Maximal chains of closed prime ideals for discontinuous algebra norms on C(K)

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Maximal chains of closed prime ideals for discontinuous algebra norms on C(K)

Jean Esterle

To cite this version:

Jean Esterle. Maximal chains of closed prime ideals for discontinuous algebra norms on C(K). Math-

ematical Proceedings of the Royal Irish Academy, 2012, 112A (2), pp.101-115. �hal-00773663�

Maximal chains of closed prime ideals for discontinuous algebra norms on C ( K )

J. Esterle

ABSTRACT: LetK be an infinite compact space, letC(K)be the algebra of continuous complex-valued functions ofK, let F be a well-ordered chain of nonmaximal prime ideals of C(K), let IF be the smallest element of F and let MF be the unique maximal ideal of C(K) containing the elements of F.

Assuming the continuum hypothesis, we show that if |C(K)/IF|= 2^ℵ0, and if there exists a sequence (Gn)n≥1 of subsets of F ∪ {MF} stable under unions such thatF ∪ {MF}=∪n≥1Gn,then there exists a discontinuous algebra norm ponC(K)such that the set of all nonmaximal prime ideals ofC(K)which are closed with respect topequalsF.

AMS classification: 46H40 (primary), 46J10, 03E50 (secondary)

1 Introduction

LetK be an infinite compact space, and letC(K)denote the algebra of contin- uous complex valued functions on K. An algebra seminormk.k on C(K) is a seminorm satisfyingkf gk ≤ kfkgkfor everyf, g∈ C(K),and such a seminorm is said to be continuous if there existsk >0 such that kfk ≤kkfkK for every f ∈ C(K), where kfkK := maxt∈K|f(t)| denotes the usual norm on C(K). A classical result of Kaplansky [17] shows that ifk.kis any algebra norm onC(K) we have

kfk ≥ kfkK for everyf ∈ C(K).

The existence of a discontinouous algebra seminorm onC(K),which is equiv- alent to the existence of a discontinuous algebra norm onC(K)and to the exis- tence of a discontinuous homomorphism fromC(K)into a Banach algebra, is the well-known Kaplansky’s problem, which turns out to be undecidable in ZFC.

Badé and Curtis obtained in [2] some partial continuity results, but H.G. Dales and the author [3], [10], [5] proved independently that if2^ℵ0 =ℵ1, which means that the continuum hypothesis (CH) is assumed, then discontinuous algebra seminorms exist onC(K)for every infinite compact spaceK. Those commuta- tive Banach algebrasAfor which a discontinuous homomorphismφ:C(K)→A

does exist under CH were characterized in [11], see also [4]. On the other direc- tion Solovay and Woodin constructed models of set theory including the axiom of choice and Martin’s axiom in which all algebra seminorms onC(K)are con- tinuous, see [6] for details. Notice that models of set theory in which2^ℵ0 =ℵ2

and in which discontinuous algebra seminorms onC(K)exist for every infinite compact spaceKwere constructed independently by Frankiewicz- Zbierski and Woodin [14] [26].

The structure of closed ideals ofC(K)for discontinuous algebra seminorms was investigated by A.M. Sinclair [24] and later, independently, by the author [8], who showed that the closure of an ideal is the intersection of all closed prime ideals which contain it. Also a chain of nonmaximal closed prime ideals is well- ordered with respect to inclusion, see [8]. If K is an F-space, which means that f and |f| generate the same ideal of C(K) for every f ∈ C(K), then the familyP rim(q) of all nonmaximal prime ideals which are closed with respect to an algebra seminormq on C(K) is a finite union of well-ordered chains of nonmaximal prime ideals.

Pham [19], [20], [21] showed that the situation is much more complicated in the general case. We refer to the survey paper [12] for a discussion of his deep contributions and of problems which remain open.

This paper is part of a program intended to describe, assuming the contin- uum hypothesis, the familiesU of nonmaximal prime ideals ofC(K)such that U =P rim(q)for some algebra seminormqonC(K)(or, equivalently, for some algebra normqonC(K)), whereP rim(q)denotes the set of nonmaximal prime ideals ofC(K)which are closed with respect toq.A neccessary condition is re- lated to the notion ofpseudo-finite familyintroduced by Pham in [19]: a family (Eλ)λ∈Λof subsets of a setEis said to be pseudofinite if the set{µ∈Λ|x /∈Eµ} is finite for everyx∈ ∪λ∈ΛEλ.Using results of [8] and ideas from [19] and [21]

it is possible to show that every sequence of elements ofP rim(q)must contain a pseudofinite subsequence (this means thatP rim(q)is "relatively compact" in the sense of [21]), which implies that every union of elements ofP rim(q) is a finite union of prime ideals, see [21].

Also{Iζ}ζ<ω is a well-ordered family of elements ofP rim(q),withIη (Iζ

forη < ζ,then∪ζ<σIζ ∈P rim(q)if[0, σ[has no countable cofinal subset, since every sequence of elements of∪η<ζIη is contained inIη for someη < ζ.

Similarly if(Iλ)λ∈Λ,is a pseudo-finite family of elements of P rim(q), with Iλ * Iµ for λ 6= µ, and if Λ is uncountable, then ∪λ∈ΛIλ ∈ P rim(q), since every sequence of elements of∪λ∈ΛIλ is contained inIλ for someλ∈Λ.We do not know whether these three necessary conditions are together sufficient for a familyU of nonmaximal prime ideals ofC(K)satisfying|C(K)/IU|= 2^ℵ0 to be equal toP rim(q)for some algebra seminormqonC(K).

In this paper we will focus attention on chains of nonmaximal primes. Our main result is the fact that if a well-ordered chain (Iζ)ζ≤ω of prime ideals of C(K), with Iω maximal, is "almost stable under unions", and if |C(K)/I0| = 2^ℵ0, then there exists a discontinuous algebra seminorm q (or, equivalently, a discontinuous algebra norm q) on C(K) such that {Iζ : ζ < ω} = P rim(q).

We do not know whether all maximal chains of closed prime are "almost stable

under unions" in the sense of definition 1, but proposition 2.6 shows that the construction of a counterexample would require methods very different from the method used in the paper.

The author has known for many years that if (Iζ)ζ<ω is a chain of non- maximal prime ideals of C(K), and if |C(K)/I0| = 2^ℵ0, then there exists an algebra seminormqonC(K)such that{Iζ :ζ < ω} ⊂P rim(q),but was look- ing for a "natural" proof: it is known that given two prime ideals I and J of C(K) such that I ⊂ J there exists a subalgebra AJ of C(K)/I such that C(K)/I = AJ ⊕π(J), where π : C(K) → C(K)/I denotes the canonical sur- jection. It it were possible to construct a family (AJ)J⊃I,Jprime such that AJ2 ⊂ AJ1 for J1 ⊂ J2, then we hould have PJ1 ◦PJ2 = PJ2 ◦PJ1 = PJ2, wherePJdenotes the projection ofC(K)/IontoAJ such thatKer(PJ) =π(J).

Given any chainF of nonmaximal prime ideals of C(K)containingI and any algebra normponC(K)/I the formula

q(f) =maxJ∈F(p◦PJ◦π)(f)

would define an algebra seminorm q on C(K) such that F ⊂ P rim(q). A variant of this formula using a suitable nonincreasing sequence of algebra norms onC(K)/I converging pointwise to zero would allow to prove directly our main result. Unfortunately, the author has been unable so far to construct such a family(AJ)J⊃I,Jprime. So we present an indirect proof, based on the obvious existence of an analogous family for the "universal" algebra of power seriesCω1

used in [9] and [10], see section 2.

We discuss at the end of the paper a natural extension of the main result which, if true, would allow to obtain a complete characterization (under CH) of the continuity ideals associated to discontinuous homomorphisms fromC(K).

2 Algebra norms on the universal algebra C

and chains of closed primes

We introduce some objects used by the author in his construction of discontinu- ous homomorphisms fromC(K).Letω1be the smallest uncountable ordinal. We denote bySω1 ⊂ {0,1}^ω1the set of all transfinite dyadic sequencesx= (xζ)ζ<ω1

for which there exists η(x)< ω1 such thatxη(x) = 1and such that xζ = 0for everyζ > η(x).

Equiped with the lexicographic order, Sω1 is a linearly ordered set, and a classical result of Sierpiński [23], see also [4] shows that every linearly ordered set of cardinal≤ ℵ1 is order-isomorphic to a subset ofSω1.

Denote byGω1 ⊂S_ω^R₁ the set of all real-valued functionsφonSω1 such that Supp(φ) := {s ∈Sω1 | φ(s) 6= 0} is well-ordered and at most countable. For φ∈Gω1\ {0},denote byρ(φ)the smallest element ofSupp(φ).By definition, a nonzero elementφ∈Gω1 is said to be strictly positive ifφ(ρ(φ))>0.Equipped with the linear structure inherited from the linear structure of S^R_ω1, Gω1 is a

linearly ordered real vector space, which contains a copy of every linearly ordered group of cardinalℵ1.

Now letGbe a linearly ordered group, and letkbe a field. We will denote by F(G, k)the set of all functionsf :G→ksuch thatSupp(f) :={τ∈G|f(τ)6=

0} is well-ordered, and we set

F₍₁₎(G, k) :={f ∈ F(G, k)| |supp(g)| ≤ ℵ0}.

Now let f, g ∈ F(G, k), and let τ ∈ G.If τ /∈Supp(f) +Supp(g) :={α+ β}α∈Supp(f),β∈Supp(g),set (f g)(τ) = 0.Otherwise set

(f g)(τ) = X

α∈Supp(f),β∈Supp(g) α+β=τ

f(α)g(β).

Thenf gis well-defined, since the set{(α, β)∈Supp(f)×Supp(g)|α+β = τ} is finite for everyτ ∈Supp(f) +Supp(g), andf g ∈ F(G, k).In fact Hahn observed in 1907 in [16] thatF(G, k) is a field. Set v(f) = inf(Supp(f))for f ∈ F(G, k)\{0}.Thenvis a valuation on the fieldF(G, k),and the valued field F(G, k) is maximal: if U is a field containing F(G, k), and if wis a valuation onU with values inGsuch thatw(f) =v(f)for everyf ∈ F(G, k)\ {0},then U =F(G, k).

Mac Lane showed in [18] that F(G, k) is algebraically closed if k is alge- braically closed and if the equationnt=τ has a solution in Gfor everyτ∈G and every integer n≥ 2. In particular, the fields F(Gω1,C)and F₍₁₎(Gω1,C) are algebraically closed.

We set, with the conventionv(0) = +∞> τ for everyτ∈G,

Cω1 :={f ∈ F(1)(Gω1,C)|v(f)≥0},Mω1:={f ∈ F(1)(Gω1,C)|v(f)>0}, so that the radical complex algebraMω1 is the unique maximal ideal of the commutative unital complex algebraCω1. The algebraCω1 is universal: if the continuum hypothesis is assumed thenCω1 contains a copy of every commuta- tive unital complex algebra of cardinality2^ℵ0 which is an integral domain and possesses a character.

Letr∈ Mω1. Since the field of fractions ofCω1 is algebraically closed, the equation yⁿ = 1 +r has n solutions of the form e^{2ikn π}(1 +rn) ∈ Cω1 where 0≤k≤n−1.Set

(1 +r)¹ⁿ = 1 +rn. (1)

Now let p be any algebra seminorm on Cω1. Let R be the completion of Mω1/Kerpwith respect top,and letπ=Cω1 →Cω1/Ker(p)be the canonical surjection. ThenA:=R⊕C.1is the completion ofCω1 with respect top.Since Ris a commutative radical algebra, the series

∞

P

n=1

(−1)^n+1π(r)_nⁿ converges inR.

Seta:=

∞

P

n=1

(−1)^n+1π(r)_nⁿ,so that1 +π(r) =exp(a).We have

0 =exp(a)−(1+π(rn))ⁿ = (exp(a/n)−1−π(rn))

n−1

X

j=0

exp(ja/n)(1+π(rn))^n−j−1.

We have χ Pn−1

j=0 exp(ja/n)(1 +π(rn))^n−j−1

= n, where χ denotes the unique character onA,and soPn−1

j=0 exp(ja/n)(1 +π(rn))^n−j−1is invertible in A.We obtain

1 +π(rn) =exp(a/n), limn→∞p(rn) = 0. (2) The author’s construction of a discontinuous homomorphism of C(K)was based on the existence of algebra norms onCω1, suggested by Allan’s embedding ofC[[X]],the algebra of all formal power series in one variable, into some Banach algebras [1](Dales and Woodin discuss in [7] the normability of algebras of formal power series larger thanCω1,which remains an open problem). We will need a precise form of this result, given by the following theorem.

Theorem 2.1 There exists a sequence(k.kn)n≥1 of algebra norms onCω1 sat- isfying the following properties

(i) kukn+1≤ kukn for everyu∈Cω1, and limn→∞kukn= 0for everyu∈ Mω1, (ii) for everyτ∈G⁺_ω1 and everyr∈ Mω1,we have

limn→∞kX^{τ /n}k1= 1,and

limn→∞kw−wX^{τ /n}(1 +r)ⁿ¹k1= 1 for everyw∈ Mω1.

Proof: Denote by A(D)the usual disc algebra of functions holomorphic on the open unit discDwhich have a continuous extension to the closed unit disc D,which is a Banach algebra with respect to the normkfk:=max|z|≤1|f(z)|= max|z|=1|f(z)|.SetM ={f ∈ A(D)|f(1) = 0},and setΩ :={f ∈M |[f M]⁻ = M}.SinceM possesses a bounded approximate identity, it follows from [10] that there exists a one-to-one mapδ:G⁺_ω1 →Ω∪ {1}such thatδ(σ+τ) =δ(σ)δ(τ) forσ∈G⁺_ω1, τ ∈G⁺_ω1, and such that δ(σ)∈Ωfor σ >0, withδ(τ)(0) >0 for everyτ ∈G⁺_ω1.

Setθm(z) =exp(_m(z−1)^z+1 )for|z| ≤1, z6= 1, so thatθm is an inner function which is continuous onD\ {1}.ThenθmM is a closed ideal ofA(D)contained in M, the quotient algebraM/θmM is radical, and∪m≥1θmM is dense in M.

Denote byπm :A(D)/θ1M → A(D)/θmM the canonical surjection. It follows from [9] that there exists a one-to-one homomorphismφ : Cω1 → A(D)/θ1M such thatφ(X^τ) = (π1 ◦δ)(X^τ)forτ∈G⁺_ω1.Set, foru∈Cω1, n≥1,

kukn=k(πn◦φ)(u)k.

Setπ =π1, letu∈Cω1\ {0}, and setτ =v(u). We haveu=X^τ(λ+w), where λ ∈ C\ {0}, w ∈ Mω1, and so φ(u) = π(δ(X^τ))(λ+φ(w)). Clearly, φ(w) ∈ π(M), and so πn(λ1 +φ(w)) is invertible in A(D)/θnM, (πn◦φ)(u) generates a dense ideal of M/θnM and(πn◦φ)(u)6= 0.So k.kn is an algebra norm onCω1. SinceθnM ⊂θn+1M forn≥1,and since ∪n≥1θnM is dense in M,the sequence(k.kn)n≥1 satisfies (i).

Sinceδ(τ)is an outer function, there exists a function f analytic onDand continuous onD\ {1} such thatδ(τ)(z) =exp(f(z))for|z| ≤1, z6= 1,and we can choose f so that f(0) ∈ R. Set gn(z) = exp_f(z)

n

for m ≥ 1, with the conventiongn(1) = 0. Then gn ∈ A(D), and g_nⁿ =δ(τ). Since gn(0) > 0 and δ(_n^τ)(0)>0,we havegn=δ(^τ_n) =φ(X^τn).

Sincef(z)/n→0 uniformly on every compact subset ofD\ {1}as n→ ∞, we have limn→∞kh−hgnk= 0for everyh∈M.Hence limn→∞kw−wX^nτk1= 0 for everyw∈ Mω1.The fact that limn→∞kw−wX^nτ(1+r)ⁿ¹k1= 1forr∈ Mω1

follows then from (2).

We will use the following easy lemma.

Lemma 2.2 Let Pω1 be the set of all prime ideals of Cω1. For I ∈ Pω1, set WI :={v(u) : u∈ I\ {0}}, and for u∈ Cω1 define θI(u) : G⁺_ω1 → C by the formulae

θI(u)(τ) = 0 if τ∈WI

θI(u)(τ) =u(τ)ifτ /∈WI

Then θI(u) ∈ Cω1, θI : Cω1 → Cω1 is an algebra homomorphism, and we have the following properties:

(i) For everyI∈ Pω1,Ker(θI) =I, u−θI(u)∈Iforu∈Cω1, θI(X^τ) =X^τ forτ ∈G⁺_ω1\WI,and[θI(1 +r)]^1/n=θI((1 +r)^1/n)for r∈ Mω1, n≥1.

(ii) If I1, I2∈ Pω1, and ifI1⊆I2,then θI2◦θI1 =θI1◦θI2=θI2.

(iii) If u ∈ Cω1, I1 ∈ Pω1, I2 ∈ Pω1, and if θI1(u^m) = θI2(u^m) for some m≥1, thenθI1(u) =θI2(u).

(iv) IfF ⊂ Pω1is well-ordered with respect to inclusion, then the set{θI(u)}I∈F

is finite for everyu∈Cω1.

Proof: SinceSupp(θI(u))⊂Supp(u),we haveθI(u)∈Cω1for everyu∈Cω1. Clearly,θI(X^τ) =X^τ for everyτ∈G⁺_ω1\WI.Also if r∈ Mω1, n≥1,we have θI(1 +r)−1 = θI(r)∈ Mω1, and θI

[1 +r]^1/nn

= θI(1 +r), which shows that[θI(1 +r)]^1/n=θI([1 +r]^1/n).

The map θI : Cω1 → Cω1 is linear. Each u ∈ Cω1 \ {0} can be written under the formu=X^v(u)w, where w∈Inv(Cω1), and sou∈I if and only if v(u)∈WI,andGω1\WI is stable under sums. This shows thatθI is an algebra homomorphism, that Ker(θI) = I and that u−θI(u) ∈ I everyu ∈ Cω1. It follows then from the definition ofθI1 andθI2 that θI2◦θI1 =θI1◦θI2 =θI2 if I1⊆I2.

Now assume thatθI1(u^m) =θI2(u^m)for somem≥1,withI2⊂I1.Ifu∈I1, there is nothing to prove. Ifu /∈I1,thenu^m−1∈/ I1,and

m−1

P

j=0

θI1(u^j)θI2(u^m−j)−

mu^m−1∈I1, and so

m−1

P

j=0

θI1(u^j)θI2(u^m−j)6= 0.We have

0 =θI1(u^m)−θI2(u^m) = (θI1)(u)−θI2(u))

m−1

X

j=0

θI1(u^j)θI2(u^m−j),

and soθI1(u) =θI2(u).

LetF be a well-ordered chain of elements of Pω1, let(In)n≥1 be a strictly increasing family of elements of F, set I := ∪n≥1In, and let u ∈ Cω1. Since u−θI(u)∈I, there existsm≥1 such thatu−θI(u)∈Im.Hence forn ≥m we have

0 =θIn(u−θI(u)) =θIn(u)−(θIn◦θI)(u) =θIn(u)−θI(u).

SinceF is well-ordered, this shows that the set{θI(u)}I∈F is finite for every u∈Cω1.

We will need the following simple observation.

Lemma 2.3 Let A be a unital subalgebra of Cω1 which possesses the following properties

(i) If a, b∈A, and ifa²∈/bA,then b²∈aA,

(ii) the equationxⁿ=ahas a solution inAfor everya∈Aand everyn≥1.

If I is a prime ideal of A, denote by I⁻ the the ideal of Cω1 generated by I,and denote byI⁺ the union of all idealsJ of Cω1 such that J∩(A\I) =∅.

Then I⁻ ⊂I⁺, I⁻ andI⁺ are prime ideals ofCω1,andI=I⁻∩A=I⁺∩A.

Proof: Since the ideals of Cω1 form a chain,I⁺ is an ideal ofCω1,which is maximal among all ideals J of Cω1 such that J ∩(A\I) = ∅. Since A\I is stable under products, a standard result of elementary algebra shows thatI⁺is prime.

Letu∈I⁻\ {0}.We haveu=a1b1+. . .+ambm,wherea1, . . . , am∈Iand whereb1, . . . , bm∈Cω1.

Letj0∈ {1, ..., m}be such thatv(aj0bj0) =min1≤j≤mv(ajbj).Thenv(u)≥ v(aj0bj0) = v(aj0) +v(bj0) ≥ v(aj0), and so v(ua⁻¹_j0 ) ≥ 0, ua⁻¹_j0 ∈ Cω1, and u∈aj0Cω1.HenceI⁻=∪a∈IaCω1.

Letu∈Cω1 satisfyinguⁿ ∈I⁻ for somen≥1, and leta∈I andb∈Cω1

such thatuⁿ=ab.There existsc∈Aandd∈Cω1 such thatcⁿ=aanddⁿ=b, anduⁿ =cⁿdⁿ.Henceu=λcdfor someλ∈Candu∈cCω1 ⊂I⁻,which shows thatI⁻ is semiprime. HenceI⁻ is prime since the ideals of Cω1 form a chain.

Leta∈A\I,and letb∈I.There exists c∈Asuch thatc⁴=b,andc∈I.

Thena² ∈/ cAsincea²∈/ I,and so c² ∈aAand v(b) = 2v(c²)> v(c²)≥v(a),

which shows thata /∈bCω1. Hence a /∈I⁻. This shows that I⁻∩A=I, and I⁻ ⊂I⁺.SinceI⁺∩A⊂I,we haveI⁻∩A=I⁺∩A=I.

Notice that It follows from (i) that the prime ideals ofAform a chain, which is also a consequence of the fact thatI =I⁻∩Afor every prime idealI ofA.

Hence∪λ∈ΛIλ is a prime ideal ofA for every family(Iλ)λ∈Λ of prime ideals of A.SinceI⁻=∪a∈IaCω1,we have[∪λ∈ΛIλ]⁻=∪λ∈ΛI_λ⁻.

It follows from the definition of the ideals I⁺ that ∪λ∈ΛI_λ⁺ ⊂ [∪λ∈ΛIλ]⁺. But if there exists a sequence(In)n≥1of nonmaximal prime ideals of A such that M=∪n≥1In, we have∪n≥1I_n⁺ (Mω1 = [∪n≥1In]⁺, since the set of strictly positive elements ofGω1 has no countable coinitial subset.

If A is a family of linear subspaces on a complex linear space E we will denote byU(A)the set of all unions of subfamilies ofA,and we will denote by V(A)the set of all linear subspaces ofEwhich belong toU(A).We will say that Ais stable under unions ifV(A) =A.We will need the following notion.

Definition 1LetEbe a linear space and letFbe a family of linear subspaces ofE.We will say that the familyF is almost stable under unions if there exists a sequence(Fn)n≥1 of subfamilies ofF such that ∪n≥1V(Fn) =F.

A countable family of linear subspaces of E is indeed almost stable under unions. Also, sinceV(V(A)) =V(A)for every familyAof linear subspaces ofE, we see that ifF is almost stable under unions there exists a sequence(Fn)n≥1

of subfamilies ofF stable under unions such thatF=∪n≥1Fn.

Notice that ifFis almost stable under unions, and ifGis a chain of elements ofF which doest not admit any countable cofinal subset, thenG ∩ Fn is cofinal inG for somen≥1,and so∪{L :L∈ G} ∈ F.

The main results of the paper are a direct consequence of the following lemma.

Lemma 2.4 Let A be a unital subalgebra of Cω1 satisfying conditions (i) and (ii) of lemma 2.3, letMbe the unique maximal ideal ofA,letFbe a well-ordered family of elements of nonmaximal prime ideals of A, and set F˜ := {I⁺}I∈F. Then there exists an algebra norm q on Cω1 such that every element of F˜ is closed inCω1 with respect toq, so that every element of F is closed in A with respect toq.

If, further, F ∪ {M} is almost stable under unions, and if the ideals of A closed with respect to any algebra seminorm on A are prime, then there exists an algebra seminorm on Cω1 such that the set of all nonmaximal prime ideals ofA which are closed with respect toqequalsF.

Proof: Letpbe any algebra norm onCω1,and setq(u) =max_J∈F˜p(θJ(u)) foru∈ Cω1, where (θJ)J∈Pω1 is the family of homomorphisms constructed in lemma 2.2. Since F˜ is well-ordered with respect to inclusions, it follows from lemma 2.2 (iv) thatqis well-defined, and, obviously, every element ofF˜is closed in with respect toq,so that everyI∈ F is closed inA with respect toq, since it follows from lemma 2.3 thatI =I⁺∩A.

Now assume thatF ∪ {M}is almost stable under unions, and let(k.kn)n≥1

be a sequence of algebra norms onCω1 satisfying the conditions of theorem 2.1.

We can write F ∪ {M} = {Iζ}ζ≤ω, and there exists a sequence (Ωn)n≥1

of subsets[0, ω]satisfying[0, ω] =∪n≥1Ωn such thatFn :={Iζ}ζ∈Ωn is stable under unions for everyn≥1.Forζ≤ω,letn(ζ)be the smallest positive integer such thatζ∈Ωn,and set, foru∈Cω1,

qζ(u) =kθ_I⁺

ζ(u)k_n(ζ),

so thatqζ is an algebra seminorm onCω1 such thatKer(qζ) =I_ζ⁺.It follows again from lemma 2.2 that the set{θ_I⁺

ζ(u)}ζ<ωis finite for everyu∈Cω1.Hence the set{qζ(u)}ζ<ω has a largest element for u∈Cω1.Now set

qF(u) =maxζ<ωqζ(u).

We see again that every element ofF˜ is closed inCω1 with respect toqF,so that every element ofF is closed inA with respect toqF.

Let u∈ M, and let Iσ be the smallest element of F ∪ {M} containingu.

We want to show that uCω1 is dense in I_σ⁺ and that uA is dense in Iσ with respect to qF. We can restrict attention to the case where u= X^τ(1 +r)for some strictly positiveτ ∈Gω1 and some r∈ Mω1. Setun =X^τn(1 +r)ⁿ¹ for n ≥ 1. Since the complex algebra A satisfies condition (ii) of lemma 2.3, we haveun∈A,and since the ideals ofAwhich are closed with respect toqF are prime,unbelongs to the closure ofuAinAwith respect toqF for everyn≥1.

Leta∈I_σ⁺.Then there exists a finite family{ζ1, . . . , ζk}of elements of[0, σ[

such that for everyζ ∈ [0, σ[ there exists j ≤k satisfyingθIζ(a) =θI_ζj(a). It follows from lemma 2.2 (i) that we have

θIζ(X^nτa) =X^nτθIζ(a) =X^nτθI_ζj(a) =θI_ζj(X^τna).

HenceqF(a−X^nτa) =maxζ<σqζ(a−X^τna) =maxj≤kqζj(a−X^τna).We obtain

lim sup_n→+∞qF(a−X^τna)≤limn→+∞maxj≤kkθI_ζj(a)−X^τnθI_ζj(a)k1= 0.

It follows from (2) that limn→+∞qF((1 +r)ⁿ¹ −1) = 0,and we have limn→+∞qF(a−aun) = 0, (3) which shows thatuCω1 is dense inI_σ⁺anduAis dense inIσ with respect to qF.

Now letIbe a prime ideal ofAwhich is closed with respect toqF,and letσ be the smallest element of[0, ω]such thatIσ containsI.Since the prime ideals ofAform a chain, we haveIζ (I for everyζ < σ.

If ∪ζ<σIζ (I,letu∈I\ ∪ζ<σIζ. Then σis the smallest element of [0, ω]

such that Iσ contains u, and so uA is dense in Iσ with respect to qF, which shows thatI=Iσ∈ F ∪ {M}.Hence I∈ F ifI6=M.

Now assume thatI=∪ζ<σIζ.IfΩn∩[0, σ[is cofinal in[0, σ[for somen≥1, thenI∈ V(Fn)⊂ F ∪ {M}.

IfI=∪ζ<σIζ,and ifΩn∩[0, ζ[is not cofinal in[0, ζ[ for anyn≥1,thenσ is a limit ordinal, and for everym≥1there existδ < σsuch thatn(ζ)≥mfor δ < ζ < σ.

Leta∈Iσ.The set{θIζ(a)}ζ<σ is finite, so we have limm→∞maxζ<σkθIζ(a)km= 0,

and there exists m1 ≥ 1 such that kθIζ(a)km < ǫ/3 for every ζ < σ and everym≥m1.Letδ < σsuch thatn(η)≥m1forδ < η < σ,and letu∈I\Iδ. Applying 2.3 to the family{Iζ}ζ≤δ∪ Mω1,we obtain

lim sup_n→+∞maxζ≤δqζ(a−aun) = 0.

Hence there exists m2 ≥ m1 such that maxζ≤δqζ(a−aun) < ǫ for every n≥m2, and we can assume that kunk1 <2 for n≥m2,so that we have, for δ < ζ < σ, n≥m2

qζ(a−aun)≤qζ(a)(1 +qζ(un))≤qζ(a)(1 +kunk1)< ǫ.

We thus see that qF(a−aun)< ǫ for n≥m2. Hence I =∪ζ<σIζ =Iσ ∈ F ∪ {M},and I∈ F ifI6=M.

In the case whereA=Cω1,we obtain the following result, which does not depend on the continuum hypothesis.

Theorem 2.5 LetF be a chain of prime ideals ofCω1 such thatF ∪ {Mω1} is almost stable under unions. Then there exists an algebra seminorm onCω1 such that the set of nonmaximal prime ideals ofCω1 which are closed with respect to qequalsF.

We do not know whether the set of prime ideals of Cω1 which are closed with respect to an algebra seminorm onCω1 is necessarily almost stable under unions. The following simple observation shows that this is indeed the case for algebra seminorms constructed via the method used in the proof of lemma 2.4.

Proposition 2.6 Let (E,k.k)be a normed vector space, and let F be a chain of linear subspaces ofE stable under unions. Assume that there exists a family (PF)F∈F of endomorphisms ofEsatisfyingKer(PF) =F such thatPF1◦PF2 = PF2 for F1, F2 ∈ F, F1 ⊆F2. Let (k.kn)n≥1 be a sequence of norms on E,and letH be the set of linear spaces F ∈ F such that PF : (E,k.k)→(E,k.kn) is bounded for some n ≥1. Then H is almost stable under unions, and F is a closed subspace of(E,k.k)for every F ∈ H.

Proof: Clearly,F=P_F⁻¹(0)is closed in(E,k.k)for everyF∈ H.Form, n≥ 1setHm,n:={F∈ H | kPF(x)kn≤mkxk ∀x∈E},so thatH=∪m,n≥1Hm,n. Let (Fλ)λ∈Λ be a family of elements of Hm,n, and set F := ∪λ∈ΛFλ, so that

F ∈ F.Letx∈E.Thenx−PF(x)∈Ker(PF) =F,and sox−PF(x)∈Fλfor someλ∈Λ.We have

PF_λ(x)−PF(x) =PF_λ−(PF_λ◦PF)(x) =PF_λ(x−PF(x)) = 0.

Hence kPF(x)kn = kPFλ(x)kn ≤ mkxk and F ∈ Hm,n, which shows that Hm,nis stable under unions.

The maps(θI)I∈Pω1 satisfy the conditionθI1◦θI2 =θI2 ifI1andI2are prime ideals of Cω1 such that I1 ⊆ I2. Let(k.kn)n≥1 be any nonincreasing family of algebra norms onCω1, letF be any well-ordered family of nonmaximal prime ideals ofCω1,and letδ:F →N be an application. LetIq be the set of prime ideals closed with respect to the algebra seminorm defined foru∈Cω1 by the formula

q(u) =maxI∈FkqI(u)kδ(I).

If follows from the proposition that Iq contains a family F ⊃ F ∪ {M˜ ω1} which almost stable under unions. Hence other ideas would be needed to con- struct, if possible, an algebra seminormq onCω1 such thatIq∪ {Mω1}is not almost stable under unions.

Denote byP rim(q)the set of nonmaximal prime ideals of C(K)which are closed with respect to an algebra seminormq on C(K). We also deduce from lemma 2.4 the following result.

Theorem 2.7 (CH) Let K be an infinite compact space, and let F be a well- ordered chain of nonmaximal prime ideals of C(K) such that|C(K)/I0|= 2^ℵ0, whereI0 denotes the smallest element of F,and letMbe the maximal ideal of C(K)containing the elements of F.

If F ∪ {M}is almost stable under unions, then there exists a discontinuous algebra normq onC(K)such thatF =P rim(q).

Proof: There exists a one-to-one homomorphism φ : C(K)/I0 → Cω1. Set A=φ(C(K)/I0).Denote by C_R(K)the algebra of continuous real valued func- tions onK,and setJ =I0∩ C_R(K).Letπ:C(K)→ C(K)/I0be the canonical surjection. It is a standard fact that the quotient order onC_R(K)/J is a linear order. Letf, g∈ C(K).If, say,π(|f|)≤π(|g|),thenπ(f)²∈π(g)C(K)/I,see for example lemma 2.1 of [22]. It is also a standard fact that the equationxⁿ =a has a solution inC(K)/I0for everya∈ C(K)/I0(more generally ifI is a prime ideal ofC(K)the equationxⁿ+an−1xⁿ⁻¹+. . .+a0= 0has a solution inC(K)/I fora0, . . . , an−1∈ C(K)/I,use for example exercise 13A of [15] to adapt to the complex case the argument used in the proof of theorem 13.4 of [15] to prove an analogous result in the real case fornodd). SoAsatisfies conditions (i) and (ii) of lemma 2.3.

Ifpis any algebra seminorm onA, thenq:=p◦φ◦πis an algebra seminorm onC(K),and it follows from [24] or [8] thatf belongs to the closure off²C(K) with respect toq for everyf ∈ C(K). Hence a belongs to the closure of a²A

with respect to pfor every a ∈ A, and every ideal of A which is closed with respect topis semiprime, hence prime. So we can apply lemma 2.4 toAand to the familyG :={(φ◦π)(I)}I∈F. We obtain a seminormponA such that the set of nonmaximal ideals of A which are closed with respect to pequals G (in factpis a norm since{0}= (φ◦π)(I0)is closed with respect top). Clearly, the set of nonmaximal ideals ofC(K)which are closed with respect toq:=p◦φ◦π equalsF.Now set q(f˜ ) =max(q(f),kfkK)forf ∈ C(K).Thenq˜is an algebra norm onK,and it is well known thatP rim(˜q) =P rim(q)(see for example [8]), so thatP rim(˜q) =F.

If U is a family of nonmaximal prime ideals of C(K), denote by MU the set of maximal ideals ofC(K)which contain some element of U. Theorem 2.7 suggests the following conjecture

Conjecture 1(CH)LetU be a family of nonmaximal prime ideals ofC(K) satisfying the three following properties

(i) every sequence of elements ofU has a pseudo-finite subsequence, (ii) U ∪ MU is almost stable under unions,

(iii)|C(K)/I|= 2^ℵ0 for every minimal elementI of U.

Then there exists an algebra norm onC(K)such thatU =P rim(q).

It is known that the "continuity ideal" of a discontinuous homomorphism fromC(K)can be written as the intersection of a "relatively compact" family of nonmaximal prime ideals in the sense of [21], or equivalently, is a pure semiprime Badé Curtis ideal in the sense of definition 1.2 of [12] (details about this equiv- alence can be found in [13]). If the conjecture were true, it would imply that, conversely, every idealIofC(K)satisfying|C(K)/I|= 2^ℵ0 which can be written as the intersection of a "relatively compact" family of nonmaximal prime ideals is the continuity ideal of some discontinuous homomorphism fromC(K),which would allow to remove the countability condition of therorem 6.7 (ii) of [21] and show that the continuity ideals of discontinuous homomorphisms fromC(K)are exactly the pure semiprime Badé-Curtis ideals. This result would have some heuristic interest since the fact that continuity ideals are pure semiprime Badé- Curtis ideals is a consequence of the two main tools of automatic continuity theory, the main boundedness theorem of Badé-Curtis and Sinclair’s stability lemma. These tools would thus provide all possible information about discon- tinuous homomorphisms fromC(K)if the continuum hypothesis is assumed.

Now let I be a prime ideal ofC(K),and let J(I)be the intersection of all minimal prime ideals ofC(K)contained inI.Following definition 5.8 of [12], we will say that a subalgebraB ofC(K)is aliftingof the quotient algebraC(K)/I ifB satisfies the two following conditions

(1)B∩I=J(I) (2)C(K) =B+I.

IfI is a minimal prime ideal of C(K)thenC(K)itself is obviously a lifting ofC(K)/I,and ifMτ ={f ∈ C(K)|f(τ) = 0}is a maximal ideal ofC(K)then

the algebra of functions constant on some neighbourhood ofτ is a lifting of the one-dimensional quotient algebraC(K)/Mτ.

It is shown in [13] that the quotient algebra AI does possess a lifting for every prime ideal I of C(K). More precisely let I be a prime ideal of C(K), and letJ be a prime ideal of C(K)containing I. It proved in [13] that every lifting ofC(K)/I contains a lifting ofC(K)/J,and that, conversely, every lifting of C(K)/J is contained in some lifting ofC(K)/I. This suggests the following problem.

Problem 1 Let I be a prime ideal of C(K) and let F be the set of prime ideals ofC(K)containingI. Does there exist a family {BJ}J∈F of subalgebras ofC(K)which possesses the following properties?

(i)BJ is a lifting of the quotient algebra C(K)/J for everyJ ∈ F.

(ii) IfJ1∈ F, J2∈ F, and ifJ1⊆J2, thenBJ2⊆BJ1.

A positive answer to this problem, at least in the case where|C(K)/I|= 2^ℵ0, would imply a result analogous to lemma 2.2 for the quotient algebraC(K)/I, which would give a direct proof of theorem 2.7. This would also open the gate to a proof of conjecture 1. As mentioned above, this would give a complete characterization under CH of idealsI ofC(K)satisfying|C(K)/I|= 2^ℵ0 which are equal to the continuity ideal of some discontinuous homomorphism from C(K). Unfortunately, the author was not able so far to solve this question, despite periodic attempts to solve a slightly weaker form of problem 1 during the last 35 years. The answer might depend on axioms of set theory.

Notice that since a maximal ideal of l^∞ contains a unique minimal prime ideal, and since the quotient algebral^∞/I is isomorphic toCω1 for every non- maximal minimal ideal of l^∞, it follows from lemma 2.2 that the answer to problem 1 is positive forβNif CH is assumed. We do not know the answer to this problem forβNin ZFC.

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Jean Esterle IMB, UMR 5251 Université de Bordeaux 351, cours de laLibération 33405 - Talence (France) esterle@math.u-bordeaux1.fr