the Jacobson radical.
Proof: Let 1Z be the Jacobson radical and let x £ 1Z. Let J be a prime ideal of A, let B = £ and let 9 be the canonical surjection from A onto B.
196
Ultrametric
Banach Algebra
Since B has no non zero divisors of zero different from 0, by Corollary 32.2 it admits a continuous absolute value cj>, hence
The semi-norm || . || si is a norm if and only if A is A is a Jacobson ring.
Proof: Let J be a prime ideal of A and let 9 be the canonical homomorphism from A onto the quotient algebra 5 = 4 . Since B has no non zero divisors of zero, its radical is {0}, and since it is a fC-affinoid algebra, the intersection of all its maximal ideals also equals {0}. But the maximal ideals of B are the images by 6 of the maximal ideals of A containing J. Consequently, we have 1
J = e-
(o) = 9-1l
f| ri
\HeMax(B)
n\ J
= n («)H£Max(B)
n
M
-
JCM,M€Max(A)
Thus J is equal to the intersection of the maximal ideals that contain it.
□
Chapter 33
Salmon's Theorems
The chapter is designed to show that topologically pure extensions are factorial, a theorem due to Paolo Salmon [45]. Definitions a n d notation: Recall that U = d(0,1), M = d(0,1") and the residue class field of K is K.. Here, Tn denotes the topologically pure extension L{Xi,..., Xn} and we put Vn = {x € Tn \ \\x\\ < 1}. Let f(Xi,... ,Xn) € Tn. We denote by / the residue class of / in jjfe-Then / is said to be k-regular if it is of the form eg with c € K and g G Vn such that g is monic as a polynomial in X^. Lemma 33.1: other inTn{Y}.
Let P, Q e Vn[Y] be monic polynomials associated to each Then P = Q.
Proof: By hypothesis there exists u invertible in Tn{Y} such that Q = uP. By Theorem 31.14 u is of the form I + h with I e K, he Tn{Y} and \\h\\ < \l\. Since the Gauss norm is multiplicative, and since both P, Q axe monic, we have ||P|| = \\Q\\ = 1 and then ||u|| = |/| = 1. Suppose P ^ Q. Since both P, Q are monic and associated in jjj^t-Xn], they have same degree, and therefore we check that I = 1 = u. Consequently, u is of the form 1 + / , with / e Tn{Y} and ||/|| < 1. Therefore Pf = Q - P. Let A € K be such that ||A(P - Q)\\ = 1. Then the degree in Y of X(P - Q) is at most q — 1. But since P is monic, and since ||A/P|| = 1, the degree in Y of XPf is at least q, a contradiction. Hence P = Q. □ L e m m a 33.2: Let f € Tn be n-regular. There exists one and only one monic polynomial P e V^_i[X„] associated to f inTn. 197
198
Ultrametric
Banach Algebra
Proof: Since / is n-regular of order q, it is of the form YITLo aj-^n' w ^ n Oj € V„_ 1 and aq invertible in Vn~i- Hence by Corollary 7.11 / is associ ated in Tn to a monic polynomial in Xn. Suppose we can find two monic polynomials P and Q associated to / in Tn. In particular, Q is associated to P in Tn. Without loss of generality we can clearly assume that both P, Q belong to Vn. Then by Lemma 33.1 they are equal. □ Definition: Let / € Tn be fc-regular. The unique monic polynomial P e UK{Xi,...,Xk-i,Xk+i,---,Xn}[Xk], associated to / in Tn will be called the k-canonical associate to f ■ Let / € Tn be fc-regular and let f = eg with g the /c-canonical associate to / . By definition, the degree q of g in Xk is such that |ao,...,o,g,o,...,o — 1 | < !. laji,..,jfc-i,WM-i,..,j'J < * V ( i i , . . . , j „ - i ) ^ ( 0 , . . . , 0 ) , and \aju...jj < 1 whenever j n > q. The integer q will be called the k-order of f. Theorem 33.3: Let f e Tn be k-regular of k-order q. There exists a unique monic polynomial P(Xk) & K{Xi,... ,Xk-\,Xk+i,. ■ ■ ,Xn}[Xk] which is associated to f inTn. Moreover, the degree of P in Xk is q. Proof: Without loss of generality, we can obviously assume that k = n, what we do for convenience. We put T n _i = E and Y = Xn. Without loss of generality, we can also assume that the coefficient ao,...,o,<j of V in / is 1. Let q be the n-order of / . In E{Y} f is of the form ^f=0}jY^ with I i m ^ o o / j = 0, H/j-ll < 1 Vj e N, / , = 1 and \\fj\\ < 1 Vj > q. Then / is a monic polynomial LI of degree q, hence by Corollary 7.11 there exists Q e E[Y] such that Q = J and h e £ { F } such that 7i = 1 and / = Q/i. Since Q = / , the coefficient Z of Y 9 in Q is of the form 1 + e with ||e|| < 1, therefore by Theorem 6.13 Q is invertible in the Banach K-algebra Tn. In the same way, since h = 1, h is invertible in Tn. Consequently, the ^-degree monic polynomial P — l_1Q is associated to / in Tn. Now, we will show such a factorization is unique. Let wS be another factorization of / , with w invertible in Tn and S a monic polynomial in E[Y]. Thus in E[Y] P and S are two monic polynomials associated to each other. Consequently, by Lemma 33.1 they are equal and this shows the uniqueness of the polynomial P. □ Theorem 33.4: Let / € T n be k-regular of k-order q and factorize in Tn in the form /1/2. Then, each fj is k-regular of order qj (j = 1, 2), such that q = qi + q2. Proof: Without loss of generality, we can assume that / , f\, fi lie in Vn and that / is monic of degree q. Then / = fxf2Putting
Salmon's
Theorems
199
qj = deg(fj(Xk)), (j = 1,2), we have q = qi + <72- Since / is monic, each / has a coefficient of degree qj in Xk which is constant, hence each fj is fc-regular, of/c-order qj. □ Theorem 33.5: Let f E Vn-i[Xn] be monic and irreducible in Then it is irreducible in Tn^i{X}.
Tn-i[Xn).
Proof: Suppose / is not irreducible in Tn. By Theorem 33.4, there exists ?i-regular elements / i , ^ e T n such that / = /1/2, and therefore by Theo rem 33.3 there exists a unique monic polynomial g\(Xn) e Tn-\[Xn] asso ciated to /1 in T„ and a unique monic polynomial g2(Xn) G r„_![X n ] associated to f2 in Tn. Then g\g2 is a monic polynomial in Xn and is associated to / hence it is equal to / , a contradiction to the hypothesis: / irreducible in T n _i[X„]. □ Notation: In N™ we will denote by On the order relation defined as it fol lows: given (ii,...,in), (ji,---,3n) £ N n , weput (h,...,in) < (ji,---,jn) if for certain t < n we have it-\ < jt-i and im = j m \/m — t,...,n, and we put (ii,...,in) < ( j i . . . . , jn) if either (iu...,in) < ( j i , . . . , jn) or (ii,... ,in) = (ji,..., j n ) . Then this relation is a total order relation in Nn. Proposition 33.6: Let h € Vn be such that in h, the coefficient of maxi mal index, with respect to On is equal to 1. There exists a K-automorphism ifi ofTn preserving Vn, such that ip(h) is a nxonic polynomial in Xn. Proof: Let (fci,..., fcn) be the maximum index (with respect to On) whose coefficient in h is not 0. We can clearly find t2, ■ ■. ,tn € N such that h (Xi, X2 + X\2,..., Xn 4- X[") is a monic polynomial in X\ of degree k\ + &2<2 + .. • 4- kntn. Let
200
Ultrametric Banach
Algebra
loss of generality we can obviously assume that g belongs to Vn and that in 5 the coefficient of maximal index, with respect to On, is equal to 1. Consequently, by Proposition 33.6 there exists a K-algebra automorphism ip of Tn preserving Vn, such that tl>(g) is a monic polynomial in Xn. Thus, without loss of generality, we can also assume that g~ is a monic polynomial in Xn. Then, by Theorem 33.2, g is associated to a monic polynomial P(Xn) e Vn-i[Xn]. Let q = deg(P). Suppose that P admits a factorization in V n - i ^ n ] in the form Q1Q2, with Qi{Xn) = Y,%oaj,ixl (* = 1>2)Then qi+q2 = Q and aqiaq2 = 1. Since g is irreducible in T„, so is P, hence one of the Qi is invertible in Tn. Both polynomials — are monic and Ql®2 aqi
aqia,q2
divides P in Tn. Now, we consider Tn as E{Xn}. By Theorem 33.3 there exists only one monic polynomial in Xn associated to a n-regular series such as P. Hence one of the polynomials — is equal to 1, and therefore P is irreducible in £[X„]. Now, suppose that P divides a product fife in E{Xn} with fe, fe € E{Xn}. And according to Theorem 7.5, let i?, be the rest of the Euclidean division of /* by P (i = 1,2) in E{Xn}. Then P divides RiR2 in B. But by Theorem 7.5 the Euclidean division in J5[Xn] is induced by the one in E{Xn). Consequently, P divides .R1.R2 in jB[Xn]. And since, by induction hypothesis, E is factorial, so is -E[X„]. Consequently P must divide one of the Ri, say # 1 , in £LY„]. But since deg(i?i) < q, of course Ri = 0, hence P divides / 1 , and therefore generates a prime ideal in £[X„] and so does g in E{Xn}. Thus every irreducible element of Tn generates a prime ideal, and consequently, by Lemma 1.2 we know that E{Xn} is factorial, hence the theorem holds when m = n. □
Chapter 34
Separable Fields
When the field K is separable (in the topological meaning) we can prove that, given a if-affinoid algebra A, the topology of simple convergence on Mult(A, || . ||) is metrizable, and we construct an equivalent metric, a study first made by N. Mainetti [39]. Of course, such a metric defines a topology weaker than this defined by 6 in the case of one variable, as it was seen in Chapter 13. N o t a t i o n : In this chapter, the field K is supposed to be separable, i.e. to have a dense countable subset that we will denote by S, and we will denote by S[Zi,... Zq] the set of polynomials in q variables with coefficients in K. A will be a iiT-affinoid algebra. As in previous chapters, we will denote by || . || the Gauss norm on an algebra of polynomials K[Yi,..., Yq] and on K{Y\,... ,Yg}. In order to avoiding any confusion, || . ||0 will denote the norm of if-affinoid algebra of A. We will denote by K[x\,..., xq\ a .ftT-algebra of finite type which is not necessarily a K-algebra of polynomials, i.e. the Xj are not supposed to be algebraically independent. R e m a r k : A field which is separable is weakly valued. But a weakly valued field is not necessarily separable. For instance, it is known that the spherical completion of a C p is not separable, but is weakly valued as C p . Lemma 34.1 and 34.2 are immediate: L e m m a 3 4 . 1 : Let f e K[xi,...,xq\. The mapping v defined on (Mult(K[xx,...,xq]))2 by v{4>,ip) = min(|(/) - V(/)|oo,l) is a semidistance. 201
202
Ultrametric Banach
Algebra
Lemma 34.2: There exists a sequence (Pn)neN' in S[Z\,..., Zq\ satisfy ing Pn = Zn Vn < q, such that, given Q 6 K[x\,... ,xq] and e > 0, there exists m 6 N such that \\Q - Pm\\ < e and degh(Q) = degh(Pm) V/i = l,...,q. Lemma 34.3: A admits a dense countable S[xi,...,xq], with \\xj\\a < 1 V j = l,...,q.
subset
of the
form
Proof: Since A is a K-afRnoid algebra, it is of the form T m [j/i,..., yt] with Tm = K{Yi,...,Ym}. Consequently, the K-algebra of finite type K[Y\,... ,Ym,yi,.. .yt] is dense in A, and of course S[Xi,...,Xm,yi,...,yt] is dense in K[Yi,... ,Ym,yi,... ,yt], hence in A. By hypothesis the Yj satisfy ||Yj|| 0 = 1 V j = 1 , . . . , m and of course we can take the yj such that \\yj\\a = 1 V j = l , . . . , l D Definitions and notation: In K[Zi,... ,Zq], we consider a sequence (Pn(Zi,.. .,Zq))neN. in S[Zi,...,Zq] satisfying Pn = Zn Vn < q, such that, given Q e K[Zi,..., Z?] and e > 0, there exists m £ N such that [|Q(Z1,...,Z,)-Pm(Zi,...,Z,)||<e (with respect to the Gauss norm in K[Z\,...,
Zq\) and
d e g h ( Q ( Z i , . . . , Zq)) = d e g h ( P m ( Z 1 ) . . . , Zq)) Vh =
l,...,q.
Such a sequence will be called a a S-appropriate sequence of K[Z\,..., Zq\. On Mult(K[xi,... ,xq\), for <j>,xp 6 Mult(K[xi,... ,xq\) we put oo
2 > ( ^ ) = V -2 min(|0(P n ) - V(^n)|oo, !)■ n=l
Then £> will be called the metric associated to the sequence (Pn)nen- on Mult(K(xi,...,xq}). In the same way, in A, we consider a dense if-subalgebra of finite type of the form K[xi,...,xq] and an appropriate sequence of K[Zi,... ,Zq\. Then by Theorem 6.8 there is a natural injective map ping from Mult(A, || . || a ) into Mult(K[xi,... ,xq}) which lets us consider Mult(A, j| . |j a ) as a subset of Mult(K[xi, ■.. ,xq}). Consequently the metric on Mult(K[xi,... ,xq]) associated to the S-appropriate sequence (-Pn)neN* is defined on Mult(A, || . || a ), and then, as a semi-distance defined on Mult(A, \\ . \\a), it will be called the metric associated to the S-appropriate sequence (Pn)nen- on Mult (A, || . || a ).
Separable Fields
203
In Mult(K[x\,... ,xq)) and in Mult(A, || . || a ) an open ball of center (f> and diameter r will be denoted by B(
1) =c E ^ I W n ) ~ ^ —' n
P
^
2
n-K
71=1
e
2 And OO
.,
OO
£ ^min(|^(Fn)-^(Pn)U,l)< E n=s+l
-
^
n=s+l
Consequently, V(4>,tp) < e therefore V(, P i , . . . , Ps, n) is included in 5 ( 0 , e). Thus, a ball of center <j>, with respect to T>, is a neighborhood of (j) with respect to the topology of simple convergence, and therefore is an open set with respect to the topology of simple convergence, so the topol ogy defined by the semi-distance V is weaker (in the large sense) than the topology of simple convergence (we notice that the reasoning holds in the same way in K[x\,... ,xq) and in ^4). Conversely, we now consider a neighborhood V{<j>,Qi,... ,Qt,e), with Qj £ K[x\, ...,xt] (resp. Qj e A)) and e €]0, l[ and we will look for a ball B(4>,rj) included in V(<j>,Qi,..-,Qt,e)We first consider the case of Mult(K[xi,... ,xq}). For each j = 1 , . . . ,t, let Uj be the total degree of Qj and let p > 1 + maxi<^<, 4>(xh)- On the ff-algebra of polynomials K\Z\,..., Zq], we can consider the Gauss norm j| . ||, and then, with respect to this norm, for each j — 1 , . . . , t there exists
204
Ultrametric Banach Algebra
rij € N* such that \\Pnj(Z1,...,Zq)-Qj(Z1,...,Zq)\\pui<e-
(34.1)
and degh(Qj) = deg A (P n .) V/i = 1 , . . . ,g.
(34.2)
Let Z = m a x ( n i , . . . ,nt,q) and let n = gfj (hence Z2n < 1). Let ip e B((/>,n). Then we have ' x 5 ] - 5 min(|^(P n ) - i>(Pn)\oo, 1) < n. n=l
In particular for each n = 1 , . . . , Z we obtain ^ min(|^(P„) - ^(P„)|oo, 1) < »7
(34.3)
and therefore min(|(P„) - V(-Pn)|<x>,l) < n2rf = ^ < 1 Vn = 1 , . . . ,Z. Consequently, min(|(Pn) - V(Pn)|oo,l) = |
|0(P n ) - V(Pn)U < n 2 n = | ^ < | Vn = 1 , . . . , Z. Thus, we have proven \<j,(Pn) - V(Pn)|oo < | Vn = 1 , . . . , Z.
(34.4)
Now, consider the obvious inequality | V ( Q i ) -
+ |^(Pn,)-0(Qj)|oo. On one hand, since p > 4>{xh) V/i = 1 , . . . , £, by (34.2) we check that tf(P„,(Zi, ...,Zq)-
Qj(Zu ...,Zq))<
\\Pnj - QjWp"',
and therefore by (34.1) we obtain <j)(Pnj — Qj) < f, hence \
(34.5)
On the other hand, since x/i = P/i V7i = 1 , . . . , q, and since ip £ B(4>, TJ), we have 1 eh2 -jp\^{xh) -
< 1.
Separable Fields
205
Consequently, since p > 4>(xh) + 1 , we check that ip(xh) < p. Consequently, similarly to (34.5), we obtain |V(P»i)-^(Qj)loo
(34.6)
Then, by (34.4), (34.5), (34.6) we can conclude that \ip(Qj)-<j>(Qj)\oo < e, hence ip G V(, Q i , . . . ,Qt,e), and therefore B((f>,r]) is included in ^ ( 0 ! <3i> • ■ • i Qt, e)- This finishes proving that the two topologies are equiv alent on Mult(K[xi,... ,xq]). We now consider Mult(A, \\ . \\a) and therefore assume that Qj € A. By hypothesis, the sequence (Pn(Zi,..., ZQ))n^f^ is dense in K\Z\,..., Z g ], with respect to the Gauss norm, hence the sequence Pn(xi,... ,xq)n^n is dense K[xi,..., xq] and in A with respect to the norm || . ||0 of A. Conse quently, for each j = 1 , . . . , t there exists rij e N* such that \}Pn3 ~ Qj\\a <
€
--
(34.7)
The beginning of this proof is almost similar to that in the case pre viously studied of Mult{K[x\,... ,xq]). Let I = m a x ( n i , . . . ,nt) and let n = g|2, (hence Z2n < 1). Let \p € B(<j),r]). Then we have E l = i n* min (l^ > (- p n) "~ V'(-Pn)|oo, 1) < V- In particular for each n = 1 , . . . , I we obtain ^ min(|<^(Pn) — ip(Pn)\oo> 1) < ?7, and therefore 2
min(\4>(Pn) - ^(Pn)loo, 1) < n2r, = ^ J < 1 Vn = 1 , . . . , I.
(34.8)
Consequently, min(|0(P n ) - ^(P„)|oo,l) = \4>{Pn) - ip(Pn)\oo , hence by (34.8) o
\4>(Pn) - ^(Pn)|oo < n2r? = — < - Vn = 1 , . . . ,1. Thus, we have proven |0(P„) - V(P 1 )U < | Vn = 1 , . . . , /.
(34.9)
Then, thanks to (34.7) and (34.9), by Lemma 34.2 we know that \ip(Qj) —
206
Ultrametric Banach
Algebra
Moreover, in both cases, since the topology of simple convergence is separated, this shows that T> is a distance. □ Corollary 34.5:
Mult (A, || . || 0 ) is sequentially compact.
Corollary 34.6: compact.
In Mult(K[xi,...
,xq]), every compact is sequentially
Corollary 34.7: Let D be a closed bounded subset of K. The topology of simple convergence on Mult(H(D),\\ . \\p) is metrizable. N o t a t i o n : By Lemma 34.2 we can assume that \\XJ \\a < 1 V j = 1 , . . . , q. Then, in Mult(A, || . || 0 ) we can also define another distance by con sidering a sequence (Rn)neN of ( S f l U)[Z\,... ,Zq] which is dense in U[Zi,...,Zq] and then we put V{
mm{\
2
'
and therefore T>(<j),ip) > — ^ ^ , which shows that the topology defined by V, and therefore the topology of simple convergence is stronger than this defined by V. On the other hand, we will show that the topology defined by V is stronger than this of simple convergence, in a similar way as in Theorem 34.5. Let Qi,...,Qt e A, let N = 1 + max 0 <j
Separable Fields
207
that \\Rnj {x\, ■ ■ ., xg) — fijQj \\a < rj, so we have
tp(Rn:J(xi,...,xq)
- fijQj) < V- (34.10)
Now, since tp G B(
j
y"^i^(i?„)-^(JR„)|00<7?. In particular for each n = 1 , . . . , Z we obtain ;^-|(-R„) — 4>(Rn)\oo < Vi therefore \
< 1 < — Vn = 1 , . . . , I.
an
d
(34.11)
Consequently, by (34.10) and (34.11) we obtain \ip(fijQj) - <j>{njQj)\oo < Ne, therefore \ip(Qj) — (Qj)|oo < e an<^ finally B'(
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C h a p t e r 35
Spectral N o r m of AfRnoid Algebras
In Chapter 31 we saw that if A is reduced, its spectral semi-norm is a norm. Then it is a natural question to ask whether A is uniform. The answer is quite easy and was soon given in [49] when the characteristic p of K is 0. It is much more difficult when p ^ 0 (such a study was also made in [3]). Notation: Throughout the chapter and in the next one we denote by p the characteristic of K. Next, considering Z" as a Z-module, for all (*) = (*ii • • • ,i ra ) S Z, and q € Z we put q(i) = (qit,... ,qin)- Let f(Yu...,Yn)
= aiu...,i7Yi1
■ ■ ■ Yt
G Tn.
We put (Y) = (Xi,-.-,Yn), ( F ) « = (Y*\...,Yi*) and f(Yu...,Yn) = £ ( W e a(0(y)W. For each n & W we denote by Ln the field of fractions of Tn and A is a i^-affinoid algebra. By Corollary 31.5 A has finitely many minimal prime ideals S\,... ,Si. For each j = 1,... ,1, we denote by Aj the if-affinoid algebra ^-, by 6j the canonical surjection from A onto A,, and by || . \\Sjtj the spectral norm of Aj. Theorem 35.1: Let A be reduced. For every f € A, we have
||/IU=max(||0 j (/)|U J ).
209
210
Ultrametric Banack
Proof:
Algebra
By Theorem 31.10 we have ||/|| S j = s u p A e s j M ^ |A|, and l|0j-(/)lki=
sup
\\\, (j =
But by Lemma 1.15 spA(f) — Dlj=1spAj(0j(f)).
i,...,i).
So, our claim is obvious.
□ x
T h e o r e m 35.2: For all x € A there exists x € X(A) such that \x( )\ = ||x|| sl . Moreover, if A is an integral domain finite over Tn, then for all f eA, we have S{irr(f,Tn)) = \\f\\si . Proof: Let f € A. First, we will show that there exists \ G X{A) such that \x(x)\ = ||a;||si- When A is a topologically pure extension, the state ment is proven in Theorem 31.12. Here we first consider a iC-affinoid algebra A without non zero divisors of zero. There exists a topologically pure extension Tn such that A is a finite over Tn. By Lemma 1.18 every x € X(Tn) admits at least one extension to an element of X(A). By Theorem 31.10 the spectral norm || . || si of A satisfies ||/|| S j = sup{|x(/)| | X € X(A)}. On the other hand, by Proposition 29.4 we have sup{|x(/)| | X € X(A)} = S(irr(f,Tn)) and there exists x e X(A) such that | x ( / ) | = sup{|x(/)| | X £ X(A)}. Consequently, there exists x G X(A) such that | x ( / ) | = ll/IUi- Thus, our claim is now proven when A is an integral domain. We now consider the case when A is just reduced. By Theorem 35.1 we have maxi<j
The algebraic closure of Ln is Ln-productal with respect
to II • IfeProof: If p = 0, the claim comes from Theorems 29.1 and 29.7. So, we suppose p / 0 . By Theorem 29.10, it is sufficient to prove that L% is Lnproductal. For each k = 1 , . . . , n, we put Y^ = tfXk and Y = ( Y i , . . . , Yn).
Spectral Norm of Affinoid Algebras
211
So, we have Y^u) = X, and more generally T^'P-W = X^u\ Let 5 e L*. with a Then 5 is of the form Y%=1 rYi'M> J ' 6 i € T„ Vj = 1 , . . . , n. And
9
~h w ~h ^)p '
Let b = rij=o fy and for each j = 0 , . . . ,p - 1 we set Cj = £■, which belongs t o T n . Then p-i
Now, for each j = 0, . . . , p — 1, CLJCJ is of the form Yl,u)enn \i),3Xl, with A(j)i:/- e # , hence (a.jCj)p = X)(i)eN» ^W.j-^'* anc * thereby 6P#P = EWeN-,0<J
^ m a x ^ m a x f l A ^ . | (i) e N"}) =
max
||a,'cJ| p
0<J"
hence \\gp\\ = max 0 <j< p _i ( imfp ) • Since || . || is an absolute value on Tn, finally we have
\\g"\\ =o<j
«*=o^1(l£l)Thus, Ln is L n -productal with respect to || . || ";.
D
Theorem 35.4 Let A be reduced. Then the norm of Banach K-algebra of A is equivalent to the norm || . | | S j . Proof: First we suppose that A is an integral domain. Let Tn be a topologically pure extension of K such that A is finite over Tn. As a finite
212
Ultrametric Banach Algebra
T„-module, A admits a basis { e i , . . . , eq}. Putting M = ma,xi<j
max ||o,-||.
(35.1)
1<J<
Let Ln be the field of fractions of Tn. Of course { e i , . . . , eq} is also a basis of Ln[xi,..., x9] as a L n -vector space. By Theorem 35.3 the algebraic closure of Ln is L n -productal with respect to || . || "t. So we have a constant V > 0 such that
ll/II^S > ^ max |KI|.
(35.2)
Thus, by (35.1), (35.2) and by Theorems 35.2 we obtain y max | K | | < \\f\\fe = H/ll = U/IU < M max |K||. Thereby, || . ||s$ is clearly equivalent to || . || on A. We now consider the general case, when A is reduced but not neces sarily an integral domain. Let B = rij=i ~§r- Let 0 be the A"-algebra homomorphism from A into B defined as
\\fj\\si,j-
On the other hand, the norm || . || B , B is provided with, is a .A-quasialgebra-norm because for each j = 1 , . . . , m, we have
(s)/;lk; < PMlUMlUj
< \\g\\si\\fi\\»i,i < llsllll/.lki-
Consequently, since B is finite over A, by Corollary 30.3 every yl-submodule of B is closed with respect to its norm || . || B . In particular, as a Asubmodule, A is closed in B with respect to || . || B . Now, given / € A, we have ||/|| s i > maxi<j< m \\0j{f)\\s,,j- The inequal ity actually is an equality. Indeed, according to Theorem 35.2, we can find x £ X{A) such that | x ( / ) | = H/IU, and there does exist an index
Spectral Norm, of Affinoid Algebras
h such that Sh C Ker(x),
213
so x factorizes in the form \h ° &h> hence si,7-
Finally, it is easily seen that the spectral norm || . ||^ of B is just the product norm B has been provided with, because X{B) = Yl^jLi XiAj)Thus, we have maxi<j< m \\&j(f)\\si,j = ||(0i(/),••• ,0m(/))llf»> a n d there fore, the spectral norm || . ||s$ of A is induced by that of B. Consequently, A is complete for both norm || . \\Si and || . ||, and therefore, by Theorem 3.4 the two norms are equivalent. □ By Theorems 15.14, 32.4 and 35.4 we can state Corollary 35.5: Corollary 35.5: Let A be reduced and let f, g (E A. Then for every (j) £ Mult(A, || . ||), we have S(f*(cp),g*((/))) < \\f - g\\. Moreover, if ip? = f*{4>), ), and if
P{f) = U, T is secant with sp(f), therefore linvr |P(x)| = supX€sp^^ |A|, thereby we get the equality. Consequently, the closure of K[f] in A is equal to K{f}. Now, consider g G A such that ||/ — g\\Si < 1. Then sp(g) is included in U, as sp(f). And since diam(sp(f)) = 1, we have diam(sp(g)) = 1, hence of course sp(g) = U. Now, suppose g £ K{f}, hence g is of the form Y^=o^m.fm, with |&o — 1| < 1, |6mj < 1 Vm € N. Without loss of generality, we can clearly assume that b0 = 1. Thus, g is of the form / + h(f), with h(Y) G K{Y}, and \\h\\ < 1. Let 1{Y) = Y + h(Y). Since K{Y} is identical to H(U), and since \\h\\ < 1, it clearly satisfies the hypothesis of Theorem 17.12, hence l(Y) is a strictly injective analytic element in U, making a bijection from U onto U. Consequently we can apply Theorem 17.14 showing that, if we put Z = l(Y), then Y is a strictly injective function in Z, hence of the form Z + t(Z), with t G K{Z} and ||i|| < 1. Consequently, applying this to / , we see that / belongs to K{g}. □
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Chapter 36
S p e c t r u m of an Element of an Affinoid Algebra
In [40] Y. Morita proved that the spectrum sp(x) of an element x of an affinoid algebra is an affinoid subset of K. However, the proof involved considerations on Tate's acyclicity Theorem [49], and spherical completion of the field K. Here we give a more simple explanation to show that such a spectrum is an affinoid set. Notation: Keeping notation introduced in Chapter 32, here we put ||(n,---,«n)||oo = m a x ( | i i | 0 0 , . . . , | i n | 0 0 ) . In Kn, (0) denotes (0, . . . , 0 ) . Proposition 36.1: Let F(X) = Y11=o fi-%-1 £ r„[X] be monic and irre ducible in Tn\X}. The set of A e K such that F{\) is not invertible is an affinoid subset of K. Proof: For each i = 0 , . . . ,q, we put /, = X ^ e N " a(i)>iT<"J) ■ So> F(X) is of the form J2(j)eN™ fyjjC-X')^^, whereas each 6^) lies in K[X] and sat isfies deg(by)) < q V(j) ^ (0), and deg(6(0)) = q because F is monic. Moreover, since F is irreducible, we notice that there exist no a € K such that 6(j)(a) = 0 V(j) e N". Further, since lim||(.,)||o__+00a(j)>i = 0 Mi = 0, . . . , g , we have lim||Q-)||=io_>_|_00 ||4>(j)|| = 0 . For every A € if we put B(X) = sup(j) eN „ \b(j)(\)\. By Theorem 31.14 F(A) is not invertible in Tn if and only if |6 (0) (A)| < sup 0 - )7 , (0) \b{j}\. Let D = {A e K | |6(0)(A) < su P(j)^(o) l^(i)(^)]} an-d s = suPfj)^(o) 11^0)11- We first notice that D is bounded. Indeed, let d(0,p) be a disk containing all zeros of 6(0), and let r = max(p, s, 1). Let A € if be such that |A| > r. Since all zeros of 6(o) lie in d(0,r) and since 6(0) is monic, by Corollary 12.8 we have l^(o)(^)l — \M9- On the other hand, for each (j) ^ (0), if t = degfr^, 215
216
Ultrametric Banach
Algebra
we have the basic inequality |6Q)(A)| < ||6(j)||-|A|* < ||fr(j)||-|A|9 1 because |A| > 1. Consequently, \b{j)(X)\ < s]^'1 < |A|«, hence A $ D. This shows that D C d(0,r). Now, we will show that there exists a finite subset S of N" \ (0) such that su P(j)^(o) \b(j)W\ = suP{j)es lfyj)MI V A e L>. Indeed, suppose this is not true. There exist injective sequences (A77l)TneN in D and (km)me^ in N" such that sup(j)^(0) \b(j)(Xm)\ = |^(fc)m(Am)| V T H E N . Since D is bounded, by Proposition 10.7 we can extract from the sequence (Am) a subsequence thinner than a circular filter T. Thus, without loss of generality, we can directly assume for convenience that the sequence (Am) itself is thinner than T. Since the sequence ((k)m) is injective, by definition of Tn we have limm_>O0 ||6(fc)m|| = 0, therefore (O)) = 0, hence ipjr is not an absolute value on if [A-]. Consequently, by Theorem 13.1 J7 is the filter of neighborhoods of a point a € D, hence 6(o)(a) = 0. But as we noticed above there exists (h) e N n such that b^){a) ^ 0, hence the inequality \b(h)(x)\ > \b(o)0*01 holds in a neighborhood of a, a contradiction to the hypothesis: sup(j-)yS(0) \b(j)(Xm)\ = |&(/t)m(Am)| V m e N. Thus we have proven this existence of 5. Now, for each (j) € S, we set D{j) = {A € K | |6 (0) (A)| < |b(j)(A)|}. Then D = U{j)eSDU). Since deg(6(0)) > deg(6 (j) ) V(j) ^ (0), by Theorem 12.10 each set D^ is affinoid and therefore by Lemma 8.14 so is D. □ Theorem 36.2:
For all x £ A sp(x) is an affinoid subset of if.
Proof: Assume that A is finite over Tn and let x £ A. First suppose that A is an integral domain and set F(X) = YH=o fiX% = irr(x,Tn). For each i = 0 , . . . , q, we put fi=
E (j)£N"
aw)iirW,P(A-) = ^o(o),iX
Now let A e K and G A P O = irr{x-A, T n ). One checks that GA(0) = F(A), so by Lemma 1.7 x — A is invertible in A if and only if F(A) is invertible in Tn. Thus, sp(x) is equal to the set of A e if such that F(X) is not invertible, and then by Proposition 36.1 this set is affinoid. We now suppose that A is not necessarily an integral domain. By Corol lary 31.5 A admits finitely many minimal prime ideals, V\,..., 7 \ . For each j = 1 , . . . , k, let A3-, — ■£: and let 8j be the canonical surjection from A to Aj. Each algebra Aj is a if-affinoid algebra without non zero divisors of zero, hence sp{9j{x)) is an affinoid set. Then by Theorem 1.15 we have
Spectrum of an Element of an Affinoid Algebra
217
sp(x) — Uj=1sp(8j(x)), and therefore by Lemma 8.14 sp(x) is an affinoid set. So our claim is proven in the general case. □ Theorem 36.3: Let A be a reduced K-affinoid algebra without non trivial idernpotents, whose spectral norm is denoted by \\ . ||, let y e A, let a e S PA(V), letr e \K\C\]0, \\y - a\\] and let S = {y - b \ b <= d(a,r~)}. Let c e C(0,r), and let B = /c_rv^a.)z)Aiz} ■ Then A is canonically identified with a K-subalgebra of the K-affinoid algebra B, S~XA is dense in B, d(a,r~) is a hole of sp^iy) and for every x € X{A) satisfying \x{y)\ > r, X has a unique extension to an element of X{B). Moreover, if r < \\y — a\\, then for every (f> £ Mult{A, || . ||) satisfying
r
\yJ
<
©
hence sup X€X(A)
H
i.e.
jnf
\x(y)\
and therefore d(0,r ) is a hole of spA{y)- Consequently, we can just take A = B. Now, suppose that D n d(0,r~) ^ 0. Let || . || B be the norm of B, let jj . || fi be the spectral semi-norm of B and let 0 be the canonical surjection from A{Z} to B. Then B is a if-affinoid algebra and the restriction of 8 to A is an injection. Since y is clearly invertible in B we have O(-) = 8(Z), hence in B we have
O'-IKf)
hence B = A{Z} = A{^}. For each b e d(0,r ), we see that the series X^^Lo ^ + r converges to —^. So S~1A is included in B, and consequently d(0,r~) n sps{y) — 0- On the other hand, since B M^}: S-*A is
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Ultrametric Banach
Algebra
obviously dense in B and more precisely, if we denote by So the set of yn, n e N, then SQ1A is dense in B. Consequently, if an element <j> of Mult(A: || . ||) has continuation to an element
But it is easily checked that \!>{h) + e 0(/Q + 2e V(/») " #&) - e ' so we obtain
4>(y)t ~ ip(y)i\r-e)
''
On the other hand, since e <
Spectrum of an Element
of an Affvnoid Algebra
219
which extends ip to B. Therefore, in B we have
= 1>(f) < hence by (36.1) we obtain
4>{y)q ~
r
Y4>{h) + 2e
( \r-eJ ^ )
4>(h)-e '
This last relation is true for all e G]0, min(
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Chapter 37
Krasner-Tate Algebras
In this chapter we consider a Banach if-algebra which is isomorphic to both a K-algebra, of analytic elements H(D), and a JsT-affinoid algebra. Such a Banach K-algebra called a Krasner-Tate algebra [18]. The issue is to characterize Krasner-Tate algebras among algebras H(D) on one hand, and among affinoid algebras on the other hand. Notation: Henceforth, for each ideal J of T n , we denote by A(J7") the set of maximal ideals of Tn containing J, and similarly, for every h € T„, we put A(/i) = A(hTn). Theorem 37.1: Let P, Q e K[X) be relatively prime polynomials. Then the ideal (P(X) - YQ(X))K{Y,X} of K{Y,X) is equal to its radical. Proof: By Theorem 33.7 K{Y,X} is factorial. Suppose its factorization into irreducible factors contains a factor g with a power q > 1. Then g divides gy — = ~~ Q(X)- Consequently, g divides both P and Q in K{Y, X}. Let M. be a maximal ideal of -ftT{Y, X } such that g € M and let x € X{K{Y,X}) be such that J^er(x) = M. Let x P O = a. Then we have x(P(X)) = P(a) = 0, x ( Q ( ^ ) ) = <3(a) = 0, a contradiction to the hypothesis " P and Q relatively prime". This shows that P(X) - YQ(X) has no multiple irreducible factors in its decomposition. But since K{Y, X} is factorial, then by Lemma 1.3 the ideal (P(X) - YQ{X))K{Y,X) of K{Y,X} is equal to its radical. □ Theorem 37.2: Let D be affinoid. Let P, Q € K[X] be relatively prime polynomials such that deg(P) > deg(Q), 5 € R(D), r;(D) = U, 221
222
Ultrametric Banach Algebra
( 5 ) (^) algebra
=
^'
Then the K-algebra H(D) is isomorphic to the K-affinoid K{Y,X} (P(X)-YQ(X))K{Y,X}
which is also isomorphic to K{h}[x], with h = QT^TProof: Since deg(P) > deg(Q), in H(D) x is integral over K[h] and satisfies P(x) — hQ(x) = 0. Now, since spn(D)(h) = h(D) = U, the norm || . \\D induces the Gauss norm on K\h\. So, by Theorem 35.6 the closure of K\h] in H(D) is isometrically isomorphic to T\ that we can denote here by K{h}. Without loss of generality we can obviously assume that ||X||D < 1. Thus in H(D), x is integral over K{h} and satisfies P(x) — hQ(x) = 0. Let B = K{h}[x\. By Corollary 31.4, B is a K-affinoid algebra isomorphic to a quotient of K{Y, X} by an ideal J that contains P(X) — YQ(X) and the canonical surjection 9 from K{Y,X} onto B satisfies 6(Y) = h, 6{X) = x. On the other hand, by definition B is a if-subalgebra of H(D). For every a € K \D since \h(a)\ > \\h\\o, by Theorem 6.13 h(a) — h is invertible in B, hence R(D) C B c H{D). In particular sps(x) = spH(D)(x) = D. We notice that H(D) is reduced, hence so is B. Consequently, I is equal to its radical and therefore, by Theorem 35.4, B is complete for its spectral norm. But since sps(x) = spn{D){x) w e c a n easily deduce that the spectral norm of B is induced by the norm || . \\D- Indeed, given \ £ X(B), since x(x) € D, x has continuation by continuity to H(D), and therefore belongs to X(H(D)), hence for all / G JB, the spectral norm || . ||^ of B satisfies H/llf. = ll/H D . Therefore, since R(D) C B, we have B = H(D). Since P(x) - YQ(X) e X, of course A(J) c A(P(X) - YQ{X)). Now, consider a maximal ideal M. which contains T and let \ e <;'::'(-^{^>^}) be such that Ker(x) = M. Let X P O = a ! X(^) = /3- Since a; - a is invertible whenever a ^ D, we see that a lies in D, and then x(Y) = ft(a), thereby P(a) - h(a)Q(a) = 0, hence P(X) - YQ(X) belongs to M. Consequently, we have A(J) = A(P(X - YQ(X)). Therefore, the ideals (P{X) - YQ(X))K{Y,X} and I have the same radical. Thus, on one hand, by Proposition 37.1 (P(X) -YQ(X))K{Y,X} is equal to its radical, on the other hand, since H{D) is reduced, I also is equal to its radical, hence (P{X) - YQ(X))K{Y, X}=X. D T h e o r e m 37.3: if D is affinoid.
A K-algebra H(D) is a Krasner-Tate algebra if and only
Krasner-Tate
Algebras
223
Proof: Indeed, if H(D) is a Krasner-Tate algebra, then by Theorem 36.2 sp(x), i.e. D, is affinoid. Conversely, suppose that D is affinoid. By Theorem 12.11 there exists h G R(D) such that h(D) = U, D = h~l(U), with h = £, P, Q relatively prime and such that deg(P) > deg(Q). Con sequently, Theorem 37.2 shows that H(D) is a Krasner-Tate algebra isomorphic to
D
{P_YQ)'K{Y,X}-
Corollary 37.4: Let D be affinoid and let / 6 H(D). to sp(f) and is affinoid.
Then f(D) is equal
Proof: Indeed by Theorem 37.3 H(D) is a K-affinoid algebra, hence by Theorem 36.2 sp(f) is an affinoid subset of K, but since if-affinoid algebras are noetherian, by Theorem 19.2 we have sp(f) = f(D). □ Another way to describe Krasner-Tate algebras H(D) when the set D is not infraconnected consists of focusing on the infraconnected components of D, in order to show a product of Krasner-Tate algebras. Theorem 37.5: Let D be affinoid and let D\,... ,Dq be its infraconnected components. For every j = 1,... ,q, let Pj, Qj € K\X] be relatively prime and satisfy:
geH(D,),g(^) = ^(|(Di))"V)=^. Letl=W)=l{Pi-YQj)K{Y,X). Then H(D) is isomorphic to —^—'-. Proof: Without loss of generality we can obviously assume that ||O;||D < 1. For each j = 1 , . . . , q, we denote by Ij the ideal (Pj — YQj)K{Y, X} of K{Y, X}. By Theorem 19.12 H(D) is isomorphic to the K-algebra H(D{)x . . . x H(Dq). Each set Dj is an infraconnected set, hence by Theorem 37.2 H(Dj) is isomorphic to the Krasner-Tate algebra without non zero divisors of zero —ij;—i. On the other hand, since D^ flfl; = 0 Vfc ^ I, we have A(/fe + Ifi = 0, therefore Ik + Ii = K{Y,X} Vfc ^ I. Consequently, by Lemma 1.1 the direct product K{Y,X} (P1-YQ1)K{Y,X}
•••
K{Y,X} {Pq-YQq)K{Y,X}
is isomorphic to —^y—L, which completes the proof.
□
R e m a r k : Given a Krasner-Tate algebra (P_YO)'K
224
Ultrametric Banach Algebra
jFsf-affinoid algebra A =
K{Y,Z} (Z 3 + AZ2Y + 5ZY2 + 2Y2 + Z + 1)K{Y, Z}'
Now, let X = Z + Y. Then A becomes K{Y,X} [(X 3 + X + 1- Y(-X2
+ 1)]K{Y, X}'
Thus, 3A appears as a Krasner-Tate algebra H{D) where D = /i _1 (C/) with u _ x +x+i 1 -
-xi+1 ■
Now, consider the afhnoid algebra B = (z^-YZ)KiY z) • Apparently, it looks like a Krasner-Tate algebra, except that the polynomials P — Z2 and Q = Z are not relatively prime. Of course, B has non zero divisors of zero. Putting X = Y — Z, we obtain B = IXY)K\Y x\' s o w e c a n c n e c k that such a /^-algebra has no idempotent different from 0 and 1. Suppose B is a Krasner-Tate algebra H(D). Since B has n non trivial idempotent, D is infraconnected, hence by Theorem 19.2 we know that B not noetherian. Consequently, it can't be a Krasner-Tate algebra. Lemma 37.6: Let A = —*- 1'["' " ; be a K-affinoid algebra without non zero divisors of zero such that the field of fractions of A is a pure degree one transcendental extension K(t) of K. Let 6 be the canonical surjection from K{Yi,..., Y„} onto A. For each j = 1 , . . . , n, let yj = 9(Yj) = hj(t), with hj(t) G K(t). Ift belongs to A, then sp(t) contains no poles of hj and no points a such that /i'(a) = 0 Vj = 1 , . . . , n . Proof: Suppose that t € A and let D = sp(t). Suppose that for some k £ { l , . . . , n } , /ifc admits a pole b in D. Let h^ = Q W , with P, Q relatively prime and let \ e X{A) be such that x(i) = b. Then x(P(t)) = x(vk)x{Q(t)) = x(Vk)Q(b) = 0, a contradiction to the hypothesis: P, Q relatively prime. Consequently no poles of the hj lie in sp(t), whenever j = 1 , . . . , n. On the other hand let F(Yi,..., Yn) € K{Y\,..., Yn} be such that t — F(yi,..., yn). Consider the mapping G from sp(t) into Kn defined as G(a) = F(hi(a),..., hn(a)). Then G is the identical mapping on sp{t). If there exists a e sp(t) such that h'j(a) = 0 \/j = 1 , . . . , n, then we have G'(a) = 0, a contradiction since G is the identical mapping on sp(t). □ Theorem 37.7: The K-affinoid algebra A = gral domain but is not integrally closed.
IY2-X3)'K/X
Y\
*5 an inte
Krasner-Tate
Algebras
225
Proof: Let 6 be the canonical surjection from K{X, Y} onto A, let x = 0(X), y = 0(y). Suppose that A is has non zero divisors of zero. Since by Theorem 33.7 K{X,Y} is factorial, there exist non invertible elements h, f2 € K{X}[Y] such that fif2 = Y2 ~ X3. Then, by Theorem 33.3 each fj is associated in K{X,Y} to a monic polynomial in Y, Pj{Y). Since both Pj are not invertible in K{X,Y}, each Pj is of the form Y + Sj(X), with Sj(X) € K{X}. Consequently, we have Y2-X3 =Y2+Y(S1(X) + S2(X))+S1(X)S2(X), hence by identification, 2 3 3 52 = — Si, and S = X . But X is not a square in K{X} because it has a zero of order 3 at 0. Consequently we are led to a contradiction proving that Y2 — X3 is irreducible in K{X,Y}, and therefore A is an integral domain. Suppose now that A is integrally closed. Let B = K[x,y\. The field of fractions of B is clearly the field K{u) with u = j . And since u is integral over B, it belongs to A. On the other hand, both x, y are not invertible in A, hence neither is u. But putting x = g(u), y = h(u) we have g'(u) = h'{u) = 0, a contradiction with regards to Lemma 37.6. □ Corollary 37.8: The K-affinoid algebra A = (yi-x'^kix Krasner-Tate algebra.
Y\ *S
rao
*
a
Proof: Indeed, a Krasner-Tate algebra without non zero divisors of zero is a principal ideal ring. □ Proposition 37.9 will be useful in further consideration in chapter 39: P r o p o s i t i o n 37.9: Let r e]0,l[n\K\, let E = d(0,r), F = d(l,r) and let D = E U F. Let u be the characteristic function of E in H(D), and A = uxiHiD\ ■ Then A is a non reduced K-affinoid algebra containing a dense Luroth K-algebra. Proof: Since D is an affinoid subset of K, H(D) is a X-affinoid algebra, hence so is A. Let 8 be the canonical surjection of H(D) onto A. Then, 9(ux) is a nilpotent element of A, hence A is not reduced. But since H(D) is a Krasner-Tate algebra, it is of the form if{/i}[x], with h £ K(x),deg(h) > 0, and h{D) = U, D = h~l(U), so K[h,x] is a dense X-subalgebra of H(D). Let T = 0(h), £ = 9{x). Then Ker{6) n K{h} is equal to {0}, and so is Ker(6) n K[h,x]. Consequently, K[T,£] is a Luroth if-subalgebra of A isomorphic to K[h,x], and is dense in A. □
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Chapter 38
Universal Generators in Tate Algebras
It is possible to find another way of characterizing Krasner-Tate algebras among isf-affinoid algebras, by using universal generators, a study made in [22]. Proposition 38.1: Let A be a K-affinoid algebra without non zero divi sors of zero, of the form K{Y}[x\, with x integral over K{Y}, let n
F(Y,X) = J2MY)V j=o
n-l n
= X
+ ] T ( f l i + hY + Y2Ci(Y))Xl i=0
= irr(x, K{Y}) e
K{Y}[X]
and suppose that \\F\\ = 1. Let P(X) = Xn + YZZo
\P(X)\ < \Q(X)\).
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Ultrametric Banach
Algebra
Proof: For each A € if, we denote by GX(X) = Xn + J2"Zo 9\j(Y)XJ the minimal polynomial of x — A over K{Y}. We can easily check B that gx,o = F(Y,X) = E " = o / j ( y ) A J Y Theorem 19.2 K{Y} is principal, so the equivalence between (i) and (ii) are immediate. Now, by Theorem 17.6 F(Y,X) is invertible in K{Y} if and only if |P(A)| > max(|Q(A)|,sup 0 < i < n _ 1 (||e j (r,A)||). In the same way, F(Y,\) is irreducible in K{Y} if and only if |P(A)| < |Q(A)| and |Q(A)| > sup 0 < •<„_!(||CJ(Y", A)ID). Thus, Condition (Hi) is equivalent to "for every A G sp(x), we have |P(A)| < |Q(A)| and |Q(A)| > s u p o ^ ^ G M Y , A)|Q", which shows that Condition (Hi) is equivalent to Conditions (ii) and (iv). Now, suppose the four conditions satisfied. Since every maximal ideal of A has codimension 1, given A G K, it lies in sp(x) if and only if there exists x £ X(A) such that x(x) = A, i.e. there exists fx G U such that F(fi,X) = 0. Let ft = A(F), let ft' = A(P(X) - YQ(X)), and let n be the canonical projection from U x U to U denned as 7r(/i, A) = A. Then ?r(ft) = sp(x), and Tr(ft') = {A | |P(A)| < |Q(A)|}. On the other hand, given A G K, x — X is not invertible if and only if (by (Hi)) if |Q(A)| > |-P(A)|, i.e. 7r(ft') = sp(x) and this shows the last statement. □ Remark: Proposition 38.1 gives a piece of answer to this general problem: given the quotient of K{Y,X} by two different principal ideals, are these quotients be isomorphic when the generators are very close? Lemma 38.2: Let A be a K-affinoid algebra without non zero divi sors of zero, of the form K{Y}[x\, with x integral over K{Y} and let F(Y,X) = irr(x, K{Y}) e K{Y}[X\. Let 1 be an ideal of K{Y,X} con taining F(Y, X) and included in infinitely many maximal ideals of A. Then 1 = F(Y, X)K{Y, X}. Proof: Without loss of generality, we can assume that ||x|| is small enough to assure that ||F(y,X)|| < 1. Since A is an integral domain, it is isomor phic to -W(Y\X)K{Y\[X\ > a n d F(Y>X) i s irreducible in K{Y}[X}. But then, by Theorem 33.5 it is irreducible in K{Y,X}, hence F(Y,X)K{Y,X} is a prime ideal of K{Y,X}. Since by Theorem 32.7 K{Y,X} is a Jacobson A n d since K Y x ring, then F(Y,X)K{Y,X} is equal to C\M€A(F)Mi>} is noetherian, the family of infinite sets A(J) contained in A(I) obvi ously admits minimal elements. Let N be such a minimal element, and let S = C\fi4eN^Then S contains I. Suppose that two elements / , g G K{Y, X) satisfy fg G S,f $. S,g £ S. Then the ideal generated by / and S strictly contains V, hence A(/) does not contain L, therefore
Universal Generators in Tate Algebras
229
A(f)nN is finite. Similarly, A(g)(~)N is finite, and so is A(g)t~)N. But this is a contradiction to the hypothesis fg € S. Consequently, <S is a prime ideal of K{Y, X} which contains X. Let S' = S n K{Y}. Suppose first S' ^ {0}. Since S' is a prime ideal of K{Y} and since K{Y} is isomorphic to H(U), by Theorem 19.2 S' is of the form (V — a)K{Y}, with a G U. For every Ai € N, the unique x £ ^ ( / ^ { y , X}) such that Xer(x) = A4 must satisfy F(a, x(x)) = 0, an equa tion that only has finitely many solutions, and therefore L is finite, a contra diction. Consequently, we have <S' = {0}. Since X C <S, there exists a canon ical homomorphism 6 from A onto —^—*-. But since SnK{Y} = {0}, #(:r) is integral over K { F } and then irr(0(x), K{Y}) divides F in K{Y}[X\. But since F is irreducible in K{Y}[X], irr(9(x),K{Y}) is just F(Y,X), hence 5 = F(Y, X)K{Y, X}, and therefore X = S = F(Y,X)K{Y,X}.
a T h e o r e m 38.3: K is supposed to have characteristic 0. Let A be a K-affinoid algebra without non zero divisors of zero of the form K{Y}[x\ where x is a universal generator of A. Let n-l
F(Y, X)=Xn
+ ^(ai
+ biY +
Y2el(Y))Xi
i=0
= irr(x,K{Y})
e
K{Y}[X]
and suppose that \\F\\ = 1. Let P(X) = Xn+ Y™~o o^X1 let Q(X) = - YTCo hiXi, and let D = sp(x). Then A is isomorphic to both H(D) and (p/x)-YQ('x))KiYx\ ■ Proof: Since A is an integral domain, by Theorem 35.4 we can consider that the norm of A is its spectral norm |j . | | S j . By Condition (iv) in Proposition 38.1 Q has no zero in D, so § belongs to R(D). Then Q{x) doesn't lie in any maximal ideal of A because if there existed x £ %{A) such that x{Q(x)) — 0> then we would have simultaneously Q(x(x)) = 0 and x(x) € D. Consequently, Q(x) is invertible in A. Let Z = Q S - For each a € D, we put
ua{Y) = E7:^fY)
6 K{Y).
By Relation (iv) in Proposition 38.1 we have
E:=O^(Y) Q(a)
< 1 Va € D.
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Ultrametric Banach
Algebra
t(y) Let f(Y) = E " = £f'f - Consider Q(x)
lx(/)l
X
e # ( 4 ) and let x(ar) = a. Then
£r=~o^(x(y)) Q(«)
< -
ESL^f(y) Q(a)
< 1.
Consequently | x ( / ) | < 1 Vx e ^(-A)- Therefore by Proposition 31.9 we have ||/Q0||ai < 1. On the other hand, we have Z — Y — f(Y) hence by Theorem 35.6 K{Z} is a topologically pure extension of K and of course its residue class ring is equal to this of K{Y}. Consequently by Proposition 30.6 A is finite over K{Z}. Let B = K{Z~\\x\. Since P, Q are relatively prime, by Theorem 37.2 B is Krasner-Tate algebra, and since it has no non zero divisors of zero, it is a principal ring, hence in particular, it is integrally closed. Of course, A is finite over B so, by Theorem 1.19 x is a spectrally injective element of B. As a Krasner-Tate algebra B is semisimple. Consequently, by Theorem 1.19 we have A — B. Now, by Theorem 37.2 X{^}[x] is the Krasner-Tate algebra H(D'), with D' = Z~X(IJ). And by Proposition 38.1 we have D' = D. □ Proposition 38.4: Let A be a K-affinoid algebra without non zero divi sors of zero of the form K{Y}[x] where x is a spectrally injective element and let F(Y,X) = irr(x,K{Y}) e Jsr{Y'}[.X']. If x is not a universal gen erator of A, then we have p ^ 0, | y = 0, and further, there exists t € N such that K{YP }[x] is a Krasner-Tate algebra admitting x as a universal generator. Proof:
Let F(Y, X) = £™ =0 fjXj.
For all A e K we put OO
F(y,A) = ]Ta A , m y m . m=0
Suppose x is not a universal generator. By Proposition 38.1 there exists v G K such that F(Y,v) is neither invertible nor irreducible in K{Y}, therefore, by Theorem 17.2 there exists q > 1 such that K , | > K,i\ Vj < q
(38.1)
Let r e]0, i^imax^^di^iDt- N o w > for e v e r y 3 < Q, and for all A e d(v,r), it easily seen that \a\,q\ = \
Universal Generators in Tate Algebras
231
X'(g)
Vg e K{Y}, and x ( s ) = A, and similarly, x"{g) = X"(g) V q for every A e d{v,r). Consequently, we have 53^ = 0 A*/j(a(A)) = 0 VA G d(u,r). Therefore, the set of maximal ideals of K{Y, X} which contain both F and | £ is infinite. But since A is an integral domain, by Lemma 38.2 the ideal generated by J £ and F in K{Y,X} is just equal to F{Y,X)K{Y,X}. Consequently, F must divide f f in K{Y,X}. Hence, by Corollary 7.6 F must divide ^ in if{Y}[X], which obviously implies ^y = 0 because | y is a polynomial of degree q — 1. Hence, we have / / = 0 Vi = 0 , . . . , n, and therefore each fi is of the form gf, with g G if{Y}. Now, putting Yx = Y?, and F i ( y i , X ) = F(Y,X), by Theorem 1.19 x is a spectrally injective element of l
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Proof of Theorem 38.6: Let D = spA(x) and let B = H(D). Since A is an integral domain, and since A is complete for its spectral norm, B is clearly the completion of the if-subalgebra R(D) of A. Suppose B ^ A and let g € A \ B. By Theorem 1.19 the characteristic p of K is different from 0, and g is of the form rt/f, with / e A and t e N*. Now, let a e D . Since x is a universal generator, and since g — g(a) vanishes at a, clearly / admits a as a zero of order pl. Consequently, / ' is identically zero, hence by Theorem 16.10 $Tf belongs to B. Putting / i = ) is affinoid by Theorem 36.2. Since / is an injective function, it is a spectrally injective element of the Krasner-Tate algebra H(D). Then, since K has characteristic 0, by Theorem 38.4 / is a universal generator of H(D). On the other hand, as a Krasner-Tate algebra, H(D) is of the form K{y}[:c], where K{Y} is a topologically pure extension of K and x is the identical mapping on D. Now, since all maximal ideals of H(D) have codimension 1, by Theorem 19.2 we know that sp(f) = f(D), so by Theorem 38.5 we have H(D) = K{Y}[f], so H(D) is isomorphic to H(f(D)) because sp(f) = f{D). Then, the homomorphism cf> from H(f(D)) to H(D) defined as (j>{h) = ft o / is an isomorphism. Consequently, by Corollary 17.10 we know that f^1 belongs
to H(f(D)).
□
Chapter 39
Mappings from H{D) to the Tree Mult{K[x})
We show that every element / e H(D) has continuation to a mapping / * from Mult(H(D), || . \\D) to Mult(K[x]). Given a circular filter T e <j>{D) the mapping that associates to each / G H{D) the multiplicative seminorm f*{T) is uniformly continuous with respect to the norm of H{D) and the metric 5 on Mult(K[x\). Lemma 39.1: Let h £ K(x) and let T be a circular filter of center a and diameter r such that d(a,r~) contains no pole of h. Then h{T) is a base of the circular filter of center h{a) and diameter
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Ultrametric Banach
Algebra
the form | | f | with (P,Q) = 1 and ||P|| = ||Q|| = 1. Let K be the residue class field of K and for all X €U, let A be the class of A in K. Similarly, for every P e K[x], we denote by P the residue class of P in }C[x}. Then the equation P(x) = ~cQ(x) has no solutions, a contradiction to the fact that K. is algebraically closed. Consequently, B n d(c,s~) is not empty whenever c € d(b,s). In the same way, through a change of variable of the form u = —r- we can show that Bn(K\ d(b, s)) ^ 0. Consequently, by Lemma 10.2 B lies in H, and therefore T' = H. D Proposition 39.2: Let T be a large circular filter on K and let h € K(x). Then h(J-) is a base of a circular filter denoted by /i*(.F), such that
Proof: Let r = diam(T). Suppose first that T has a center a. With out loss of generality we can assume that h has no poles in d(a,r~) because there does exist center b such that d(b,r~) contains no poles of h. Then by Lemma 39.1 h(!F) is a base of the circular filter Q of cen ter h(a) and diameter
Mappings from H(D) to the Tree Mult(K[x])
235
considering a sequence (bn)n€^ of T(a, r, s) such that \bn — a\ < \bn+i — a\ and limn_»oo |6 n | = s, we can show that Q' is the circular filter of center 0 and diameter s' = linin^oo IM&n)| = l a m|s m . □ Corollary 39.4: Let h G K(x) have no poles in an annulus T(a, r, s) and be equal to a Laurent series ^2J^o,n(x — a)n Vx G F(a,r,s). Assume that \am\pm > Wn\pn Vn ^ 0, m, V/9 €]r, s[. Let T (resp. Q) be the circular filter of center a and diameter r (resp. s). Then, h*(.F) (resp. h*(Q)) is the circular filter of center a^ and diameter \am\rm (resp. \am\sm). Lemma 39.5: LetJ-, Q be two circular filters having a center a, such that T -< Q, let S be the class of Q(Q), T is secant with. Let h G K(x) have no zeros and no poles in E \ Q(!F) and be such that \h(x)\ is not constant in E \ Q(F)- Let w be the absolute degree of h. Then diam(h*(<3)) | I diam(h*(J-)) loo — I
/diam(Q)\w Kdiam^)/
Moreover /i*(.F) and h*(G) are comparable for the order X. Proof: Let r = diam(T) and let s — diam(Q). In T(a,r, s), h(x) is equal to a Laurent series £Z-oo ctn(x — a)n- Since h has no zeros and no poles in T(a, r, s), by Lemma 17.4 there exists a unique integer m G Z such that | a m | p m > \an\pn Vn ^ m, Vp e]r, s[, and since \h(x)\ is not constant in E \ Q(T), by Lemma 12.7 we have m ^ 0. Consequently, by Corollary 39.4, h*(J-) is the circular filter of center a 0 and diameter |a TO |r m , and h*(Q) is the circular filter of center a^ and diameter | a m | s m . Particularly, h*(T) and h*(Q) are comparable. Next, we have /diam(Q)\m Kdiam^);
diam(ht:(Q)) diam(h*(T))
Now, given a number t > 1, we notice that |1 — t\oo > \1 — ||oo, and consequently, we have 1-
diam(K(g)) diamfo^J7))
oo
<
/diam(Q)\m \diam(T)
(39.1)
On the other hand, by Corollary 12.8 m is the difference between the num ber of zeros and the number of poles of h inside Q(JF). Hence |m|oo < w,
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Ultrametric Banach
Algebra
and therefore by (39.1) we obtain 1-
diam(h*(Q)) diam(h*{!F))
/diam(G)y \ diam{T I
<
n Lemma 39.6: Let T be a circular filter of center a and diameter r. Let h G K (x) and let w be the absolute degree of h. Then there exists s < r such that every circular filter Q of center a and diameter p £ [s,r[ satisfies 1
diam(h*(Q)) diam{h*(T))
< 1
/ diam(G) y V diam{T) J
Moreover T and Q are comparable for the order ^ . Proof: We can find v G]0, r[ such that h has no poles in T(a, v, r). Hence, inside T(a,v,r), f(x) is equal to a Laurent series 5 Z - r o a « ( x ~ a)™- Let f(x) = h(x) — ao and let f(x) = X]-ro a ri( x — a ) " - There exists s €]v,r[ such that / has no zeros inside T(a, s,r). Since a'0 = 0, |/(x)| is not constant inside T(a,s,r), hence we can apply Lemma 39.5 to / and then for every circular filter G of center a and diameter p G [s, r[, we have 1
diam(h*(G)) diam{h*{T))
< 1
/ diam(Q) \ ' \diam(T))
and T and G are comparable for the order -<. On the other hand, since f*{T) and /*((?) have center 0, then by Corol lary 39.4 /i*(.F) and h*(G) have center aoO Theorem 39.7 ([11]): Let D be a closed bounded subset of K and let f G H(D). Let T G ®(D). There exists a unique circular filter /*(•?-") less thin than the filter admitting for base f[T n D) and satisfying
G Mult(H(D'),\\
.
\D')
and for all g G H(D'), it satisfies
=vA9°f)-
Proof: For every P G K[x], we put ip{P) =
Mappings from H(D) to the Tree Mult(K[x])
237
of / * and /». Let T& be the intersection of T with D. Let J-' be the filter on K admitting for base f(JrD)- Suppose first that T' is not less thin than Q. Then we can find a sequence thinner than T' which is either a monotonous distances sequence, or an equal distances sequence or a cauchy sequence, and then this sequence is thinner than another circular filters H. Consequently, T' is secant with another circular filters H. Hence we have V>(P) = tpg(P) =(pnyp e K[x], hence by Theorem 13.1 H = G, therefore T' is thinner than Q. Let D' be a closed bounded set containing f{D). By Proposition 17.8, for all g G H{D'), g o f belongs to H(D), and we check that
Let then
Corollary 39.9: Let D be a closed bounded subset of K. Let T G &(D). Given f,g e H{D), then 5(gt(F),f,(r)) < \\f - g\\D. Further, if \\f g\\D < diam{s\xp(ft(J:),gt,{J::)), then ft(F) = g*{T). R e m a r k : In [3] it is shown that when two elements / and g of H{D) are such that ||/ — g\\o is small enough, then f*{T) = g*{F)N o t a t i o n : For every tp G Mult(H(D), || . \\D), we denote by Z^ the mapping from H(D) to Mult(K[x\) defined as Z^(f) = f*(ip). Corollary 39.10: Let D be a closed bounded subset of K. Then Z^ is continuous with respect to the norm on H{D) and to both topolo gies on Mult(K[x]) and it is uniformly continuous with respect to the metric S on Mult(K[x]). Moreover, the family of mappings Z^, ( G Mult(H(D), || . ||o)) is uniformly equicontinuous with respect to the metric 5 on Mult{K[x\). Corollary 39.11: Let D be a closed bounded subset of K and for every ip G Mult(H(D), || . ||jr>), Then Z^ is continuous with respect to the norm on H(D) and to the topologies of simple convergence on Mult(K[x}).
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Chapter 40
Continuous Mappings on
Mult(K[x])
Every analytic element on a closed bounded set D has expansion to a function from Mult(H(D), || . ||JD) to Mult{K[x\) which is continuous with respect to both topologies on Mult(K[x\). The family of functions Zyr defined by circular filters secant with D as Z^(f) = f*{!F) is a uni formly equicontinuous family with respect to the <5-topology on Mult(K[x\). Results apply to meromorphic functions in K which define open continuous function with respect to the topology of simple convergence. Such a map ping is increasing with respect to the order of the tree Mult(K[x]) if and only if it is an entire function. Now, let D = h~l{U). We have a natural mapping from H(U) into H(D), and another from Mult(H(D), || . \\D) into Mult(H(U), || . ||[/). We show that the Gauss norm || . \\u on H{U) has a number of continuations to the Krasner-Tate algebra H(D) which is equal to the cardinal of E(-D). Theorem 40.1 ([11]): Let D be a closed bounded subset of K and let f € H{D). Let T € $(£>). The mapping f* from Mult(H(D), || . \\D) to Mult{K[x\) defined as f*(
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Ultrametric Banach Algebra
loss of generality. Let w be the absolute degree of / . Let e be > 0. Since / is uniformly continuous in D, we can find v e]0, r[ such that \x — y\ < 2v implies \f(x) - f(y)\ < f. Now, let r\ e]0, \\ satisfy T
II
V,, _
Or,'
2/?'
+ v(l + | l - (
2r?
V —2r/y
rioo)(u-(^?ric >)) < e. v — Ar\ /v
(40.1) Consider now f , ^ e $(D) such that <5(^, 5) < 77 and let 5 = sup(^", Q). Suppose first that diam{T) < v, diam(G) < v. Then diam(S) < v + T) < 2v. Therefore, Q(S) is a disk d(a, p) with p < 2v. such that both T, Q are secant with d(a,p). We notice that given a G D, every set A included in d(a,u) is such that A C d(f(a), §). Therefore, we have S(f(a),f(T)) < \ and finally 5 ( / , ( ^ ) , / . ( a ) ) < 5(/,(.F), /(a)) + «5(/(a), /*(£)) < e. Now, suppose that diam{T) > v. Consider the ball Bs of center
<
r + 2r)
therefore r + 2r] r — 2rj
5(S',f')
(40.2)
Similarly, we obtain 5{S': Q') = \diam(S') - cZiam(£')loo = ?\1 - (^] Now by the hypothesis on s' and t' we check that 1 -
f°'\
u)
< 1CO
(t'\ b V / s
< 00
t + 2T] t-2r)
<
r + 4r) r — Ar)
therefore we obtain
s(s',g')
r + 4rj r — 4T]
(40.3)
Continuous
But by (40.2) we have \t' - r'\ < r'\l
-
rr+2r] 2V>
we obtain S(S',G') < r ' ( l + |1 - ( £ ^ ) sequently, by (40.2) again, we have 5(F',g')
1
)dl
241
», therefore by (40.3) ( £ g r i o o ) , and con-
r + 2r,y r-2ri)
OO
\ W
1 +
Mappings on Mult(K[xJ)
(r + 2r \r-2r )
J(
Ar\ r — Arj
But by hypothesis, r' < T and f < r, hence
1 +
v ~2r\
i/ + A-q v — Ar)
Consequently, by (40.1), we obtain 5'(^F',Q') < e, and so much the more £(■?-"',§') < e. Thus we have proven in every cases that &{T,Q) < T\ implies 5(F',G')<*-
□
Theorem 40.2: Assume K is separable, let D be a closed bounded sub set of K and let f e H(D). The mapping f* from Mult(H{D),\\ . \\D) to Mult(K[x]) defined as f*(
242
Ultrametric Banach
Algebra
to the complexity to define distances such as V. And yet, the answer might depend on the distance we consider defining the topology of simple conver gence. Anyway, we can state Theorem 40.3. Theorem 40.3 Assume K is separable and let D be a closed bounded subset of K. The family of functions Z^,, <j> € Mult(H(D), || . ||D) is uniformly equicontinuous on every ball of H(D) with respect to any metric defining the topology of simple convergence on Mult(H(D), || . \\D). Proof: Let d be a distance defining the topology of simple conver gence on Mult(H(D),\\ . \\D). Suppose that the family {Z,p \ (j> e Mult(H(D), || . | | D ) } is not uniformly equicontinuous on certain ball {/ e H(D) | II/HD < TO}- There exists a sequence (v^„) n eN of Mult(H(D), || . ||D), t > 0 and asequence (fn,gn)ne(<s of pairs of elements of H(D) such that l i m ^ ^ ||#„-/„||z> = 0 and d(fZ(iprn),gZ(iprJ) >
f(A).
Continuous
Mappings on Mult(K[x])
243
Proof: Let Q € $(B). Let (6 n ) n6 pj be a sequence of B thinner than Q and let (a„)„SN be a sequence of A such that f(an) = bn Vn G N. Since the sequence (an)neN is bounded, by Corollary 10.8 we can extract a subsequence which is thinner than a circular filter T. Then T G $(A). And then by Theorem 39.7 the sequence (6„)„gN is thinner than f*{J-), hence /»(.F) = 5□ T h e o r e m 40.6: Let f G .M(K') and let T be a circular filter on K which is not the filter of neighborhoods of a pole of f. Then f(J-) is a base of a cir cular filter f*^) such that
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Ultrametric Banach
Algebra
let A be a jF-affinoid containing no poles of / . Let B — f(A) and let Q = /,(?). Then B lies in g. Let us put W = Mult(H(A), || . \\A) and Z = Mult{H(B), || . | | B ) . Thus, Z is a neighborhood of
Mult(K[x})\
Proof: Let / e A(K). Let T, g € Mult{K[x\) be such that T < g. Let s = diam(g) and let r = diam(T). We can find a center a of g such that 5(a,J-) < s. Let p — 5{a,!F). Without loss of generality we can assume a = 0. Let f{x) = Y^=oanxnThen we know that fj^(f) < sup„ e N \an\pn < sup„ e N |a„|s n =
{r)-\H{E)).
Continuous Mappings on Mult(K[x])
245
Proof: If T is not secant with D, then /*(J-) is not secant with E because D = f~1(E). Now, suppose that T is secant with D. Then f*(T) is obviously secant with E because so is the filter admitting for base f(J-). First we suppose T to be D-bordering. Suppose that f*{T) is not Ebordering. Then, f*(T) is not secant with K\E There exists A G f*(J-) such that A C E. But since D = f~l(E) of course we have f~l(A) C D, -1 but / (^4) G T, a contradiction to the hypothesis F «s D-bordering. Now, suppose that F is not D-bordering. Then it admits an element A c D. Hence f{A) C E therefore f*(F) is not D-bordering. Consequently S(JB) = /*(£(!>)) and E(D) = ( D " 1 ^ ) ) D Corollary 40.9: Let Q be a circular filter of center a and diameter r G \K\, let h G K{x) and let D = h,-l{d{a,r)). Then {h*)-l{{Q}) = £(£>). R e m a r k s : In [41] E. Motzkin claims that the tree of a quasi-connected set is conserved through a conformal mapping. According to our definitions, this suggests that given a bounded closed infraconnected set D and an injective element of H(D) might (under certain hypothesis) define a strictly monotonous continuous mapping from &(D) onto $(/(£>)). And under more restrictive hypothesis, / _ 1 should belong to H(f(D)). Theorem 40.10 [11]: Let H(D) be a Krasner-Tate algebra of the form K{h}[x], with h e K{x), deg(h) > 0, and D = h'1^). Let S be the set of D-bordering filters. Then S is finite and the restriction of § to S is a bisection from S onto the set of ip G Mult(H(D), || . \\D) expanding the Gauss norm on K{h} to K{h}[x]. Proof: Let Q be the {/-bordering circular filter. Then the Gauss norm on K{h} is ipg. Let 7 be the homomorphism from H(U) into H{D) defined as 7(/) = foh, / G H(U). By Corollary 40.9 7 -1 ({Vff}) i s t h e s e t o f ¥>^ where T is a D-bordering filter. On the other hand, h~^{U) is an affinoid set, hence by Lemma 10.10 the set of D-bordering filters is finite. Hence, when considering H(D) as a finite extension of K{h}, the extensions of the Gauss norm on K{h} to H(D) are the p? when T is a D-bordering filter.
□ Remark: Let D be an infraconnected affinoid set and let H(D) = K{h}[x], with h G K{x), dega(h) > 0, and D = h~l{U). Since D has finitely many holes, so much the more it has finitely many envelopes of holes d{a,j,rj), (1 < j < q). Let D = d(a0,r0). The D-bordering fil ters, and therefore the Shilov boundary of i
246
Ultrametric Banach Algebra
with j = 0 , . . . ,q. Then in the Krasner-Tate algebra H(D) = K{h}[x], the Gauss norm on K{h} admits q + l different extensions to K{/i}[x] which are the ipy?.. In particular the Gauss norm admits only one extension if and only if every holes of D have the same diameter as D itself. On the other hand, it is well known that an absolute value denned on a complete field X admits a unique expansion to any algebraic extension. Here, we don't consider a complete field, but a complete ring K{h}, and a finite extension AT{/i}[x]. The Gauss norm on K{h} can admit several distinct expansions to jF£f{/i}[:r] because the field of fractions F of if{/i}[:r] is not complete, so an element of Jlf{/i}[a;] can admit a conjugate over F which actually lies in the completion of F. Example: Let h{x) = x + j , with a € K, such that \a\ < 1, let r = \a\, and let D = {x | r < \x\ < 1}. Then we check that D = /i _ 1 (f/), and there are two /^-bordering circular filters: the circular filter T' of center 0 and diameter r, and the circular filter T" of center 0 and diameter 1. Then, the Krasner-Tate algebra H{D) is obviously equal to A"{/i}[x], and the Gauss norm on K{h) admits two extensions to K{h}[x\: ipjn and ipr». Now, if we take \a\ = 1, then T' = T": the Gauss norm on K{h} has a unique extension to H(D).
Chapter 41
Examples and Counterexamples
In this chapter, several typical examples let us answer natural questions. Consider a commutative uniform Banach K-algebva. with unity A contain ing a dense iiT-algebra of finite type: A is not necessarily a K-affinoid algebra. It is also a natural question to ask whether a noetherian Banach K-algebra H(D) such that ||/||o S \K\ V/ € H(D), containing a dense Luroth A'-subalgebra, is a K-affinoid algebra. The answer is no in the general case, again. Other examples will be useful to Chapter 43. Theorem 41.1: Let A be a noetherian commutative ultrametric Banach K-algebra with unity satisfying Property (p). Let x e A be such that P(x) ^ 0 VP 6 K[x\. Then sa{x) is infinite. Proof: Suppose sa(x) is finite and equal to {ai,...,aq}. Let g = n ' = i ( x ~ aj)- T h u s w e h a v e IMIsa = 0. hence by Property p), \\g\\si = 0. Let I be the ideal of the / € A such that fgn = 0 for some n € N, let B = y and let 9 be the canonical surjection from A onto B. Let t = 9{g). Clearly t ^ 0 because by hypothesis g is not nilpotent. We will show that t is regular in B. Let z € B be such that tz = 0 and let w € A be such that 9(w) = z. Thus wg £ I, so there exists n € N such that (gw)gn = 0, thereby w e I and z = 0, so t is regular. Since 7 is a closed ideal of A, B is a Banach K-algebra with respect to the quotient norm j[ . \\q. Now, since \\g\\Si = 0 and since ||0(/)Hg < ||/|| V/ £ A, in B we have lim„_ (X) ^/|]t"|| g = 0. Consequently, there exists an increasing sequence of integers un such that lim™-^ iit„ .. 247
9
= 0 because otherwise
248
Ultramelric
Banach
Algebra
there would exist a > 0 such that | | i n + 1 | | g > fl||tn||g Vn € N which would lead t o linin^oo \ / | | i " | | g > a. Since t is regular, the mapping ip defined in B as tp(z) = tz is a con tinuous K-vector space isomorphism from B onto tB. But since a quotient of a noetherian ring B is noetherian, by Corollary 30.4 tB is closed in B with respect t o || . || g . Consequently, tB is complete for || . ||g, therefore by Banach's Theorem, ip is bicontinuous. Now, let (Zn)n€N be a sequence in K such that 1 < ||i n * u " II? < 2. Clearly, lim„_oo ||*u**+1||g = 0? s o the sequence (yn) defined as yn = lntUn satisfies Ill/nil? > 1> a n d li m rwoo llV'd/n)!!? = 0, a contradiction to the bicontinuity ofV»□ We shall establish Proposition 41.2, already stated in [18] as proposition IV.8. Proposition 41.2: Let («„)ngN be a sequence in K such that \an\ < |a„+i| , |«n| £ \K\, Vn € N and lim„_,oo |a n | = r e \K\. For each n e N, let dn = \an\, let rn e \K\ be such that 0 < rn < dn, let Dn = {£ € K | |f - o„| < r n } andte£D = C(0,r) U (u£° =0 .D n ). Then H(D) is a non noetherian Banach K-algebra such that \\f\\o € \K\ V/ € H(D), and its K-subalgebra K[x, j ] is dense in H(D). Proof: Since D has infinitely many infraconnected components, by Corol lary 16.13 we know that H(D) is not noetherian. We will show that II/HD € \K\ V / € i/(£>). Let / e H{D) ( / ^ 0) and let h e R(D) be
such that ||/ — h\\o < | | / | | D : of course | | / | | D = ||/I||D, SO we just have to show that \\h\\o € \K\. We can find s €]0,r[ such that h has neither any zeros, nor any poles inside 17(0, s,r). Let T = T(0, s,r). Let q e N be such that rq > s and D' = (XL,j + i Dn and D" = D \ D'. Suppose first that \\h\\D = \\h\\D»- Each set C(0, r ) , Z?„ (n e N) is calibrated and strongly infraconnected. Hence, by Theorem 16.6 we check that there exists a 6 D" such that ||/I||D" = \h(a)\, so ||/i||z>" € \K\. We now suppose that ||/I||ZJ > ||/i||£>", hence ||/i|| D = ||/i||i?'. Since /i has neither any zeros nor any poles inside T , by Corollary 12.4 we know that \h(£)\ = ip0^\(h) V£ e T. Next, we notice that D' C T , and more precisely, D' C U^_g_)_iC'(0,(iri). Consequently, (||/I||D' — su P£eTn£>' l^(£)l = su Pn>? =
Examples and Counterexamples
249
of course, by Corollary 12.4 v{h{£)) = v(h, — logr) is true in all classes of C(0, r ) , except at most in finitely many ones, which contradicts our hypoth esis \\h\\D > \\h\\D„. This finishes showing that \\f\\D e \K\ V/ e H(D). Now, we will show that the K-algebra B = K[x, A] is dense in H(D), which is not easy. Let A be the closure of B in H(D). Actually it is sufficient to show that for every a £ K \ D, -^^ belongs to A. So, we fix a € K \ D. By Lemma 16.4, we just have to show that there exists ha € R{D) fl A such that |/i 0 (a)| > ll^allo- When \a\ > r, we just take ha = x. And since C(0,r) C D, it only remains us to consider the case when |o| < r. Let q be the biggest of the integers n such that \a\ > dn. Then we can check that either \an — ai\ > dn Vn € N or \aq — ai\ < dq. We first suppose that \an —
(41.1) (41.2)
<1V£€DJ. k=j
In order to constructing such fo,-..,fq, to, • • • tq satisfying
we first define positive integers
k=j+l
jAI/
VdJ
(41.4)
< 1.
Indeed, since Tj < dj Vj, we can easily define such integers tj by a downing induction on j beginning at tq. Now we put fj = [^~-J ' (0 < j < g). Clearly / 0 ,..-/<, satisfy (41.1). We will check they satisfy (41.2). Indeed, let £ e -D,. We have l/*(OI = ( j ) ^ V * = j + l , . . . « , and | / , ( 0 | < ( ^ ' , Consequently we obtain
"(on/*® k=j
<
|A|
and | u ( 0 | = %■
ng©"^-
k=j+l
Thus we have constructed / 0 , . . . , / , satisfying (41.1) and (41.2).
250
Ultrametric Banach
Algebra
Now, let ha = uYl1=0fk- We will check that ha satisfies ||ft0||z> < 1 < \ha(a)\. Indeed, let £ G C(0,r). We have |u(f)| = ^ < 1 and |/,-(£)| = 1 Vj = 0 , . . . ,q thanks to (41.1). Consequently, when £ G C(0,r) we have |/i0(£)l < 1- Now, if £ G Dm with m > q, we have |u(£)l = ^ < 1 a n d |/j(£)| = 1. And finally, when £ G D m with m < q, then by (41.1) we have I/; (01 = 1 Vj = 0 , . . . , g - 1, hence |A o (0l = K O I I L , - A ( 0 I < L which finishes proving that ||/ia||D < 1- O n * n e other hand, since by hypothesis dq < \a\ < dq+1, we have |u(£)l = |^| > 1, so by (41.1) we check that l/7'(C)l = 1 Vj = 0 , . . . , 9 , which shows that |ft 0 ( a )| > 1We now suppose \aq a\ < dq. First, let us choose w G N such that
(dq y ^ rq \da+\)
(41.5)
d„
Let v G K be such that (41.6)
M = (dq+ly and let y = -^. We will define g\,...
,gq G R(D) satisfying (41.1) and
'(HySglliV 1 ) < \y(a)gg(a)\
(41.7)
and q-1
v(Z)l[9i(t) < 1 V£ G U A-
(41.8)
i=0
In order to constructing go,...,gq, So,...,sq such that sq = 1, and
±_ ^ "^
n ( ^ r ( i=n+l
we first choose positive integers
^ ) - < l V n = 0,.
..,q-l.
(41.9)
. It is easily seen that go,...,
gq
satisfy (41.1). We will show that they satisfy (41.7) and (41.8). Let n <
Examples and Counterexamples
251
q — 1. Given £ € D, we have 1*01 = ^
,
l*(OI = ( £ ) " v i = n + l , . . . , * and l*.(fll = ( g ) '
Thus, by (41.9) Relation (41.8) is satisfied. Finally by (41.5) we obtain
^^
=
\u\ \ f\a-an\\
(
r„ \
f\a-a„
[<^){^r)>{t^){^f)>i-
On the other hand we have
(dq)wJ
V d, *q
> \y(Ogg(0\ V£ e Dq,
which shows that gq satisfies (41.8). Consequently, we have constructed g0,---,gq satisfying (41.7) and (41.8). We can now conclude. Let ha(x) = yYln=o9n- Clearly ha satisfies \ha(a)\ = \y(a)gq(a)\ and \ha(0\ = |j/(Off«(0l v £ € D« h e n c e b y ( 41 - 7 ) w e have \ha{a)\ > max(||/i a ||0 , !)■ On the other hand we have
1^(01=^)11^) I
< 1 V£ € Dn Vn ^ q-
i=l
Consequently, \ha(a)\ > \\ha\\D.
a Theorem 41.3: Let A be a commutative ultrametric Banach K-algebra with unity, satisfying Property (q), containing a dense Luroth K-subalgebra. Then A satisfies Property (p). Proof: Let B = K[h,x] be a dense Luroth .ftT-subalgebra of A, with h e K{x). We will first show that sa{x) = sp(x). let D = sp(x), let a & D, and let \ be defined on B by x(f) = f(a)- Suppose there exists / s B such that ll/H > |/(a)|. Then by Theorem 6.13 / ( a ) - / is invertible in A. Let g = (/(a) - / ) - 1 . Then, by Lemma 1.8 x — a belongs to K[g,x\, hence to A, a contradiction. Consequently, we have | x ( / ) | < ||/|| V/ € B, which shows that \ is continuous, and therefore has continuation to A. So, sa(x) = sp(x). This is immediately generalized to every / e B because given / e B, by Theorem 23.4 we know that sp(f) = f(sp(x)) and sa(f) = f(sa(x)). Thus, we have ||/|| s a = T ( / ) V/ e B. But since A satisfies
252
Ultrametric Banach
Algebra
V / e B , and since B is dense
Property (q), actually we have ||/|| s a = in ^4, finally the equality holds in all A. Theorem 41.4: norm
□
Let K[x] be provided with the ultrametric K-vector space
Y^anxn
sup|a n | n
1 n+l
This norm is a K-algebra norm and the completion A of K\x] for this norm is a Banach K-algebra satisfying Property (p), such that P{x) ^ 0 VP € K[x] and such that sp(x) = {0}. Moreover, A is not noetherian. Proof: By definition, the elements of A are power series X ) ^ l o Q " 1 " such that linin-xx, | a n | ( ^ - j - ) n = 0. Consequently, given such a series X^n=o anXn, there exists q € N such that oo
£
= \a„
OCr>X
71=0
1 9+1
We first have to check that the norm is a if-algebra norm. Let S^Lo briXn and fg = X^=o cnxn- Let g e N be such that||£r=o^"l! = | C 9 | ( ^ Y ) 9 . By noticing that the inequality i W J + lJ \q-i
i
.
<
+ 1,
q+l
is true for every i < q we obtain II/5II < sup (| a i ||6,
-i + \/ <
sup
1 i+ \q- 1
—V(-
,TT)'>
■ 1911
Now, it easily seen that for all b e K we have linin^^ ||(frr)™|| = 0. In — 0. Consequently the particular, for every o S K* then, limn_> series 5^L 0 (f)™ is the inverse of 1 — ^ which proves that x — a is invertible in A for all a £ K*. Therefore, sp(x) = {0} and A is a local ring whose maximal ideal is xA and has codimension 1. More precisely, we can check that ||5]| si = 0 Vg € xA. Indeed let / 6 A, let A € K and let v e K be such that v > \\f\\. Then ||(Ax/)"|| < ||(Ai/a;)n||, hence as we just saw, limn^oo ||(Aj/:r)n|| = 0. Since this is true for all A € K, by Theorem 6.9 we have ||x/|| 5 j = 0. Consequently, given any / = Y^=o anxTl £ A we have
\ir
Examples and Counterexamples
253
HI2£LifflnZn||si = 0, therefore || X^Loa™x™IU = |«o|- Now, consider the (unique) homomorphism x from A onto K denned as x(IZnLo anXn) = a®: it shows that ||/|| s a = ||/IUi, so A satisfies property (p). By Theorem 41.1, we notice that A is not noetherian. □ Notation: We will denote by (d) the following property, in a normed commutative if-algebra: (d) for all f £ A such that \\f\\ < 1, the sequence \\fn\\ is bounded. According to Theorem VII.7 in [18] we have Theorem 41.5: Theorem 41.5: Let \\ . \\ be the ultrametric K-vector space norm on K[x] defined by ||^n=o a n x "ll = su Po
+ 1) < loiKt + l)|6,|(j + 1) < \ak\(k + l)\bi\{l + 1) = ||/||.|| 5 ||,
therefore \ct\(t + 1) < ||/||.|| 5 || and finally ||/ 5 || < ||/||.|| ff ||. The identical mapping from (if[x], || . ||) into (if [a;], || . \\u) being obvi ously continuous, the completion A of (if[a;],|| . ||) is a if-subalgebra of H(U). For every a £ K \ U, it is seen that the series Yl'jLoia)" converges in A, so x — a is invertible in A, and consequently R(U) is included in A. Since ||/|| > || . || y , of course we have ||/|| s i > || . || y , (just because || . \\u is semi-multiplicative). On the other hand, we have ||x||Si = 1, hence < sup \aj 3=0
0<j
UiXJ j=o
and finally the equality holds for all / £ A. For every a £ U, we have a if-algebra homomorphism Xa G X(A,K) defined as Xa(f) = f(a), and this shows that ||/|| s o > \\f\\u, hence ||/|| s o > | | / | U : thus A sat isfies Property (p). Of course, A does not satisfy Property (d) because l i m ^ o o liar" II = +oo.
254
Ultrametric Banach
Algebra
Let / € A be invertible in H{U). By Theorem 17.6 / is of the form E ° l o a n x n > w i t h \a0\ > sup„ > 0 (|a r a |). Since || . || si is induced by || . \\u, then we check that |ao| > II X^"Li anXn\\si, and therefore, by Theorem 6.13 / is also invertible in A. Consequently A is a full .KT-subalgebra of H(U). Now, we will show that each ideal of A is generated by a polynomial. First, we notice that for every a € U, g e A we have \\xg\\ > \\g\\, and therefore ||(x-a)fl||>|M|.
(41.10)
Let / G A satisfy / ( a ) = 0 and let / „ be a sequence in K[x] such that limn^oo ||/ - / n | | = 0. Then we have | / „ ( o ) | = \f(a) - fn(a)\
= \Xa(f ~fn)\<
11/ - / „ | | , i < 11/ " /nil-
Consequently, the sequence (fn—/(a))n<=N converges to / in A and obviously satisfies fn — f(a) = (x — a)gn, with gn £ K[x]. Hence, by (41.10) we have ||3n|| < ||/n — /n(o)||) and therefore the sequence (gn) converges in A. Let g be its limit. Thus, we have / = (x — a)g. Consider now an element / of A. As an element of H(U), it has a decomposition of the form P(x)h(x), with P a polynomial whose zeros lie in U and h an invertible element of H(U). By an immediate induction, we can see that h belongs to A, so this decomposition holds in A. And since A is a full if-subalgebra of H(U), then h is invertible in A. Now, according to a classical process, we can easily deduce that every ideal of A is generated by a polynomial whose zeros lie in U. Let / be an ideal of A, let P be the g.c.d. of polynomials which belong to / and let / € I. As we have seen, / factorizes in the form Q{x)h{x) with Q a polynomial whose zeros belong to U and h an invertible element of A. Thus Q also belongs to A, and therefore P divides Q. Hence P divides / in A, which proves that / = PA. □
Chapter 42
Associated Idempotents
Throughout the chapter, A is a commutative ultrametric Banach /^-algebra with unity. We will show that given t € A, if there exists a £-clear annulus, then A admits idempotents corresponding to the two subsets of Mult(A, || . ||) defined by the annulus. Lemma 42.1: Let x € A, and let r > \\x — a\\si with r £ \K\, let sp(x) admit an x-clear annulus T = r(a,r',r"). Let s',s" e]r',r"[ with s' < s". The restriction of Qx to R(d(a,r) \T(a,s',s"),\\ . ||) is continuous and expands to H(d(a, r) \ F(a, s', s")). Proof: Let A 6 C(0, ±), let D = d(a,r)\T(a,s',s"), let be if such that \b\2 = s's" and let c € K such that |c| 2 = s"r. Let t(x) = A ( x ~ff~ c ) 2 . It is easily seen that t{D) = U, D = t~l(U). By Theorem 37.2 H(D) is a Krasner-Tate algebra of the form iiT{i}[x]. Now, let s = \\x\\Si and let D' = d(a, r ' ) u A ( a , r", s). By hypothesis, for every ip
256
Ultrametric Banach Algebra
and 4>(u) = 0, V<£ € Mult{A, || . ||) such that tj>{x) < r". Moreover, u is unique satisfying there properties. Proof: Let T = T{a, r', r"). Let r > | | x - a | | s i , let s', s" e]r', r"[n|K| with s' < s" and let D = d(a, r) \ T(a, s', s"). By Lemma 42.1 the restriction of Qx to R(D) is continuous and expands to H{D). Now, by Theorem 17.15 H(D) contains idempotents h and I such that h(z) = 1 Vz 6 d(a,s'),h(z) = 0 Vz £ d(a,r) \d(a,s"~), and l(z) = 0 Vz e d(a,s'),2(.z) = l V z £ d(a,r) \ d ( a , s " ~ ) . Therefore its image i4' = QX{H{D)) contains an idempotent u = Qx(h). Let <j> £ Mult(A, || . ||). Either ^ x is secant with d(a,s'), and then (u) = 0 x (/i) = 1, or (/^ is secant with d(a,r) \ d(a,s"~), and then 4>{u) =
Associated Idempotents
257
of C(a,r) which contains no poles of h and let / G K[h,t]. Let O be the partition of d(a,r) consisting of the family of classes of d(a,r). The Mittag-Leffler decomposition of / in H{0) is of the form J3?=O/J> w ^ /o e R(d(a,r)), and fj e # o ( # \ < % , ? - ) ) (1 < j < ?)■ Then by Theorem 21.1 we know that 1
+ f
>
1
x—co x—c But on the other hand, for all g{x) € R(sp(t)), we have | = limg \g(x)\ = 4>t{g) =
+ /(*)
t-c
1
>
x—c which proves that K[h,t] is not dense in A. Suppose now that T is secant with D. Let / = diam^). Then T must be secant with a hole T = d(b,p~) of D such that I < p. Hence J- is (6,s)approaching with I < s < p. Let r €]s,/3[n|/f| and let Q be the circular filter of center b and diameter r. Since there exists no £-clear annulus, we can find £ e Mult(A, || . ||) such that Ct = fg- We can now go on similarly to the previous case. Let d(c,r~) be a class of C(b,r) which contains no poles of h and let O be the partition of d(b, r) consisting of the family of classes of d{b,r). Let / e K[h,t]. By Theorem 21.1 we know that 1
+/ x—c
>
1 o
But on the other hand,
+ f(t)
>
which proves that K[h,t] is not dense in A again.
o
□
Theorem 42.6: Let A satisfy Property (q), let x e A and suppose that sp(x) admits an empty annulus T(a,r, s). There exists a unique idempotent u such that x(u) = 1 for all x £ %{A) such that \(x — a) < r, and xiu) = 0 for all x £ ^(-<4) such that x(x — a) > s. Proof: Indeed, since A satisfies Property (q), by Proposition 28.1 we know that Multm(A,\\ . ||) is pseudo-dense in Mult(A,\\ . ||). Let <j> e Mult(A, || . ||). Since T(a, r, s) n sp(x) - 0, the circular filter T = * _ 1 ( ^ x ) of the restriction
258
Ultrametric Banach
Algebra
sp(x) n£(T). Then the existence of u and the uniqueness come from Theorem 42.2. D Corollary 42.7: Let A satisfy Property (q) and have no idempotents dif ferences from 0 and 1. Then for every x £ A sp(x) is infraconnected. By Theorem 8.13 we obtain this Corollary: Corollary 42.8: Let A satisfy Property (q), let x G A and suppose that sp(x) has finitely many infraconnected components Di,...,Dq. For each u j = 1 , . . . ,q, there exists a unique idempotent Uj such that x( j) — X for all x £ <^(-<4) such that x(u) £ Dj and x(uj) = 0 for °^ X S X{A) such that x(u) £ Dj. And by Theorem 25.2 we have these last Corollaries: Corollary 42.9: Let A be uniform, and have no maximal ideals of infinite codimension, let x G A and suppose that sp(x) has finitely many infracon nected components D\,..., Dq. For each j — 1 , . . . , q, there exists a unique idempotent Uj such that x(uj) = 1 for a^ X e X(A) such that x(u) S Dj and x(uj) = 0 for all x £ X{A) such that x(u) $■ Dj. Proof: Indeed, since A has no maximal ideals of infinite codimension, it obviously satisfy Property (o), and therefore, by Theorem 25.4 it satisfy Property (p). □ Corollary 42.10: Let A be uniform and have no maximal ideals of infi nite codimension, let x G A and suppose that sp(x) has finitely many infraconnected components D\,... ,Dq. For each j = l,...,q, there exists a unique idempotent Uj such that x(uj) — 1 for a^ X S X{A) such that x(u) G Dj and x(uj) = 0 for M X £ X{A) such that x{u) <£ Dj. Corollary 42.11: Let A be uniform and have no maximal ideals of infi nite codimension and have no idempotents different from 0 and 1. Then sp(x) is infraconnected. Theorem 42.12: Let A have bounded normal ratio, have no maximal ideals of infinite codimension and have no idempotents different from 0 and 1. Then, for every x G A, sp(x) is infraconnected. Proof: Indeed, since A has bounded normal ratio, by Theorem 24.3 it satisfies Property (s). But then, since no maximal ideal has infinite codi mension, then A satisfies Property (p), hence Property (q), and then by Corollary 42.7, since A has no non trivial idempotents, sp(x) is infracon nected for all x G A. □
Chapter 43
K r a s n e r - T a t e Algebras a m o n g Banach iT-Algebras
Throughout the chapter, A is a commutative ultrametric Banach K-&\gebr& with unity. Here we mean to characterize Krasner-Tate algebras among ultrametric K-Banacb. algebras through algebraic and topological proper ties denoted by (a), (6), (c), (d), (e) [18]. In a final remark, we will show that these properties are logically independent, and therefore none of them is redundant. Lemma 43.1: Let A contain a dense Luroth K-subalgebra K[t,h] with h G K(t) n A and be such that the strict t-spectral partition coincides with the canonical partition of d(0, \\t\\Si) \ sp(t). Then A satisfies Property (q). Proof: Let E = {x e X(A) | x{t) £ &}■ Let O be the strict t-spectral partition, and let / G K[h,t], with h(x) £ K(x). Since the strict i-spectral partition coincides with the canonical parti tion of d(0, \\t\\Si) \ sp(t), by Chapter 18, in R(sp(t)), we have \\f(z)\\sp(t),o = \\f(x)\\sp(t) = sup{|/(A)| | A G sp(t)} = sup{| X (/)| \xeE}=
r(f) < \\f\\si. (43.1)
On the other hand by Theorem 23.5 we have ||/|| S i < ||/|| S p(t),o, hence by (43.1) the equality ||/|| s i = T ( / ) holds for all / € K[h,i\. Then, since K[h, t] is dense in A, we can generalize the equality to all A. Indeed, let / G A, and let (fn)neN D e a sequence in K[h,t] such that limn-^oo ||/ n —/|| = 0. Let e e]0, \\f\\si[ and let N G N be such that ||/ n - / | | < e Vra > N. We notice that ||/„ - / | | s i < t Vn > N, and therefore ||/„|U = ||/|| s i Vn > N. Since the equality holds in K[h, t], we can find \ G E such that |x(/n)| > 259
260
Ultrametric Banach
Algebra
ll/n||s» — e- Thus, the sequence ||/n|U» is constant and equal to |j/||s» for all n > N and we have |x(/n)l = lx(/)l o n the other hand, the sequence x(fn) is a Cauchy sequence and therefore converges in K to x(/)- Hence x(f) belongs to K and we have | x ( / ) | > H/IU — e, which proves that \\f\\si = T ( / ) in the general case. D Theorem 43.2: If A contains a dense Luroth K-subalgebra and has no non trivial idempotents, then A satisfies Property (q). Proof: Suppose that A does not satisfy Property (q), and let K[h,t] be a dense Luroth if-subalgebra, with h G K(t) n A. By Lemma 43.1 the strict t-spectral partition does not coincide with the canonical partition of sp(t). Let T = d(b, p~) be a hole of the strict t-spectral partition which is not a hole of sp(t). There exists ip G Mult(A, || . ||) such that the circular filter T — <& -1 (^ t ) is (b, r)-approaching, and therefore is not secant with sp(t). Then by Proposition 42.5 we have a contradiction with the hypothesis K[h,t] is a dense Luroth K-subalgebra. □ We are now able to characterize Krasner-Tate algebras among ultramet ric Banach X-algebras through the following two theorems: Theorem 43.3 ([18]): A is a Krasner-Tate algebra if and only if it sat isfies the following five properties: (a) (6) (c) (d) (e)
A is noetherian, A is reduced, ||/|U € \K\ V/ G A, for every f G A such that \\f\\Si = 1, the sequence \\fn\\ is bounded, A contains a dense Luroth K-subalgebra.
Theorem 43.4: Let A have no non trivial idempotents and admit infinitely many maximal ideals of codimension 1. Then A is a KrasnerTate algebra if and only if it satisfies the following four properties: (b) (c) (d) (e)
A is reduced, ||/|U G \K\ V/ € A, For every f € A such that \\f\\Si = 1, the sequence \\fn\\ is bounded, A contains a dense Luroth K-subalgebra.
Proof of Theorems 43.3 and 43.4: A Krasner-Tate algebra obviously satisfies Properties (a), (6),(c), (d), (e). So, we just have to consider a if-Banach algebra satisfying these five properties, and prove that it is a
Krasner-Tate
Algebras among Banach K-Algebras
261
Krasner-Tate algebra. Let B = K[h,x) be a dense Luroth if-subalgebra, with h e K(x) and let D = sp(x). First, we assume that A has no non trivial idempotents. By Theorem 43.2 A satisfies Property (q). Hence, by Theorem 41.3 A satisfies Property (p) in both Theorems 43.3 and 43.4. Consequently, by Theorem 41.1 D is infinite in the hypotheses of Theorems 43.3. Now, assuming the hypotheses of Theorem 43.4, suppose that D is finite: let D = {a\,... ,as}. For each j = 1 , . . . , s, there exists Xj € X{A) such that Xj(x) = aj a n d then Xj(h) = h(cij). Given any £ G X{A), its restriction to B is equal to the restriction of one of the \j saY Xk- Then, since B is dense in A, we have £ = Xk- Consequently, A only has s maximal ideals of codimension 1, a contradiction. Thus we have proven that D is infinite in both Theorems 43.3 and 43.4. Next, B is isomorphic to a A'-subalgebra of R(D), so we will consider that B C R(D) C A. We can check that for every / € R(D), we have S PA(J) = f(D). Indeed, since R(D) C A, we have sp^(f) C f(D) because f(D) = spR(D)(f). But given A e /(£>), and a <= D such that f(a) = A, there exists x € X{A) such that x(x) = a> hence x(f) = f(a) — ^~ Therefore A £ SPA(J) and finally SPA(J) = f(D). Consequently, we have TAU) = WIWD, and therefore by property q) we have ||/|| D = ||/|| s i V/ e R(D). Consequently, A and H(D) are the completions of R(D) for the norms || . || and || . || jr>, respectively. On the other hand, since A has no non trivial idempotents, by Theorem 43.2 it satisfies Property (q). Then by Corollary 42.7 SPA{J), hence D, is infraconnected. Let d(a,r) = D, with r G \K\. Since B is dense in H(D), by Lemma 16.15, D has finitely many holes: d(a,j,r~) (1 < j < q). Since \\x — a\\si = ||x — a||u = r, by Property c) we notice that r lies in \K\. Similarly, since j 1 la: — aj
1 si
1
\X — dj
we have rj € \K\. Hence by Theorem 37.2 H(D) is a Krasner-Tate algebra K{h}[x] where h € R(D) satisfies deg(/i) > 0, U = h{D), D = h^iU). Then, by Property (d) the canonical mapping ip from K[h,x] into A defined as tp(F(h,x)) = F((h(x),x) is continuous. Consequently, it has continua tion to a continuous homomorphism ip from H(D) to A, and this homo morphism is obviously bicontinuous because | | / | | D = ||V;(/)IUi < IIVK/)!!Thus, H(D) is a if-subalgebra of A, containing J5, hence is dense in A
262
Ultrametric Banach Algebra
and such that both norms || . |£> and || . || are equivalent on H(D). Con sequently, A = H(D). This finishes proving that A is the Krasner-Tate algebra H{D) when A has no non trivial idempotents. In particular, this finishes the proof of Theorem 43.4. We now consider the general case in the proofs of Theorems 43.3. Clearly A has finitely many idempotents because A is noetherian. So, we can find a family of idempotents u\,...,Uk satisfying UiUj = 0 Vi ^ j , and S j = i ui = 1- For each i = 1 , . . . , k, we put Ai = UiA and Di = sp(ttjx). Then A is isomorphic to the direct product Ai x . . . x Ak. Then we can check that each K-algebra Ai (1 < i < k) also satisfies Properties (b), (c), (d), (e). Let us fix k < q, and suppose first that Dk is infinite. Then UiB is a Luroth .ftT-algebra again because UkX is transcendental over K. Consequently, UkA satisfies Property (e) and we can make the same reasoning as we did when D is infraconnected and we conclude that H(Dk) is a Krasner-Tate algebra. Suppose now that Dk is finite. Since it is infraconnected, it is a singleton {a}. Consequently, UkA is of the form (Y-a\»K\Y} • ^ u ^ s m c e -A is reduced, we have s = 1, hence UkB is isomorphic to K and so is UkA. Finally, if we denote by Di,..., Dm the infinite infraconnected components of D and by Dm+i,..., Dq those which are singletons, we can see that A is isomorphic to A i x . . . x Am x K'1 m which itself is a Krasner-Tate algebra isomorphic to Hiyi^Dj). D By Theorem 43.3 we obtain Corollary 43.5: Corollary 43.5: If A is uniform, it is a Krasner-Tate algebra if and only if it satisfies the following four properties: (a) A is noetherian, (b) A is reduced,
(c) u/iu e \K\ v/
6
A,
(e) A contains a dense Luroth K-subalgebra. Corollary 43.6: A reduced K-affinoid algebra is a Krasner-Tate algebra if and only if it contains a dense Luroth K-algebra. Theorem 43.7: Let A be a K-affinoid algebra without non zero divisors of zero, containing a K-subalgebra of finite type B, dense in A, whose field of fractions is a pure degree one transcendence extension of K. Then A is a Krasner-Tate algebra if and only if A is integrally closed. Proof: We just have to assume that A is integrally closed and show that A is a Krasner-Tate algebra. Let K(x) be the field of fractions of B and let
Krasner-Tate
Algebras among Banach K-Algebras
263
B = K[yx,..., yq\. For each j = 1 , . . . , q, we put y$ = fj(x) (fj e K[x)). There obviously exists a linear fractional function h such that deg(/i oh) > 0. So, we put z = h~1{x) and gj = fjohVj = I,... ,q. So, z is obviously integral over B, and therefore belongs to A. Let B' = B[z). Then B' is a Luroth /^-algebra hence by Corollary 43.5 A is a Krasner-Tate algebra.
□ Remark: By Proposition 41.2 there exist Banach K-algebras H(D) sat isfying Properties (6), (c), (d), (e) but not satisfying Property (a). By Proposition 37.9, there exist K-affmoid algebras satisfying (a), (c), (d), (e) but not satisfying Property (b). If \K\ ^ R, any Banach AT-algebras H(d(0,r)), with r <£ \K\ satisfies Properties (a), (6), (d), (e) but does not satisfy Property (c). Theorem 41.5 shows the existence of commutative ultrametric Banach .K"-algebras with unity satisfying Properties (o), (fo), (c), (e), but not satis fying Property (d). There obviously exist commutative ultrametric Banach Jsf-algebras with unity satisfying Properties (a), (b), (c), (d) but not satisfying Property (e): we can just consider Tn for n > 1. Thus, none of the five Properties (a), (6), (c), (d), (e) is a consequence of the four others in the category of commutative ultrametric Banach Kalgebras with unity. However, if K is such that \K\ = [0, +oo[, Condition (c) is obviously trivial and must be cut. Remark: We don't know examples of noetherian uniform Banach Kalgebras without non zero divisors of zero, satisfying Property (c) and containing a dense i^-subalgebra of finite type, which are not iiT-affinoid algebras.
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References
1. Amice, Y. Les nombres p-adiques, P.U.F. (1975). 2. Berkovich, V. Spectral Theory and Analytic Geometry over Non-archimedean Fields. AMS Surveys and Monographs 33, (1990). 3. Bosch, S., Guntzer, U., and Remrnert, R Non-Archimedean analy sis, Grundlerhen Wissenschaften, Bd. 261, Springer, Berlin-HeidelbergNew York, (1984). 4. Bourbaki, N. Topologie generate, C%. 3 Actualites scientifiques et industrielles, Hermann, Paris. 5. Bourbaki, N. Theorie spectrale Ch. 1 et 2., Actualites scientifiques et indus trielles, Hermann, Paris. 6. Bourbaki, N. Algebre commutative, Ch. 2., Actualites scientifiques et indus trielles, Hermann, Paris. 7. Boussaf, K. and Escassut, A. Absolute values on algebras of analytic elements, Annales Mathematiques Blaise Pascal 2, n2 (1995). 8. Boussaf, K. Shilov Boundary of a Krasner Banach algebra H(D) Italian Journal of pure and applied Mathematics, N.8, pp. 75-82 (2000). 9. Boussaf, K., Hemdahoui, M., and Mainetti, N. Tree structure on the set of multiplicative semi-norms of Krasner algebras H(D). Revista Matematica Complutense, vol XIII, N. 1, pp. 85-109 (2000). 10. Boussaf Image of circular filters, International Journal of Mathematics, Game Theory and Algebra, Volume 10, N. 5, pp. 365-372. 11. Boussaf, K., Escassut, A., and Mainetti, N. Mappings in the tree Mult{K[x\). To appear in Bulletin of the Belgian Mathematical Society Simon Stevin. 12. Cherry, W. Non-Archimedean analytic curves in Abelian varieties, Mathematishe Annalen 300, pp. 396-404 (1994). 13. Decomps-Guilloux, A. and Motzkin, E. Fonctions analytiques univalentes dans un corps ultrametrique complet algebriquement clos. C.R.A.S. Paris t. 268, pp. 1531-1533 (1969). 14. Diarra, B. Ultraproduits ultrametriques de corps values, Annales Scientifiques de FUniversite de Clermont II, Serie Math., Pasc. 22, pp. 1-37, (1984). 265
266
Ultrametric Banach Algebra
15. Dwork, B. Lectures on p-adic differential equations. Springer-Verlag, (1982). 16. Escassut, A. Algebres d'elements analytiques au sens de Krasner dans un corps value non archimedien complet algebriquement clos, C.R.A.S. Paris, A 270, pp. 758-761 (1970). 17. Escassut, A. Complements sur le prolongement analytique dans un corps value non archimedien, complet algebriquement clos, C.R.A.S.Paris, A 271, pp. 718-721 (1970). 18. Escassut, A. Algebres de Banach ultrametriques et algebres de Krasner-Tate, Asterisque n. 10, pp. 1-107, (1973). 19. Escassut, A. Algebres d'elements analytiques en analyse non archi- medienne, Indag. math.,t.36, pp. 339-351 (1974). 20. Escassut, A. T-filtres, ensembles analytiques et transformation de Fourier p-adique, Ann. Inst. Fourier 25, n 2, pp. 45-80, (1975). 21. Escassut, A. Spectre maximal d'une algebre de Krasner, Colloquium Mathematicum, XXXVIII, fasc. 2, pp. 339-357, (1978). 22. Escassut, A. Elements spectralement injectifs et generateurs universels dans une algebre de Tate, Collectanae Mathematica (Barcelona), Vol XXVIII, Fasc.2, pp. 131-148 (1977). 23. Escassut, A. The ultrametric spectral theory, Periodica Mathematica Hungarica, Vol. 11, (1), pp. 7-60, (1980). 24. Escassut, A. Maximum principle for Analytic Elements and Lubin-Hensel Theorem for series with Analytic Coefficients, Ann. Mat. Pura Appl. vol. CXXXV, pp. 265-278 (1983). 25. Escassut, A. Integrally closed algebras of analytic elements, Communication in Algebra, 19, pp. 1565-1584 (1991). 26. Escassut, A. and Diarra, B. Non integrally closed algebras H(D), p-adic functional analysis, Lecture Notes in Pure and Applied Mathematics n. 137, pp. 63-74 (1992). 27. Escassut, A. and Sarmant, M.-C. Sufficient Conditions for injectivity of Ana lytic Elements, Bull. Sci. Math. vol. 118, n. 1, pp. 29-46, (1994). 28. Escassut, A. and Sarmant, M.-C. Injectivity, Mittag-Leffler Series and Motzkin Products, Annales des Sciences Mathematiques du Quebec, 16, (2), pp. 155-173 (1992). 29. Escassut, A. and Mainetti, N Spectral semi-norm of a p-adic Banach algebra, Bulletin of the Belgian Mathematical Society, Simon Stevin, vol. 8, pp. 79-61, (1998). 30. Escassut, A. Analytic Elements in p-adic Analysis, World Scientific Publish ing Inc., Singapore (1995). 31. Escassut, A. Shilov boundary for ultrametric algebras, preprint. 32. Garandel, G. Les semi-normes multiplicatives sur les algebres d'elements ana lytiques au sens de Krasner, Indag. Math., 37, n. 4, pp. 327-341, (1975). 33. Guennebaud, B. Algebres localement convexes sur les corps values, Bull. Sci. Math. 91, pp. 75-96, (1967). 34. Guennebaud, B. Sur une notion de spectre pour les algebres normees ultrametriques, these Universite de Poitiers, (1973).
References
267
35. Krasner, M. Prolongement analytique uniforme et multiforme dans les corps values complets: preservation de I'analycite par la convergence uni forme, Theoreme de Mittag-Leffler generalise pour les elements analytiques, C.R.A.S. Paris, A 244, pp. 2570-2573, (1957). 36. Krasner, M. Prolongement analytique uniforme et multiforme dans les corps values complets. Les tendances geometriques en algebre et theorie des nombres, Clermont-Ferrand, pp. 94-141 (1964). Centre National de la Recherche Scientifique (1966), (Colloques internationaux de C.N.R.S. Paris, 143). 37. Mainetti, N. Spectral properties of p-adic Banach algebras. Lecture Notes in Pure and Applied Mathematics n. 207, pp. 189-210, (1999). 38. Mainetti, N. Sequential Compactness of some Analytic Spaces, Journal of Analysis, 8, pp. 39-54 (2000). 39. Mainetti, N. Metrizability of some analytic affine spaces Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, (2001). 40. Morita, Y. On the induced h-structure on an open subset of the rigid analytic space P1{k). Mathematische Annalen 242, pp. 47-58, (1979). 41. Motzkin, E. Un invariant conforme p-adique, Groupe d'Etude d'Analyse Ultrametrique de FIHP, 1981-82, 42. Rivera-Letellier, J. Espace hyperbolique p-adique et dynamique des fonctions rationnelles, preprint. 43. Robert, Alain Advanced calculus for users North-Holland, Amsterdam, New York, Oxford, Tokyo (1989). 44. Van Rooij, A.CM. Non-Archimedean Functional Analysis, Marcel Decker, Inc. (1978). 45. Salmon, P. Sur les series formelles restreintes, Bulletin de la Societe Mathematique de France, 92, pp. 385-410 (1964). 46. Sarmant, M.-C. et Escassut A. T-suites idempotentes, Bull. Sci. Math. 106, pp. 289-303, (1982). 47. Sarmant, M.-C. Prolongement analytique a travers un T-filtre, Studia Scientiarum Mathematicarum Hungarica 22, pp. 407-444, (1987). 48. Schikhof, W.H. Ultrametric calculus. An introduction to p-adic analysis, Cambridge University Press (1984). 49. Tate, J. Rigid analytic spaces, Inventiones Mathematicae 12, pp. 257-289 (1971).
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Definitions Index
(a, r)-approaching circular filter 60 v4-affinoid algebra 183 >4-polnorm of B 175 ^4-quasi-algebra norm 181 ^4-quasi-algebra normed A-module 181 affinoid set 49 analytic element 93 analytic element meromorphic at a 95 analytic element admitting a as a pole of order q 95 analytic element meromorphic in a hole 95 analytic element strictly vanishing along T 110 analytic element vanishing along T 110 arcwise connected topological space 85 ascending chain 117
calibrated set 102 canonical base of a monotonous filter 52 canonical homomorphism associated to an element t of a /
base of a filter 51 beach of T 52 boundary for (B, 0) 25 bounded normal ratio (for an element) 136 bounded normal ratio (for a /("-algebra) 136
Z?-bordering circular filter 60 ^-peripheral circular filter 60 269
270
Ultrametric Banach Algebra
decreasing distances sequence 51 decreasing distances holes sequence 54 decreasing distances holes sequence that runs a monotonous filter 54 decreasing filter of center a and diameter R 52 decreasing filter with no center, of canonical base (Dn)neN 52 decreasing T-filter 106 degree of a rational function 2 dense valuation 14 diameter of an ascending chain 117 diameter of a monotonous distances holes sequence 54 diameter of a weighted sequence 106 discrete valuation 14 distinguished circular filter 115 empty annulus 47 equal distances sequence 51 ^-affinoid 57 /-hole 98 /-hole (for an analytic element on a classic partition) 124 filter secant with D 51 filter thinner than a filter Q 51 Gauss norm 37 hole of a set 46 idempotent weighted sequence of O 119 idempotent weighted sequence 106 increasing distances sequence 51 increasing distances holes sequence 54 increasing distances holes sequence that runs a monotonous filter 54 increasing filter of center a and diameter R 52 increasing T-filter 106 infraconnected set 46 infraconnected component 47 intersection of a filter with D 51
irregular distinguished circular filter 115 Jacobson radical 1 Jacobson ring 1 fc-regular power series 197 fc-canonical associate to / 198 fc-order of / 198 L-productal vector space 17 large circular filter 57 Luroth F-algebra 2 metric associated to the sequence (Fn)n€N- on Mult(K(xi,...,xq}) 202 metric topology associated to / 11 Mittag-Leffler series of an analytic element 98 Mittag-Leffler term of / associated to a hole on (D, O) 124 Mittag-Leffler series of / on (D, O) 124 monotonous distances sequence 51 monotonous filter 52 monotony of a monotonous distances holes sequence 54 monotony of a weighted sequence 106 multbijective commutative ultrametric Banach /("-algebra 167 multiplicative function 22 multiplicity order of a zero of an analytic element 96 nilradical 1 O- bordering circular filter 127 O-minorated annulus 120 O-set 120 perfect field 175 pierced filter 53 piercing of a monotonous distances holes sequence 54 piercing of a weighted sequence 106
Definitions Index principal term of / (for an analytic element on a classic partition) 124 polynomial of the poles of / in a hole 95 principal term of an analytic element 98 pseudo-dense subset of Mult(A, || . ||) 171 punctual multiplicative semi-norm 75 quasi-invertible analytic element 96 radical of an ideal 1 reduced ring 1 regular distinguished circular filter 115 residue class field 14 residue characteristic of L 14 restricted power series 37 returning filter of an ascending chain 117 5-appropriate sequence of K[Z-L,...,Zq) 202 S-multiplicative function 22 semi-group 22 semi-multiplicative function 22 semi-simple ring 1 sequence («n)ngN thinner than a filter G 51 Shilov boundary for (B, 9) 25 simple monotonous distances holes sequence 54 spectrally injective element 4 spherically complete field 52 strictly /^-bordering circular filter 60 strictly O-bordering circular filter 127 strictly injective analytic element 103 strict x-spectral partition 136 strongly infraconnected set 46
271
strongly valued field 15 submultiplicative function 22 superior gauge of a monotonous distances holes sequence 54 i-clear annulus 138 T-optimal partition of an infraconnected set 114 T-specific partition of an infraconnected set 115 T-sequence 107 T-sequence of O 120 topologically pure extension of L of dimension n 183 tree 9 [/-submodule defining the topology of E 181 uniform Banach L-algebra 34 universal generator 5 valuation associated to an absolute value 13 valuation group 14 valuation ideal of B 14 valuation ring 14 valuation ring of L 14 weakly valued field 15 weighted sequence 106 weighted sequence of a classic partition 120 well pierced monotonous distances holes sequence 54 whole distance associated to / 11 wide a>spectral partition 136 x-normal partition of center a and diameter r 136 x-spectral partition of center a and diameter r 136
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Notation Index
C h a p t e r 1: Maxa(A) 1 Max{A) 1 X{A,E) 1 X{A) 1 SPA (x) 1 SOA (X) 1 sp(x) 2 sa(x) 2 irr(x, B') 3
Mult'(A,T) 23 MM/t m (A,T) 23 Multa(A,T) 23 /us(a;) 25 C h a p t e r 6: 4>* 31 <£ 3 2
II • II,< 3 3 C h a p t e r 7: E{Xi,...,Xn} 37 Z(gi,...,gg) 39 £(ff,A) 40
C h a p t e r 3:
M 13 u(a;) 13 | . |oc 13
C h a p t e r 8: d(o,r), d(o,r~) 45 C(a, r) = {a; £ I | |a; - o| = r} 45 T(a,ri,r2) 45 A.(o,ri,r 2 ) 45 D 46 D 46 diam(D) 46 5 46 <5(a,£>) 46 72-D 47 O(D) 47 J D ( r ( a , r 1 ) r 2 ) ) 47 £D(r(a,n,r2)) 47
C h a p t e r 5: V ( v , a i , . . . , o n , e ) 22 B(D,g) 22 Af m ( A 0 ) 2(5,5) Ker(
274
Ultrametric Banach Algebra
J ( r ( a , n , r 2 ) ) 47 £(r ( a , r i , r a ) ) 47 Chapter 9: TV\D 51 VD(T) 52 C{T) 52 P(.F) 52 CD{T) 52 diam(T) 52 Chapter 10: G(.F) 57 $(£>) 58 $ ' ( £ ) 58 Chapter 11: <, < 65 A(jF,g) 68
6(F,g), 8'(F,g) 68 Chapter 12: II • \\D 71 fl(D) 71 Rb(D) 71 1/5
71
72 73
Chapter 13: * 75
Chapter 16: 93 H(D), Hb(D), H0(D)
WD
/T./o
^(a)
9
8
99
Chapter 18: T(D,a,r) 105 S(D,a,r,q) 105 7D(a.r.9)
105
Chapter 19: W 114
;y0 (JO ii4 incT(D) 116 decT(.D) 116 Chapter 20: $(£>,£>) 121 $(C?) 121 II • || D O 121 H(D,0) 121 Chapter 21: R'(D,0,(bi)iej) «"(£>, e>,(k) i€ .7) H'(DtO,{bi)i€j) #"(£>, 0 , ( b i ) i e j ) Chapter 22: E(O) 127 Eo(C) 127
5(
E(L>), E 0 (D) !SO Chapter 15: ||z||«. 87 \\x\Um 87 r(.) 87 Properties (o), (p), (
Chapter 23: 0t 137 ipt 137 Chapter 25: T(r) 153 II ■ Ik 154 Chapter 29: S(P,V) 175 S(P) 175
124 125 125 125
93
Notation Index Chapter 30: Tn 183 / 184 Chapter 31: (i) 190 X^ 190 Chapter 33: Vn 197 £>„ 199 Chapter 35: q(i) 209
Chapter 36:
1K0IU 215 Chapter 37: A(J-) 221 A(ft) 221 Chapter 39: / . 234 Chapter 40: M(K) 242 A(K) 242
275
