UNCOUNTABLE FAMILIES OF PRIME z-IDEALS IN C (R) 0 HUNGLEPHAM 8 0 0 Abstract. Denote by c= 2ℵ0 the cardinal of continuum. We construct an 2 intriguing family (Pα : α ∈c) of prime z-ideals in C0(R) with the following properties: Jan •• TIfif6=i∈0PPii06⊂foPri0sofmoreeia0c∈hcı0,t∈hecn. f ∈Pi forallbutfinitelymanyi∈c; We also construct a well-ordered increasing chain, as well as a well-ordered 1 decreasing chain,of ordertypeκofprimez-idealsinC0(R)foranyordinalκ ofcardinalityc. ] A R h. 1. Introduction t a Let Ω be a locally compact space. In [5], we introduced the notion of pseudo- m finite family of prime ideals as follows. [ Definition1.1. Anindexedfamily(P ) ofprimeidealsinC (Ω)ispseudo-finite i i∈S 0 1 if f ∈Pi for all but finitely many i∈S whenever f ∈ i∈SPi. v Apseudo-finitefamily(P : i∈S)ofprimeidealsinC (Ω)hasmanyinteresting 5 i S0 properties,forexample,whenS isinfinite, theunion P isagainaprimeideal 1 i∈S i 3 and any infinite subfamily of (Pi) gives rise to the same union. S 0 Apseudo-finite family ofprime ideals(Pi : i∈S)is saidto be non-redundant if . for every proper subset T of S, P 6= P . Non-redundancy is equivalent 1 i∈T i i∈S i 0 to either of the following ([5, Lemma 3.4]): T T 8 (a) P 6⊂P (α6=β ∈S); α β 0 (b) P 6⊂P for each α∈S. : β6=α β α v Note that (a) is apparently weaker, whereas (b) is apparently stronger than the i T X non-redundancy. Thus, in this case, P cannot be written as the intersection i∈S i r oflessthan|S|primeidealsand|S|≤|C0(Ω)|. Furthermore,foreverypseudo-finite a family of prime ideals, the subfamilyTconsisting of those ideals that are minimal in the family is non-redundantandpseudo-finite andhas the same intersectionas the original family. The notionof pseudo-finitenesshas a connectionwith automatic continuity theory. It is provedin [5] that, assuming the Continuum Hypothesis, for eachpseudo- finite family (P : i ∈ S) of prime ideals in C (Ω) such that C (Ω)/ P = c, i 0 0 i∈S i there exists a homomorphism from C (Ω) into a Banach algebra whose continuity 0 (cid:12) T (cid:12) ideal is i∈SPi. Recall that the continuity ideal is the larg(cid:12)est ideal of C0(Ω(cid:12)) on which the homomorphism is continuous, and it is always an intersection of prime T ideals (see [1] for more details). 2000 Mathematics Subject Classification. Primary46J10; Secondary13C05. Key words and phrases. Algebraofcontinuous functions,primeideal,locallycompact space. This research is supported by a Killam Postdoctoral Fellowship and a Honorary PIMS Post- doctoralFellowship. 1 2 HUNGLEPHAM Suppose that Ω is metrizable and that ∂(∞)Ω♭ 6= ∅; see §2 for the definition. Examples of such spaces include many countable locally compact spaces and all uncountable locally compact Polish spaces. For such Ω, it is known that there exists an infinite non-redundant pseudo-finite sequence of prime ideals in C (Ω) 0 ([5]). Here, we are going to show that there exists even a non-redundant pseudo- finite family (P :i∈c) of prime ideals in C (Ω) (Theorem 3.9). As a consequence, i 0 assuming the Continuum Hypothesis, there exists a homomorphism from C (Ω) 0 into a Banach algebra whose continuity ideal cannot be written as intersection of countably many prime ideals. Note that when Ω is metrizable and ∂(∞)Ω♭ = ∅, then every non-redundant pseudo-finite families of prime ideals in C (Ω) is finite and the continuity ideal of 0 everyhomomorphismfromC (Ω)intoaBanachalgebraistheintersectionoffinitely 0 many prime ideals. ([5]) In §4, we shall construct, for every ordinal κ of cardinality at most c, a well- ordered decreasing chain of order type κ of prime z-filters on any uncountable locally compact Polish space. In particular, there exists a well-ordered decreasing chain of order type κ of prime z-filters beginning with any non-minimal prime z-filters on R. (This was shown for κ ≤ ω2, where ω is the first uncountable 1 1 ordinal, in Theorems 8.5, 13.2 and Remark 13.2 of [3].) We also show that there are various countable compact subspaces of R on which there is a well-ordered decreasing chain of order type c of prime z-filters (c is identified with the smallest ordinal of cardinality c). In [3], it was asked whether there exists an uncountable well-ordered increasing chain of prime z-filters on R. We shall construct in §5, for every ordinal κ of cardinality at most c, a well-ordered increasing chain of order type κ of prime z-filters on any uncountable locally compact Polish space. We also construct well- ordered increasing chains of order type c of prime z-filters on various countable compact subspaces of R. Allthreeconstructionshaveasacommoningredientthe resultdue toSierpinski that N can be expressed as the union of c ”almost disjoint” infinite subsets. 2. Preliminary definitions and notations For details of the theory of the algebras of continuous functions, see [2]. Let Ω be a locally compact space; our convention is that the topological spaces are always Hausdorff. The one-point compactification of Ω is denoted by Ω♭. For each prime ideal P in C (Ω), either there exists a unique point p ∈ Ω such 0 that f(p)=0 (f ∈P), in which case, we say that P is supported at the point p; or otherwise, we say that P is supported at the (point at) infinity. It is an important fact that, for each prime ideal P in C (Ω), the set of prime 0 ideals in C (Ω) which contain P is a chain with respect to the inclusion relation. 0 Foreachfunctionf continuousonΩ,thezeroset off isdenotedbyZ(f). Define Z[Ω] = {Z(f): f ∈C(Ω)}. For each closed subset Z ⊂ Ω, we have Z = Z(f) for some function f ∈ C (Ω) if and only if Ω\Z is σ-compact. An ideal I of C (Ω) is 0 0 a z-ideal if g ∈I whenever g ∈C (Ω), Z(g)⊃Z(f) and f ∈I. 0 Az-filter F onΩisanon-emptypropersubsetofZ[Ω]thatisclosedunderfinite intersection and supersets. Each z-filter F associates with the ideal Z−1[F]={f ∈C(Ω): Z(f)∈F} of C(Ω). PRIME IDEALS IN C0(R) 3 A z-filter P is a prime z-filter if Z ∪Z ∈/ P whenever Z ,Z ∈Z[Ω]\P. 1 2 1 2 Let P be a prime z-filter on Ω. Then we say that P is supported at a point p∈Ω,ifp∈Z foreachZ ∈P;ifthere existsnosuchp,wesaythatP is supported at the (point at) infinity. The supportpointof eachprimez-filterP coincides with the support point of the prime z-ideal C (Ω)∩Z−1[P]. 0 Let Ω be a compact space. Define ∂(1)Ω = ∂Ω to be the set of all limit points of Ω. Since Ω is compact, ∂Ω is non-empty unless Ω is finite. We then define inductively a non-increasing sequence ∂(n)Ω:n∈N of compact subsets of Ω by setting ∂(n+1)Ω = ∂ ∂(n)Ω for each n ∈ N. Set ∂(∞)Ω = ∞ ∂(n)Ω. By the (cid:0) (cid:1) n=1 compactness, either ∂(∞)Ω is non-empty or ∂(l)Ω is empty for some l∈N. (cid:0) (cid:1) T A Polish space is a separable completely metrizable space. Every separable metrizable locally compact space is a Polish space. 3. Pseudo-finite families of prime ideals and prime z-filters Let Ω be a locally compact space. First, we shall make a connection between pseudo-finite families of prime ideals andpseudo-finite families of prime z-idealsin C (Ω). For each closed subset E of Ω, we define the z-ideal 0 K ={f ∈C (Ω): E ⊂Z(f)}. E 0 The following is [4, 3.3 and 3.4], we shall give here a combined proof. Lemma 3.1. [4] Let I be an ideal in C (Ω). Then 0 Iz = {K : E is closed in Ω and K ⊂I} E E is the largest z-ideal con[tained in I. If I is a prime ideal then so is Iz. Proof. ItiseasytoseethatIz containseveryz-idealcontainedinI. Since thesum of two z-ideals is again a z-ideal, we see that Iz = {K : E is closed in Ω and K ⊂I}; E E where the sum is algebXraic. Thus Iz is the largest z-ideal contained in I. Now,suppose that I is prime. Letf ,f ∈C (Ω)\Iz. Then there existsg ,g ∈ 1 2 0 1 2 C (Ω)\I such that Z(g ) ⊃ Z(f ). Then Z(g g ) ⊃ Z(f f ). The primeness of I 0 i i 1 2 1 2 implies that g g ∈/ I. So f f ∈/ Iz. (cid:3) 1 2 1 2 The following strengthens the implication (a)⇒(c) of [5, Lemma 8.4]. Proposition 3.2. Let Ω be a locally compact space. Let (P : i∈S) be an infinite i non-redundant pseudo-finite family of prime ideals in C (Ω). Then P = P is 0 i∈S i a prime z-ideal, and (Pz : i∈S) is a non-redundant pseudo-finite family of prime i z-ideals whose union is P such that Pz ⊂P (i∈S). S i i Proof. We shall need another theorem of [4] which say that the sum of two non- comparable prime ideals in C (Ω) is indeed a prime z-ideal ([4, 3.2]). 0 We know that P must be a prime ideal. Assume toward a contradiction that P is not a prime z-ideal. Choose α 6= α ∈ S arbitrary. Then P +P is a prime 1 2 α1 α2 z-ideal. Suppose that we already have distinct indices α ,...,α ∈ S such that 1 n n n P is a prime z-ideal. Then P 6= P , and so we can find α ∈ S i=1 αi i=1 αi n+1 P P 4 HUNGLEPHAM n suchthat P 6⊂ P . The induction can be continued. However,this gives αn+1 i=1 αi a contradiction since then P ∞ ∞ n P = P = P αn αi n=1 n=1i=1 [ [ X is a z-ideal. Hence, P is a prime z-ideal. We claim that (Pz : i ∈ S) is a pseudo-finite family with union P. Indeed, i assumetowardacontradictionthatthereexistsf ∈P anddistinctα ∈S (n∈N) n such that f ∈/ Pz (n ∈ N). For each n, we can then find f ∈/ P such that αn n αn Z(f ) ⊃ Z(f); we can further assume that 0 ≤ f ≤ 2−n. Define f = ∞ f . n n ∗ n=1 n Then we see that f ≤f so f ∈/ P (n ∈N), and that Z(f )⊃Z(f) so f ∈P. n ∗ ∗ αn ∗ P∗ This is a contradiction to the pseudo-finiteness of (P : i∈S). i It remains to prove the non-redundancy of (Pz : i ∈ S). So, assume that i Pz ⊂Pz for some α6=β ∈S. Then Pz is containedin both P and P , and so P α β α α β α and P are in a chain. This contradicts the non-redundancy of (P : i∈S). (cid:3) β i Conversely, it is obvious that if (Q : i ∈ S) is a pseudo-finite family of prime i (z-)ideals and P is a prime ideal containing Q and contained in Q (i∈S) i i α∈S α then (P : i∈S) is a pseudo-finite family of prime ideals. i S We define a similar notion of pseudo-finite families of prime z-filters. Definition 3.3. An indexed family (P ) of prime z-filters Ω is pseudo-finite if i i∈S Z ∈P for all but finitely many i∈S whenever Z ∈ P . i i∈S i A pseudo-finite family of primez-filters(Pi : i∈SS)is saidto be non-redundant ifforeverypropersubsetT ofS, P 6= P . Similarto [5, Lemma3.4]we i∈T i i∈S i have the following. T T Lemma 3.4. Let (P : α ∈ S) be a pseudo-finite family of prime z-filters on Ω. α Then the following are equivalent: (a) (P ) is non-redundant; α (b) P 6⊂P (α6=β ∈S); α β (c) P 6⊂P for each α∈S. β6=α β α Proof. OTbviously, (c)⇒(a)⇒(b). We now prove (b)⇒(c). Fix α ∈ S. By condition (b), P 6⊂ P (β ∈ S \{α}). β α Choose Z ∈ P \P for some β ∈ S \{α}. Then, by the pseudo-finiteness, 0 β0 α 0 we have Z ∈ P for all but finitely many β ∈ S. Let β ,...,β be those indices 0 β 1 n β ∈S\{α} such that Z ∈/ P . For each1≤k ≤n, choose Z ∈P \P , andset 0 β k βk α n Z = Z . Then Z ∈ P (β ∈ S \{α}), but Z ∈/ P , by the primeness of P . k=0 k β α α Thus (c) holds. (cid:3) S The following definition and proposition are adapted from [5]. Definition 3.5. LetΩbealocallycompactspace,andletS beanon-emptyindex set. Let F be a z-filter onΩ, and let (Z : α∈S)be a sequence ofzero sets onΩ. α Then F is extendible with respect to (Z : α∈S) if both the following conditions α hold: (a) Z ∈/ F, and Z ∪Z ∈F (α6=β ∈S); α α β (b) for each Z ∈Z[Ω], if Z ∪Z ∈F for some α ∈S, then Z ∪Z ∈ F for α0 0 α all except finitely many α∈S. PRIME IDEALS IN C0(R) 5 Proposition 3.6. Let Ω be a locally compact space. Suppose that there exist a z-filter F and a family (Z : α ∈ S) in Z[Ω] such that F is extendible with α respect to (Z : α∈S). Then there exists a pseudo-finite family of prime z-filters α (P : α∈S) such that Z ∈ P \P for each α∈S. α α γ6=α γ α Proof. We see that the unionTof a chain of z-filters, each of which contains F and is extendible with respect to (Z ), is also extendible with respect to (Z ). Thus, α α by Zorn’s lemma, we can suppose that F is a maximal one among those z-filters. For each α, set F ={Z ∈Z[Ω]: Z∪Z ∈F}, and set P = F . By the α α α∈S α extensibility of F, we see that whenever Z ∈P then Z ∈F for all except finitely α S many α∈S. Thus, in particular, the set P is actually a z-filter. Claim 1: For each Z ∈ Z[Ω] \ P, we have {Z ∈Z[Ω]: Z∪Z ∈F} = F. 0 0 Indeed, we see that Z ∈/ G = {Z ∈Z[Ω]: Z∪Z ∈F} (α ∈ S); for otherwise, α 0 Z would be in P. It then follows easily that G is extendible with respect to (Z ). 0 α This and the maximality of F imply the claim. Claim 2: P is a prime z-filter. We have to prove that, whenever Z ,Z ∈ Z[Ω] 1 2 are such that Z ∪Z ∈ P, but Z ∈/ P, then Z ∈ P. Indeed, let α ∈ S be such 1 2 1 2 0 that Z ∪Z ∪Z ∈F. Then Z ∪Z ∈F, by the first claim, and so Z ∈F . α0 1 2 α0 2 2 α0 Now, for each α∈S, define D ={Z ∪Z : Z ∈Z[Ω]\P}. α α Then,by Claim2,the setD isclosedunder finite union. Obviously,D ∩F =∅. α α α Thus, there exists a prime z-filter P containing F such that D ∩P =∅. α α α α We see that F ⊂P ⊂P and Z ∈/ P (α∈S). The result then follows. (cid:3) α α α α We now define a “prototype” space Ξ. Denote by ∞ the point adjoined to N to obtain its one-point compactification N♭. The product space (N♭)N is a compact metrizable space. Define Ξ to be the compact subset of (N♭)N consisting of all elements (n ,n ,...) with the property that there exists k ∈ N such that n ≥ k 1 2 i (1≤i≤k) and such that n =∞ (i>k). The convention is that ∞>n (n∈N). i Lemma 3.7. [5, Lemma 9.2] Let Ω be a locally compact metrizable space. Suppose thatthereexistsapointp∈∂(∞)(Ω♭). Thenthereexistsahomeomorphicembedding ι of Ξ onto a closed subset of Ω♭ such that ι(∞,∞,...)=p. AkeytoourconstructionistheresultduetoSierpinskithatthereexistsafamily {E : α∈c} of infinite subsets of N satisfying the following properties: α (i) N= E , and α∈c α (ii) E ∩E is finite for each α6=β ∈c. α β S We sketch the nice construction of such family as follows (cf. [7]): The set N is isomorphic to ∞ C = {f :{1,...,n}→{1,2}}. n=1 [ For each f :N→{1,2}, define C ={the restrictions of f to {1,...,n}: (n∈N)}. f We see that C = C and that C ∩C is finite for each f 6=g. We can f:N→{1,2} f f g thenmapbackfromC toN. Inspectingtheconstruction,weseethat{E : α∈c} S α enjoys the following property: (i’) The cardinality of {α∈c: n∈E } is c for each n∈N. α 6 HUNGLEPHAM Lemma 3.8. There exists a non-redundant pseudo-finite family {Q : α∈c} of α prime z-filters on Ξ such that each z-filter is supported at the point (∞,∞,...). Proof. Let {E : α∈c} be the family of infinite subsets of N as in the previous α paragraph. For each α∈c, define N ={(j ,j ,...)∈Ξ: j =∞ (n∈E )}. α 1 2 n α Let F to be the z-filter generatedby allN ∪N (α,β ∈c, α6=β). We claim that α β F is extendible with respect to (N : α∈c); the proof will then be completed by α applying Lemma 3.6. Obviously,N ∪N ∈F (α6=β). Weclaim thatN ∈/ F (α∈c). Indeed,assume α β α thecontrary. Thenthereexistγ ,...,γ ∈c\{α}suchthatN ⊃ m N . Since 1 m α i=1 γi m E is infinite whereas each E ∩E is finite, there exists l ∈ E \ E . We α m α γi αT i=1 γi see that (j )∈ N \N where j =l and j =∞ (i6=l); a contradiction. Finally,isuppois=e1thaγit N ∈α Z[Ξ] sulch that Ni∪N ∈ F for someSα ∈ c. Then, α there exist γ ,.T..,γ ∈c\{α} such that 1 m m m N ∪N ⊃ N , and so N ⊃ N \N . α γi γi α i=1 i=1 \ \ As above, there exists l ∈ E \ m E . We can then choose γ ,...,γ ∈ c α i=1 γi m+1 n such that S n {1,...,l}⊂ E . γi i=1 [ n n Weclaim thatN ⊃ N . Indeed,let(j )∈ N . Thenj =···=j =∞. i=1 γi i i=1 γi 1 l We see that there exists k ≥ l such that j ≥ k (1 ≤ i ≤ k) and j = ∞ (i > k). i i T T For each r ∈ N, set j(r) = j (i 6= l) and set j(r) = k +r. Then, we see that i i l (j(r)) ∈ m N \N ⊂N and lim (j(r))= (j ). Thus (j )∈ N. Hence, for each i i=1 γi α r i i i β ∈c\{γ ,...,γ }, we have N ∪N ∈F. (cid:3) 1 n β T Theorem 3.9. Let Ω be a locally compact metrizable space. Suppose that p ∈ ∂(∞)(Ω♭). Then there exists a non-redundant pseudo-finite family (P : α ∈ c) of α prime z-filters on Ω, each z-filter is supported at p. Moreover, by setting P =C (Ω)∩Z−1[P ], we obtain a non-redundant pseudo- α 0 α finite family of prime z-ideals in C (Ω), each ideal is supported at p, such that 0 C (Ω) P =c. 0 α (cid:12)(cid:12) (cid:30)α\∈c (cid:12)(cid:12) (cid:12) (cid:12) Proof. In this proof, we shall(cid:12)identify Ξ wit(cid:12)h a closed subset of Ω♭ such that (cid:12) (cid:12) (∞,∞,...) is identified with p; in the case where p ∈ Ω, we can further assume that Ξ⊂Ω (cf. Lemmas 3.7). Let (Q : α∈c) be the family of prime z-filters on Ξ as constructed in Lemma α 3.8. For each α∈c, set P ={Z ∈Z[Ω]: (Z∪{p})∩Ξ∈Q }. α α Note thateveryclosedsubsetofΞis inZ[Ξ], sowe cansee thateachP is a prime α z-filteronΩ. Thepseudo-finitenessof(P : α∈c)andof(P : α∈c)thenfollows α α from that of (Q ). The cardinality condition follows from the fact that |C(Ξ)|=c. α PRIME IDEALS IN C0(R) 7 By the non-redundancy of (Q : α∈c), for each α∈c, there exists α N ∈ Q \Q . α β α β6=α \ Bythe Urylson’slemma, wecanfindZ ∈Z[Ω]suchthat(Z ∪{p})∩Ξ=N ; we α α α can even require Ω\Z to be σ-compact so that Z =Z(f ) for some f ∈C (Ω). α α α α 0 Thus we see that Z ∈ P \P and f ∈ P \P . α β α α β α β∈c,β6=α β∈c,β6=α \ \ Finally, we shall prove that each P (and hence each P ) is supported at p α α (α ∈ c). Indeed, in the case where p is the point at infinity of Ω, for each x ∈ Ω, there exists N ∈Q such that x∈/ N. We can then find Z ∈Z[Ω] such that x∈/ Z α andthat(Z∪{p})∩Ξ=N. Thus Z ∈P andx∈/ Z. So P is supportatinfinity. α α Ontheotherhand,inthecasewherep∈Ω,letZ ∈P bearbitrary. ThenZ∩Ξis α closedinΞ,andsoitisinZ[Ξ]. Since{p}∈Z[Ξ]\Q ,wededucethatZ∩Ξ∈Q . α α Hence, p∈Z, and thus P is supported at p. (cid:3) α Corollary 3.10. Let p∈R♭. There exists a family (P : α∈c) of prime z-ideals α in C (R) with the following properties: 0 • If f ∈P for some α ∈c, then f ∈P for all but finitely many α∈c; α0 0 α • P 6⊂P for each α ∈c; • eaαc6=hαP0 αis supαp0orted at p. 0 (cid:3) α T There are many countable compact metrizable spaces Ω with ∂(∞)Ω 6= ∅. We note as a specific example the following countable compact subset of [0,1]: k ∆={0}∪ 2−ni : k, n ,n ,...,n ∈N and k ≤n <···

Uncountable families of prime z-ideals in C_0(R)

Similar Uncountable families of prime z-ideals in C_0(R)