Absolute values on algebras H(D)

Let K be an algebraically closed complete ultrametric field, and let D be an infraconnected set in K such that the set H(D) of the analytic elements on D is a ring. Among the continuous multiplicative semi-norms on H(D), we look for the ones that are absolute values. They are characterized by the location of the T-filters on D. Besides, we characterize the sets D such that H(D) admits at least one continuous absolute value [ . [. Notations: Let K be an algebraically closed field complete for an ultrametric absolute value. Given a E K and r > 0, d(a, r) (resp. d(a, r~), resp. C(a, r)) denotes the disk {x E a~ r} (resp. {x E K~ ~x -a~ r~, resp. the circle {x E K~ ~x a~ = r~ ). Given a E K, r’ > 0 and r" > r’, r(a, r’, r") denotes the annulus {x E K~ r’ a) r"}. Given a set A in K and a point a E K, we denote by 6(a, A) the distance from a to A. Let E be an infinite set in K, and let a E E. If E is bounded of diameter r, we denote by E the disk d(a, r), and if E is not bounded, we put E = K. Then, E B E is known to admit a partition of the form with r; = for each i E J. The disks d(a;, r-i)i~J, are named the holes of E. R(E) denotes the set of rational functions h E K(x) with no poles in E. This is a Ksubalgebra of the algebra KD of all functions from E into K. Then R(E) is provided with the topology UE of uniform convergence on E, and is a topological group for this topology. H(E) denotes the completion of R(E) for this topolgy and its elements are named the analytic elements on E [1], [2], [3], [9]. By [3], we remember that H(E) is a K-subalgera of the algebra KD if and only if E satisfies the following conditions: A) E B E as bounded,


ABSOLUTE VALUES ON ALGEBRAS H(D) by Kamal Boussaf and Alain Escassut
Ann. Math. RIa.ise Pascal, Vo1. 2, N° 2, 1995, pp.15-23 Abstract. Let K be an algebraically closed complete ultrametric field, and let D be an infraconnected set in K such that the set H(D) of the analytic elements on D is a ring. Among the continuous multiplicative semi-norms on H(D), we look for the ones that are absolute values. They are characterized by the location of the T-filters on D. Besides, we characterize the sets D such that H(D) admits at least one continuous absolute value [ . [. Notations: Let K be an algebraically closed field complete for an ultrametric absolute value.
Given a E K and r > 0, d(a, r) (resp. d(a, r~), resp. C(a, r)) denotes the disk {x E -a~ r} (resp. {x E K~ ~x --a~ r~, resp. the circle {x E K~ ~x -a~ = r) . Given a E K, r' > 0 and r" > r', r(a, r', r") denotes the annulus {x E K~ r' a) r"}. Given a set A in K and a point a E K, we denote by 6(a, A) the distance from a to A.
Let E be an infinite set in K, and let a E E. If E is bounded of diameter r, we denote by E the disk d(a, r), and if E is not bounded, we put E = K. Then, E B E is known to admit a partition of the form with r; = for each i E J. The disks d(a;, r-i)i~J, are named the holes of E.
R(E) denotes the set of rational functions h E K(x) with no poles in E. This is a Ksubalgebra of the algebra KD of all functions from E into K. Then R(E) is provided with the topology UE of uniform convergence on E, and is a topological group for this topology. H(E) denotes the completion of R(E) for this topolgy and its elements are named the analytic elements on E [1], [2], [3], [9].
By [3] Henceforth, D will denote an infraconnected set satis f ying Conditions A) and B).
In [7], [4], the continuous multiplicative semi-norms of an algebra H(D) were characterized by means of the circular filters on D. So we have to recall the definitions of monotonous and circular filters. 03C8 will be said to be punctual if Ker03C8 is a maximal ideal of codimension 1 of H(D).
We know that there exists a bijection M from D onto the set of maximal ideals of codimension 1 of H(D), defined as M(a) = f f E = 0} (indeed, this was shown in ~~~, Proposition 11.6, when D is closed and bounded, and it is easily extended to all sets D satisfying Conditions A) and B)). As a consequence, there exists a bijection S from D onto the set of punctual continuous multiplicative semi-norms of H(D) defined as S(a)( f) = whenever f E H(D).
In order to recall the characterization of the continuous multiplicative semi-norms of H(D), we first have to recall the definition of monotonous and circular filters. Given a filter 7 on D, we will denote by Z(,~') the ideal of the f E H(D) such that lim f(x) = o.
Let a E D and S E R+ be such that r(a, r, S) n D # 0 whenever r ~]0, S[ (resp. r(a, S, r) n D ~ 0 whenever r > S). We call an increasing (resp. a decreasing) filter of center a and diameter S, on D the filter .~' on D that admits for base the family of sets r(a, r, S) n D (resp. r(i, S, r) n D) . For every sequence (rn)n~IN such that rn rn+1 (resp. rn > rn+i ) and lim rn = S it is seen that the sequence r(a, rn, S) n D (resp. r(a, S, rn) ~ D) is a base of F and such a base is called a canonical base. . We call a decreasing filter with no center of canonical base (Dn)n~IN and diameter S > 0, on D a filter F on D that admits for base a sequence (Dn)n~IN of the form Dn = d(an rn) n D with Dn+1 ~ Dn , rn+l rn, lim rn = S, and ~ d(an, rn) = n-+oo ' ' n~IN Given an increasing (resp. a decreasing) filter .~' on D of center a and diameter r, we will denote by P(7) the set {x E D~ a~ > r} (resp. the set {x E a~ r} and by the set {x E a~ r} (resp. the set {x E al > r}. Besides C(F) will be named the body of F and P(F) will be named the beach of F.
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 F we will denote by diam(F) its diameter.
The field K is said to be spherically complete if every decreasing filter on ,F~ has a center in K. (The field 0152p for example is not spherically complete). However, every algebraically closed complete ultrametric field admits a spherically complete algebraically closed extension [10], [11].
Two monotonous filters F' and 9 are said to be complementary if U D.
Let F be an increasing (resp. a decreasing) filter of center a and diameter S on D . . ' is said to be pierced if for every r E~O, 5'(, (resp. r S) , r(a, r, S) (resp. r(a, S, r)) contains some hole Tm of D. A decreasing filter with no center .~', and canonical base (Dn)nEJN, on D is said to be pierced if for every m E N, Dm B Dm+1 contains some hole Tm of D.
Let a E D, let p = 6(a, D) be such that p S diam(D). We call circular filter of center a and diameter S on D the filter ~' which admits as a generating system the family of sets r(a, r', r") n D with a E d(a, S), r' S r", i.e. F is the filter q which admits for base the family of sets of the form D n with o', E i=l q ~ IN). A decreasing filter with no center, of canonical base (Dn)n~IN is also called circular filter on D with no center, of canonical base (Dn)n~IN.
Finally the filter of the neighbourhoods of a point a E D will be called circular filter of the neighbourhoods of a on D. It will be also named circular filter of center a and diameter 0.
A circular filter on D will be said to be large if it has diameter different from 0. Given a circular filter F, its diameter will be denoted by diam(F).
The set of the circular filters on D will be denoted by ~(D). Now let ,~' be a circular filter on D. By [7], , [4], we have the following characterization of continuous multiplicative semi-norms of H(D). Theorem 0: Let F be a circular filter on D. For every f E H(D), |f(x)| admits a limit along .~', and this limit, denoted by defines a continuous multiplicative sema-norm cpf on H(D). Further, the mapping e from 03A6(D) into Mult(H(D),uD) defined as = 03C6F is a bijection.
Notations: For convenience, when ,~ is the circular filter of center a and diameter r, we also denote by the multiplicative semi-norm 03C6F.
Here, assuming H(D) to be a K-algebra, we study what continuous multiplicative semi-norms of H(D) are norms, i.e. are absolute values on H(D). Of course, this requires H(D) to have no divisors of zero. But then, as a trenscendental extension of the field K, the field of quotients L of H(D) does admit absolute values extending the one of K. Hence so does H(D ) . The problem, here, is whether such absolute values are continuous with respect to the topology of H(D), i.e. are defined by circular filters on D.
So, we will give the condition a circular filter has to satisfy in order that its continuous multiplicative semi-norm be an absolute value, and next, we will characterize the sets D such that at least one of the continuous multiplicative semi-norms is an absolute value.
All this study involves T-filters, and now we have to introduce them.
Definition: Let:F be an increasing (resp. a decreasing) filter on D, of center a and diameter s. An element f E is said to be strictly vanishing along F if there exists t s (resp. t > s) such that > 0 for all r E ~t, s~, (resp. ~s, t~). Let ,~ be a decreasing filter on D, with no center, of diameter r, of canonical base (Dn)n~IN, with Dn = d(an, rn) n D. Then an element f E is said to be strictly vanishing along F if there exists t > s such that 03C6an ,r( f) > 0 for all r E [rn, t], for every n E 1N.
Let ,~" be a filter on D, and let A C will be said to be secant with A if for every F E ,~', A n F is not empty.
T-filters are certain pierced monotonous filters satisfying particular properties linked to the holes of D, and were defined in [1], [2], [5]. Here we will only use the following characterization: A monotonous filter F is a T-filter if and only if there exists f E H(D) strictly vanishing along ,~'.
Proposition P: Let b E D, 1 > 0 and let f E H(D) satisfy f(b) ~ 0 and 03C6b,l = 0. There exists an increasing T -filter ,~ of center b and diameter t EJO, I( such that f is strictly vanishing along F and satisfies > 0 for every s ~]0, t[. Let a E D and let r, s E 1R satisfy 6(a, D) r diam(D). If f satisfies f(x) = 0 whenever x E d(a, r) n D, then f is strictly vanishing along aT-filter .~' such that d(a, r) n D C and b E C(F). Let F be a large circular filter on D. Then 03C6F is not an absolute value if and only if it s atis fies one of the following conditions: a) There exists a T-filter G on D such that F is secant with P(G), b) ,~' is aT-filter. Proof: First, suppose that there exists a T-filter G satisfying a). By Lemma 1.6 A of [5], there exists f E H(D), strictly vanishing along ~, equal to 0 in all of ~(~). Hence we have f E H(D) strictly vanishing along 7 and therefore we have lim f(x) = 0, hence cpf is not a norm. Now we suppose that there exists no T-filter ç satisfying a) and that is not a Tfilter, and we suppose that c~F is not a norm. Let f E H(D) ~ ~0} satisfying yrf( f ) = 0. Let S = diam(F). Let bED be such that f (b) ~ 0.
We first assume that ,~' has a center a.
On the first hand, we suppose that b E d(a, S). Since 0, when r approaches 0 there does exist s ~]0, S[ such that = 0 and ~ 0 whenever r ~]0, s[. Hence f is strictly vanishing along the increasing filter 9 of center b and diameter s, and therefore ,~' is secant with P(g). On the second hand, we suppose that ~a -b~ > S. Let t = la -b~. If ~pb,t( f ) = 0, there exists s ~]0, tJ such that = 0 and cp6,r( f ) ~ 0 whenever r ~]0, s[, hence f is strictly vanishing along an increasing T-filter g of center b and diameter s and therefore ,~' is secant with 7~(9). Now we may assume 0. But cpb,t(f ) = and therefore there exists s E ~S, t( such that = 0 and cpa,,.( f ) # 0 whenever r t~. . Hence f is strictly vanishing along a decreasing T-filter 9 of center a and diameter s, and F is secant with P(G). Now, assume that F is a decreasing filter with no center. Let (Dn)n~IN be a canonical base of.F, and for each n E N, let Dn = d(an, rn) n D and let un = If there exists q E 1N such that uq = 0, by Proposition P, D admits a T-filter G such that Dq is included in and therefore, ,~' is obviously secant with 'P(~ Let incT(D) (resp. decT(D)) be the set of increasisng (resp. decreasing) T-filters on D. We will denote by ~ the relation defined on incT(D) (resp. decT(D)) by .~1 -~ F2 if C(,~'2) C C{,~'1). This relation is obviously seen to be an order relation on incT(D) (resp. decT(D)) . . An increasing (resp. a decreasing) T-filter ,~' will be said to be maximal if it is maximal in incT(D) (resp. in decT(D)) with respect to this relation.
We will call an ascending chain of increasing (resp. decreasing) T -filters a sequence of increasing (resp. decreasing ) T-filters such that .Fn -~ whenever n E IN. Let (Fn)n~IN be an ascending chain of increasing T-filters. For each n E 1N let rn = diam(Fn). Since the sequence (rn)n~IN is decreasing, we put r = lim rn, and n-~oo then r will be named the diameter of the chain.
We put A := n and for each n E IN, Dn := C(Fn) B A. The sequence (Dn)n~IN is then a base of a filter ,~' on D of diameter r.
If r = 0, since D Condition Condition B), A is a point a of D, hence ,~' is the filter of the neighbourhoods of a in D. If r > 0 ~' is a decreasing filter on D of diameter r.
In both cases F will be called the returning filter of the ascending chain (Fn)n~IN. Now let (F)n~IN be an ascending chain of decreasing T-filters and let a E P(Fn) for some n E IN. The sequence (rn)n~IN is an increasing sequence of limit r and r will be named the diameter of the chain. Since D belongs to A, by Condition A) we notice that r +00, and then we will call the returning filter of the ascending chain (Fn)n~IN the increasing filter F of center a and diameter r ( it is seen that F does not depend on the point a E whenever n E 1N). We are now able to characterize the sets D such that H(D) admits continuous absolute values. Let us recall the following theorem of [6]: The algebra H(D) has no divisors of zero if and only if D does not admit two complementary T.. filters.
We will use comparison between filters. Here, a filter .~' will be said thinner than a filter 9 every element of G belongs to F.

Theorem 2:
Let H(D) have no divisors of zero. Then Mult(H(D),UD) contains no norm if and only if D admits an ascending chain of T -filters whose returning filter is either a T -filter or a Cauchy filter.
Proof : On the first hand, we suppose that D admits an ascending chain of T-filters (Fn)n~IN whose returning filter F is either a T-filter or a Cauchy filter and we will prove that Mult(H(D),UD) contains no norm. We denote by r the diameter of this ascending chain (Fn)n~IN. Let G be a circular filter on D, of diameter s > 0.
By Theorem 1, we only have to show that either 9 is a T-filter, or G is secant with the beach of a T-filter.
First we suppose F is a Cauchy filter. Then the are increasing T-filters. Let q E 1N be such that rq s. Then G is clearly secant with P(Fq). Now we suppose that F is a T-filter. Then we just have to consider the case when g is secant with C(.~'} and is not equal to ,~'. First we suppose increasing, of center b and diameter r. Hence we have C(,~') = d(b, Then G has a diameter s ~]0, r[, and then it admits elements E of diameter t E]s, r[, included in d(b, r") n D. Since E n C(,~) ~ 0, given a point a E E n C(~) we have E C d(a, t). Let q E lN be such that rq > max(t, b|). Then E is included in d(b, rq) and therefore is included in Hence G is secant with P(Fq). On the second hand, reciprocally, we suppose that Mult(H(D), UD) contains no norm and we will show that D admits an ascending chain of T-filters whose returning filter is either a T-filter or a Cauchy filter.
We denote by R' the set of the diameters of the F E incT(D), by R" the set of the diameters of the F E decT(D), and we put R = ?Z' U R". Since H(D) has no norm, by Theorem 1 7Z is not empty. Since D belongs to A, by Condition A) R is obviously bounded. We put t = sup(R). Let a E D. We will show that incT(D) ~ 0. Indeed, suppose incT(D) = 0. First let D be bounded, of diameter S. Any decreasing filter on D has a diameter r S, and therefore the circular filter 9 of center a and diameter S is secant with D, but (of course) is not a T-filter on D, and is not secant with the beach of any decreasing T-filter on D. As a consequence, by Theorem 1 cpĩ s a norm. Thus we see that D is not bounded. Then, any circular filter g of center a and diameter r > t is not a T-filter and is not secant with the beach of any T-filter.
Finally this shows that is a norm again. Thus we see that incT(D) is not empty, and neither is ~Z'. Now, we put s = inf(R'). First we suppose s E R'. Let T E incT(D) satisfy diam(T) = s, and let b E C(T). Then for every r ~]0, s(, the circular filter of center b and diameter r is not a decreasing T-filter, and therefore is secant with the beach of a decreasing T-filter. Hence there exists a decreasing T-filter ,~' of center b and diameter .~ > s. But since H(D) has no divisors of zero, ,~" is not complementary with T, hence we have s i, i.e. s ~ r.
So, we clearly deduce the existence of a sequence of decreasing T-filters such that each one admits b as a center and has a diameter rn satisfying rn rn+i s, lim rn = s. Therefore the sequence (Fn)n~IN n-;oo is an ascending chain of decreasing T-filters such that fQ P(Fn) = C(T), hence the nE1V returning filter of the ascending chain (Fn)n~IN is a T-filter. Now, we suppose s ~ R'. Let (Tn)n~IN be a sequence in incT(D) such that lim diam(Tn) = s, with diam(Tn+1) diam(Tn) for all n E N. For every n E N, n-oo we put An = C(Tn), and rn = diam(An). By Lemma 1 the sequence C(Tn)nEIN is strictly decreasing, and so is the sequence (An)n~IN . Besides, the sequence (Tn)n~IN is an ascending chain of increasing T-filters. Obviously each An contains a hole Tn of D. _o If s = 0, then we have 6(b, Tn) rn, and therefore b does not belong to D, but then, by Condition A) b must belong to D. Thus, the sequence is an ascending chain of increasing T-filters that converges to b, and therefore the returning filter of the ascending chain (Tn)nEIN is a Cauchy filter.
Finally, it only remains to consider the case when s > 0, with .s ~ R'. Let ,~' be the returning filter of the sequence (Tn)nEIN. . If ,~' were a T-filter, or were secant with the beach of an increasing T-filter, this increasing T-filter would have a diameter inferior or equal to s. Hence F either is a decreasing T-filter or is secant with the beach of a decreasing T-filter. Of course, if ,~' is a T-filter, it is just the returning filter of the chain (Tn)n~IN . Finally if F is secant with the beach of a decreasing T-filter g, then we have diam(G) s because if diam(G) were strictly superior to s, then G would be complementary to Tn when n is big enough, and therefore H(D) would have divisors of zero. Hence we have diam(G) = s, and therefore G is just the returning filter of the sequence (Tn)nEIN. This finishes proving that D admits an ascending chain of T-filters whose returning filter is either a T-filter or a Cauchy filter, and this ends the proof of Theorem 2.