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Closed prime ideals for discontinuous algebra seminorms on C(K)
Jean Esterle
To cite this version:
Jean Esterle. Closed prime ideals for discontinuous algebra seminorms on C(K). 2012. �hal-00773356�
Closed prime ideals for discontinuous algebra seminorms on C( K ) (preliminary version)
J. Esterle
1 Introduction
Let K be a compact space, let C(K) (resp.CR(K)) denote the algebra of continuous complex valued (resp. real valued) functions onK. An algebra seminormk.konC(K)is a seminorm satisfying kf gk ≤ kfkgk for every f, g∈ C(K), and such a seminorm is said to be continuous if there exists k >0such that kfk ≤kkfkK for everyf ∈ C(K),where kfkK :=maxt∈K|f(t)| denotes the usual norm on C(K). A classical result of Kaplansky [19] shows that if k.k is any algebra norm onC(K) we have
kfk ≥ kfkK for everyf ∈ C(K).
The existence of a discontinouous algebra seminorm onC(K),which is equivalent to the existence of a discontinuous algebra norm on C(K) and to the existence of a discontinuous homomorphism from C(K) into a Banach algebra, is the well-known Kaplansky’s problem, which turns out to be undecidable in ZFC. H.G. Dales and the author [4], [11], [6] proved independently that if2ℵ0=ℵ1, which means that the continuum hypothesis (CH) is assumed, then discontinuous algebra seminorms exist on C(K) for every compact space K. Those commutative Banach algebras A for which a discontinuous homomorphism φ : C(K) → A does exist under CH were characterized in [12], see also [5]. On the other direction 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 continuous, see [7]
for details. Notice that models of set theory in which2ℵ0 =ℵ2 and in which all algebra seminorms onC(K)are continuous for every compact spaceKwere constructed independently by Frankiewicz- Zbierski and Woodin [15] [29].
The structure of closed ideals ofC(K)for discontinuous algebra seminorms was investigated by A.M. Sinclair [27] and later, independently, by the author [9], 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 [9]. IfKis anF-space, which means that f and|f|generate the same ideal of C(K)for everyf ∈ C(K),then the familyP rim(q)of all nonmaximal prime ideals which are closed with respect to an algebra seminormqonC(K)is a finite union of well-ordered chains of nonmaximal prime ideals.
Pham [22], [23], [24] showed that the situation is much more complicated in the general case. To describe his deep contributions we will need the following notions ;
Definition 1.1 A family (Fλ)λ∈Λ of subsets of a set E is said to be
– pseudo-finiteif the set {λ∈Λ |a /∈Fλ} is finite for everya∈ ∪λ∈ΛFλ. – intersection redundantif ∩µ6=λFµ⊆Fλ for some λ∈Λ.
– intersection non-redundantif no subfamily of (Fλ)λ∈Λ is intersection redundant.
Definition 1.2 Let qbe an algebra seminorm on C(K).The continuity idealI(q)of qis the set of allf ∈ C(K)such that there existskf >0 satisfyingq(f g)≤kfkgkK for everyg∈ C(K).
It follows from the definition of the continuity ideal that it contains every idealI such that the restriction ofqto Iis continuous, and it is known that the restriction ofq to its continuity ideal is in fact continuous. Also the continuity ideal is the intersection of all minimal elements ofP rim(q).
Pham proved in [22] that if there exists a discontinuous algebra seminormqonC(K)such that the continuity ideal of q is not the intersection of a finite family of prime ideals then C(K) possesses an infinite intersection non-redundant pseudo-finite family of nonmaximal prime ideals. Conversely ifI is the intersection of an infinite intersection non-redundant pseudo-finite family of nonmaximal prime ideals, and if |C(K)/I| = 2ℵ0, then assuming (CH) there exists an algebra seminorm q on C(K)(or, equivalently, an algebra normqonC(K),see [9]) such that the continuity ideal ofqequals I.
Now if K is compact define∂K to be the set of all non-isolated points of K, so that∂K 6=∅ unless K is finite. Starting with an infinite compact space K,we define a non-increasing sequence (∂nK)n∈Z+ of compact subsets ofKas follows :
– (i)∂0K=K
– (ii) for eachn∈Z+, ∂n+1K=∂∂nK.
Define ∂∞(K) = ∩n∈Z+∂n(K). Then either ∂∞K 6= ∅, or ∂mK = ∅ for some m ∈ Z+. In the former case, we say thatKhas infinite limit level ; in the latter, we say thatKhas finite limit level.
Assume that K is a compact metric space. Pham proved in [22] that if t ∈ K, then the maximal ideal Mt :={f ∈ C(K) | f(t) = 0} contains an infinite peudo-finite family of nonmaximal prime ideals if and only ift∈∂∞K. In particular ifK is countable compact metric space of infinite level then assuming CH there exists a discontinuous algebra seminormqonC(K)such that the continuity idealI(q)ofqis not a finite intersection of prime ideals. The situation is even more complicated on C([0,1]) :it follows from [23] that the maximal idealMtcontains for everyt∈[0,1]an intersection non-redundant pseudo-finite family(Iλ)λ∈Λsuch that|Λ|= 2ℵ0,and so assuming (CH) there exists a discontinuous algebra seminorm onC([0,1])such that the continutity ideal ofqis not the intersection of any countable family of nomaximal prime ideals.
In [24], Pham discusses the structure of continuity ideals in the general case. He shows that the continuity ideal of a discontinuous algebra seminorm on C(K) is always the intersection of an intersection non-redundant familyJ of nonmaximal prime ideals such that every infinite sequence of elements ofJ contains an infinite pseudo-finite subsequence. He obtains a partial converse, assuming (CH) : ifJ is an intersection non-redundant familyJ of nonmaximal prime ideals such that every infinite sequence of elements of J contains an infinite pseudo-finite subsequence, and if the two following conditions are satisfied
1. |C(K)/∩I∈J I|= 2ℵ0
2. ∩I∈JIis the intersection of a countable family of non-maximal prime ideals
then there exists a discontinuous algebra seminormqonC(K)such that the continuity ideal of qequals ∩I∈JI.
A discussion at the end of [24] suggests that condition 2 above is not necessary. We were not able to remove this condition in the present article, in which we present a slightly different approach to these questions.
In section 2, we discuss the set P rim(q)of all nonmaximal prime ideals which are closed with respect to some discontinuous algebra seminorm q on C(K), and the related set U(P rim(q)) of all unions of elements of P rim(q). Using results of [12], we show that every chain of elements of U(P rim(q))is well-ordered with respect to inclusion.
Recall that an idealI in a commutative ringAis said to besemiprimeifIcontains everya∈A such that an ∈ A for some positive integer n, which is equivalent to the fact that I equals the intersection of the prime ideals of A which contain it. We will say that a semiprime ideal I of a
commutative algebraAispureif for every maximal idealMofAcontainingIthere exists a prime idealJ ofAsuch thatI⊂J (M.
Our discussion of the continuity idealI(q)involves the following notion.
Definition 1.3 LetAbe a commutative ring. A Badé-Curtis ideal ofAis an idealIofAsuch that, for every sequence(fn)n≥1 of elements of A such thatfnfm= 0for n6=m, there existsp≥1 such that fn∈I for everyn≥p.
A standard consequence of the Badé-Curtis main boundedness theorem [2], theorem 2.1, and of Sinclair’s stability lemma, [5], 5.2.7 or [28], 1.6 is thatthe continuity ideal of a discontinuous algebra seminorm on C(K) is a semiprime pure Badé-Curtis ideal of C(K) (a proof of this result using a general property of linear seminorms on RN instead of using the stability lemma can be found in [12]). In fact we will show in section 2 that if I is an ideal of C(K), then the following conditions imply each other :
1. Iis a (pure) semiprime Badé-Curtis ideal
2. There exists a familyI of (nonmaximal) prime ideals ofC(K)such that∩{J :J ∈ I}=Iand such that every chain of unions of elements ofI is well-oredered with respect to inclusion.
3. There exists an intersection non-redundant family J of (nonmaximal) prime ideals of C(K) such that ∩{J : J ∈ J } = I and such that every chain of unions of elements of J is well- oredered with respect to inclusion.
Now letE be a set, and letF = (Fλ)λ∈Λ be a family of subsets of E.We observe in section 2 that the following conditions imply each other :
1. Every chain of unions of subfamilies of the family F is well-ordered with respect to inclusion.
2. Every infinite sequence of elements of the family F contains an infinite pseudo-finite subse- quence.
In particular, the class of semiprime Badé-Curtis ideals of C(K) is the same as the class of abstract continuity ideals of C(K) introduced by Pham in [24]. Also the fact that ifI := (Iλ)λ∈Λ
is a family of prime ideals ofC(K)satisfying condition 1 above then the union of any subfamily of I is a finite union of prime ideals, observed by the author in some unpublished notes, follows from lemma 5.7 of [24].
In section 3 we discuss a general version of a lifting result which shows that ifI andJ a prime ideal of C(K), and if I ⊂ J, then there exists a subalgebra AI,J of the quotient algebra C(K)/J such that C(K)/J = AI,J ⊕πJ(I), where πJ : C(K)→ C(K)/J denotes the canonical surjection.
It follows immediately from this result that if a nonmaximal prime ideal J is the kernel of some discontinuous algebra seminormqonC(K),and ifI is a nonmaximal prime ideal containingJ,then there exists another discontinuous seminorm q˜onC(K)such that Ker(˜q) =I.Our general version of the lifting theorem provides, given a nonmaximal prime idealI ofC(K),a subalgebraBI of the quotient algebraC(K)/J(I)such thatC(K)/J(I) =BI⊕π(I),whereJ(I)denotes the intersection of all prime ideals of C(K)contained inI and whereπ:C(K)→ C(K)/J(I)denotes the canonical surjection.
If it were possible to construct the algebrasAI so that πJ−1(I′)(A′I) ⊂ πJ−1(I)(AI) for every pair (I, I′) of nonmaximal prime ideals of C(K) such that I ⊂ I′, it would be possible to remove the condition that I is a countable intersection of prime ideals in theorem 6.7 (ii) of [24], which would give a complete characterization of the ideals I of C(K) satisfying|C(K)/I| = 2ℵ0 which are the continuity ideal of some discontinuous algebra seminorm onC(K)(assuming (CH)).
We were not able to do this, but we conjecture that there exists (assuming CH) a discontinuous norm on C(K) satisfying P rim(q) = U for every family U of nonmaximal prime ideals of C(K) satisfying the three following conditions
– every sequence of elements ofU contains a pseudo-finite subsequence,
– U ∪ MU is "almost stable under unions", whereMUdenotes the set of maximal ideals ofC(K) containing some element ofU,
– |C(K)/I|= 2ℵ0 for everyI∈ U.
This conjecture is proved in [14] in the case whereU is a chain (in this situation the first condition just means thatU is well ordered).
A major success in the works related to Kaplansky’s problem in the seventies was the fact that, if the continuum hypothesis is assumed, every complex non-unital commutative algebra U of cardinality 2ℵ0 possesses an algebra norm. There were two ways to reach this result. The first one, followed by the author, consisted in showing that a ’big’ algebra of power series, denoted by Cω1, does have an algebra norm, and then use the "fundamental theorem on extension of places"
to embedd U in the maximal ideal of a complex valuation algebra which, assuming the continuum hypothesis, can be embedded into the normable algebra Cω1. The second one, followed by Dales and Woodin, consisted in proving that there always exists an ultrafilterU onNsuch thatU embeds into the quotient algebrac0/U,and then use Dales’ contruction [4] to show that, if CH is assumed, the quotient algebrac0/U is normable. Pham uses in theorem 6.5 of [22] and proposition 6.2 of [24]
arguments based on Woodin’s embeddings of nonunital integral domains of cardinality2ℵ0, based on the existence of "almost disjoint" infinite families of integers and on Hilbert’s Nullstellensatz, to obtain the major results of [22] and [24].
In section 4 we propose an alternative to this approach. We show that, if I and J are prime ideals of C(K), with I ⊂ J, and if B is a subalgebra of the quotient algebra C(K)/I such that C(K)/I = B ⊕πI(J), where πI : C(K) → C(K)/I denotes the canonical surjection, then every one-to-one homomorphism φ : B → Cω1 into a "small" subalgebra of Cω1 can be extended to a one to one homomorphismψ :C(K)/I → Cω1. Using this result it is for example easy to see that if (Iλ)λ∈Λ is a pseudo-finite family of ideals of C(K), and if|C(K)/Iλ| = 2ℵ0 for λ∈ Λ,then the quotient algebraC(K)/∩λ∈ΛIλhas an algebra norm if CH is assumed, a result slightly stronger than corollary 7.3 of [22]. The proof of our "extension theorem" is based on the "fundamental theorem on extension of places" and on Kaplansky’s embedding theorem of valued fields into maximal valued fields [18].
A survey of the current state-of-the-art concerning Kaplansky’s problem and Michael’s problem on continuity of characters on Fréchet algebras is proposed by the author in [13], where some of the results of the present paper are announced without proof.
2 Nonmaximal prime ideals closed for a discontinuous algebra norm on C(K)
In what followsK denotes an infinite compact space. Let P rim(q) be the set of nonmaximal prime ideals ofC(K)which are closed with respect to a discontinous algebra seminorm onC(K),and letI(q) be the continuity ideal ofq, see definition 1.2. As indicated in the introduction, is follows from automatic continuity theory that we have the following properties.
Theorem 2.1 (i)I(q) =∩{I:I∈P rim(q)}.
(ii)I(q) is the largest idealI of C(K)such that the restriction ofq toI is continuous.
(iii)I(q)is a pure semi-prime Badé-Curtis ideal of C(K).
IfIis an ideal ofC(K),setZ(I) :={t∈K :I⊂ Mt},whereMt={f ∈ C(K) :f(t) = 0}.The fact thatZ(I(q))is finite was proved by Badé and Curtis in their seminal paper [2]. In fact ifZ(I) is infinite for some ideal I ofC(K)then it is possible to construct by induction a sequence (tn)n≥1
of distinct elements ofZ(I)and a sequenceUn of disjoint open subsets ofK such thattn∈Un for n≥1.There exists for everyn≥1a functionfn∈ C(K),withSupp(fn)⊂Un,such thatfn(tn) = 1.
Hencefnfm= 0forn6=mandfn∈/ Ifor eveyn≥1,which shows thatIis not a Badé-Curtis ideal (this argument applies to Badé-Curtis ideals of semi-simple commutative regular Banach algebras).
Let Ωq be the Stone-C˘ech compactification of K\Z(I(q)), identify the closure I(q) := {f ∈ C(K) : f(t) = 0 ∀t ∈ Z(I(q)} of I(q) in C(K) to the set of functions f ∈ C(Ωq) vanishing on Ωq\(K\Z(I(q))), and denote byI˜(q)the set of all functionsf ∈ C(Ωq)such thatf g ∈ I(q)for everyg∈ I(q). Using similar arguments, it is shown in [9] that the setZ(˜I(q))is a finite subset of Ωq\(K\Z(I(q))).
Recall that a linear subspaceLofC(K)is said to beabsolutely convexifg∈Lfor everyg∈ C(K) satisfying|g| ≤ |f|for somef ∈L. We will use the following lemma.
Lemma 2.2 Let L be a family of absolutely convex linear subspaces of C(K), and let U(L) be the family of all sets of the form SN :=∪{L :L∈ N }where N ⊂ L. If U(L) contains a chain which is not well-ordered with respect to inclusion, then at least one of the two following conditions holds
(i) there exists a sequence(Ln)n≥1 of elements of L such thatLn+1(Ln for everyn≥1, (ii) there exists a sequence(Ln)n≥1 of elements ofL such that Ln*∪m6=nLm for everyn≥1.
Proof : There exists a sequence(Ln)n≥1of subsets ofLsuch that∪{L :L∈ Ln+1}(∪{L :L∈ Ln},and we can find a sequence(Ln)n≥1of elements ofLsuch thatLn∈ Ln andLn*∪m≥n+1Lm
forn≥1.LetS be the set of all integersn≥1such thatLn does not containLmfor anym6=n.
IfS is empty or finite there exists p≥1 such thatn /∈S forn≥p.Letn≥p,and letm 6=n such thatLm ⊂Ln. SinceLm is not contained in ∪k>mLk, we havem > n, and we can construct by induction a strictly increasing sequence (nk)k≥1 of positive integers such that Lnk+1 is strictly contained inLnk fork≥1.
Now if S is infinite, let (nk)k≥1 be a strictly increasing sequence of positive integers such that nk ∈S for everyk≥1. SetL˜k =Lnk,and let k≥2. ThenL˜k is not contained in∪l>kL˜l,and so there exists gk ∈L˜k such thatgk ∈ ∪/ l>kL˜l. Now let j ∈ {0, ..., k−1}. Since nj ∈S, L˜j does not containL˜kand so there exists f a functiongk,j ∈L˜k\L˜k∩L˜j.Using absolute convexity, we see that fk:=|gk|+k−1P
j=1
|gk,j| ∈L˜k,but thatfk ∈/ L˜l forl6=k, which completes the proof of the lemma.
The link with Pham’s discussion of continuity ideals is given by the following observation.
Lemma 2.3 LetE be a set, letF be a family of subsets ofE,and letU(F)be the family of all sets of the form SG:=∪{F :F ∈ G},whereG ⊂ F. Then the following conditions imply each other :
(i) Every chain of elements ofU(F) is well-ordered with respect to inclusion.
(ii) Every sequence of elements ofF possesses a pseudo-finite subsequence.
(iii) Every sequence of elements ofU(F)possesses a pseudo-finite subsequence.
Proof : Assume that a sequence (Fn)n≥1 of elements of F does not possess any pseudo-finite subsequence. Then we could construct by induction a sequence(An)n≥1of infinite subsets ofZ+and a sequence (un)n≥1 of elements ofE such thatAn+1 ⊂An, un ∈ ∪n∈AnFn and un ∈ ∪/ n∈An+1Fn, which contradicts the fact that every chain of elements of U(F) is well-ordered with respect to inclusion. Hence (i) implies (ii).
Now assume that some chain of elements ofU(F)is not well-ordered with respect to inclusion.
Then there exists a sequence Gn of subsets of F such that ∪{F : F ∈ Gn+1} (∪{F : F ∈ Gn} for n ≥ 1. We can then construct a sequence (Fn)n≥1 of elements of F such that Fn ∈ Gn and Fn *∪{F :F ∈ Gn+1}. Now let(np)p≥1 be a strictly increasing sequence of positive integers. We
haveFnp∈ ∪/ q≥p+1Fnq forp≥1.In particular there existsa∈Fn1 such thata /∈Fnq forq≥2,and so the sequence(Fnp)p≥1 is not pseudo-finite. Hence (ii) implies (i).
Clearly, (iii) implies (ii). Now assume that (i) holds, and let(Un)n≥1 be a sequence of elements of U(F). There exists an infinite subsetW0 ofNsuch that∪n∈WUn =∪n∈W0Un for every infinite subsetW ofW0,which means that the family{Un}n∈W0 is pseudo-finite. Hence (i) implies (iii).
It follows from [9] that every chain of nonmaximal prime ideals ofC(K)which are closed with respect to a discontinuous algebra norm on C(K) is well-ordered with respect to inclusion. The following theorem gives an improvement of this result.
Theorem 2.4 Letqbe an discontinuous algebra seminorm onC(K),letP rim(q)of all nonmaximal prime ideals ofC(K)which are closed with respect toq,and let U(P rim(q))be the family of all sets of the form SN :=∪{I :I∈ N },where N ⊂P rim(q).The setP rim(q)satisfies the following two equivalent conditions
(i) Every chain of elements ofU(P rim(q))is well-ordered with respect to inclusion.
(ii) Every sequence(In)n≥1 of elements of P rim(q)possesses a pseudo-finite subsequence.
Proof : Assume that (i) is not satisfied. Since any chain of elements ofP rim(q)is well-ordered with respect to inclusion, it would follow from lemma 2.2 that there exists a sequence (In)n≥1 of elements ofP rim(q)such thatIn *∪m6=nIm.It would then follow from lemma 3.9 of [9] that there exists a sequence(fn)n≥1 of elements ofC(K)such thatfn ∈ ∩/ m≥1Im forn≥1 whereasfmfn= 0 form6=n,which contradicts the fact that the continuity idealI(q) =∩{I :I∈P rim(q)} ofqis a Badé-Curtis ideal ofC(K).
The fact that conditions (i) and (ii) are equivalent is given by lemma 2.3.
The following result is essentially a reformulation of lemma 5.7 of [24], where Pham shows that every "compact" family of prime ideals ofC(K)admits a finite number of "roofs", in a slightly more general context.
Proposition 2.5 LetF be a family of prime ideals ofC(K)such that every sequence of elements of F has a pseudofinite subsequence, and letV(F)be the set of ideals of C(K)which belong toU(F).
(i) Let S ∈ U(F), and let GS be the set of all J ∈ V(F) contained in S. Then the set ∆S of maximal elements of GS is finite, andS=∪{J :J ∈∆S}.
(ii) IfS∈ V(F), and if(An)n≥1 is a family of disjoint maximal chains of elements ofGS\ {S}, then S=∪n≥1Jn for every sequence Jn of ideals of C(K)such that Jn∈An for n≥1.
Proof : (i) It follows from Zorn’s lemma that every element ofGS belongs to a maximal chain of elements ofGS.Since the union of a chain of elements of GS is a prime ideal contained inS,we see that every maximal chain of elements ofGS has a largest element which is a maximal element ofGS. This shows that S=∪{J :J ∈∆S}.
Assume that∆S is infinite, and letIn be a sequence of distinct elements of ∆S.It follows from lemma 2.3 that the sequence(In)n≥1admits a pseudo-finite subsequence(Inp)p≥1.SetI:=∪p≥1Inp. ThenI∈ GS,which contradicts the maximality of the idealsInp.Hence∆S is finite.
(ii) Since every sequence of elements ofV(F)possesses a pseudofinite subsequence, we can assume without loss of generality that the sequence(Jn)n≥1is pseudofinite. LetJ =∪n≥1Jn,and letL∈An. IfL⊂Jn,thenL⊂J, and ifJn⊂Lthen either L⊂J orJ ⊂L.SoAn∪J is a chain of elements ofGS forn≥1.But ifJ 6=S we would haveJ ∈ ∩n≥1An,which is impossible. HenceJ =S.
We obtain the following intriguing property, which had been already noticed by the author in some unpublished notes before he heard of the paper [24].
Corollary 2.6 Letqbe a discontinuous algebra seminorm onC(K),and letS∈ U(P rim(q)). Then there existsn≥1 andJ1, ..., Jn∈ U(P rim(q))satisfying the following properties
(i)J1, ...Jn are prime ideals ofC(K),andJi*Jj ifi6=j.
(ii)S =∪1≤j≤nJj.
Notice that if(Iλ)λ∈Λis an uncountable pseudo-finite family of elements ofP rim(q)(such families are constructed in [23] under (CH) for some compact sets) then ∪λ∈ΛIλ ∈ P rim(q). To see this assume that f ∈ C(K)satisfies limn→∞q(f−fn) = 0, with fn ∈ ∪λ∈ΛIλ for n≥1. Then the set
∪n≥1{λ ∈ Λ : fn ∈/ Iλ} is at most countable, and so there exists µ ∈ Λ such that fn ∈ Iµ for everyn ≥1, so that f ∈Iµ ⊂ ∪λ∈ΛIλ.Similarly if ω is a limit ordinal which does not admit any cofinal sequence, and if (Iζ)ζ<ω is a well-ordered chain of elements of P rim(q) satisfyingIζ (Iζ′
forζ < ζ′ < ω,then∪ζ<ωIζ ∈P rim(q).
IfI is an ideal of a commutative algebraA,we setI:a={b∈A :ab∈I} fora∈A.We now introduce a notion used by Pham in [24].
Definition 2.7 Let Abe a commutative algebra. An ideal I of A is an abstract continuity ideal if, for each sequence(an)n≥1 inA, there existsn0≥1such that
I:a1...an=I:a1...an+1 (n≥n0)
It is possible to deduce the following observation from lemmas 5.6 and 5.7 of [24]. We give a proof for the sake of completeness.
Proposition 2.8 Let(Pα)α∈Λ be a family of prime ideals of a commutative ringAsuch thatPα* Pβ for α6=β. If the family(Pα)α∈Λ is intersection redundant, then there exists a sequence(αn)n≥1
such that ∪p≥n+1Pαp(∪p≥nPαp for n≥1.
Proof : Let α be such that ∩β6=αPβ ⊂ Pα, let α1 6= α, and let f1 ∈ Pα1 \Pα∩Pα1. Since
∩β6=αPβ⊂Pα,there existsα2∈Λ\ {α, α1} such thatf1∈/Pα2.
Starting with f1, α1 and α2, we now construct two sequences (fn)n≥1 and (αn)n≥1 such that fn ∈ Pαn\Pαn∩Pα and f1...fn ∈/ Pαn+1 for n ≥1. Assume that f1, ..., fn and α1, ..., αn+1 have been constructed for somen≥1.Takefn+1∈Pαn+1\Pαn+1∩Pα.Thenf1...fn+1∈ ∩1≤p≤n+1Pαp, and since f1...fn+1 ∈/ Pα there existsαn+2 ∈Λ\ {α, α1, ..., αn+1} such thatf1...fn+1 ∈/ Pαn+2.So we can construct the sequences(fn)n≥1and(αn)n≥1 by induction.
We havefn ∈Pαn ⊂ ∪p≥nPαp. But if p≥n+ 1 we have f1...fp−1 ∈/ Pαp. A fortiori fn ∈/ Pαp
andfn∈ ∪/ p≥n+1Pαp,which concludes the proof of the proposition.
Corollary 2.9 Let L be a family of prime ideals in a commutative ring A satisfying one of the equivalent conditions of lemma 2.3. If J * L for I ∈ L, J ∈ L, J 6= L, then the family L is intersection non-redundant.
Proof : Since every subfamily of L also satisfies the equivalent conditions of lemma 2.3, the corollary follows immediately from proposition 2.8.
We will say that an abstract continuity ideal ispure if every for every maximal ideal M of A containingI there exists a prime idealJ ofAsuch that I⊆J (M.Now ifL is a family of ideals of C(K), we will denote again byU(L)the family of sets of the form SN :=∪{I :I ∈ N }, where N ⊂ L.
Using corollary 4.12 of [24], we obtain the following result.
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