Valdivia compact groups are products

V ALDIVIA COM P A CT GR OUPS ARE PR ODUCTS A. CHIGOGIDZE Abstract. It is shown that every V aldivia compact group is homeo morphic to a pro duct of metrizable compacta. 1. Introduction A compact space is called V aldivia compact if it can b e em b edded in to R A for some A in suc h a w a y that the image of this em b edding is the closure of a subset of a Σ-pro duct. These spaces ha ve b een studied b y sev eral a uthors (see [6], [7] for a nice surv ey of V aldivia compacta and related topics). A particular area o f in terest is understanding of the structure of V aldivia compact gro ups. The re are compact connected Ab elian groups whic h are not V aldivia compact [9]. On the other hand it w as sho wn recen tly [8] that ev ery V aldivia compact Ab elian group is homeomorphic to a pro duct of metrizable compacta. W e ex tend this result to the noncomm utativ e case (Theorem 3.3 ) . This answ ers negatively question from [8] whether there exist V aldivia compact groups whic h are not products. Our original pro of w as based o n a recen tly anno unced result on preserv ation of the class of V aldivia compacta b y retractions. I w ould lik e to thank the referee for p oin ting out that the pro o f of this statemen t con tained an error and that the v alidit y of the resu lt itself is still op en. The pro of of Theorem 3.3 has b een revised and a s presen ted below do es not dep end on the ab ov e men tioned statemen t. 2. Preliminaries F or the reader’s con v enience in this section we presen t some of the neede d results. 2.1. D irected sets. Let κ ≥ ω . A subse t B of a partially ordered dir ected set A is said to b e κ - c lose d in A if for eac h c hain C ⊆ B , with | C | ≤ κ , w e ha ve sup C ∈ B whenev er the elemen t sup C exists in A (sup C denotes the lo wes t upper bound of elemen ts of C ). A set A is s aid to be κ - c omplete if for eac h chain B of elemen ts o f A , with | B | ≤ κ , there exists an elemen t sup B in 1991 Mathematics S ubje ct Classific ation. Primary: 54 D30; Secondary: 54 C15. Key wor ds and phr ases. Compact group, V aldivia co mpact, in verse sp ectrum. 1 2 A. Chigogidze A . A standard ex ample of a κ -complete set is the set exp κ A of all subsets of cardinalit y ≤ κ of an y set A . The follo wing statemen t presen t s an important property of κ -complete sets ([2, Propo sition 1.1.27]). Prop osition 2.1. L et { A i : i ∈ I } , | I | ≤ κ , b e a c ol le ction of κ -close d a n d c ofinal subsets of a κ -c omplete set A . Then the interse ction ∩{ A i : i ∈ I } is also c ofinal and κ -close d in A . 2.2. I n verse Sp ectra. All limit pro j ections of in v erse spectra considere d b elo w are surjectiv e and all spaces are compact. Let κ ≥ ω . An in ve rse sp ectrum S X = { X α , p β α , A } consisting of compact spaces is a κ -sp ectrum if: (i) w ( X α ) ≤ κ , α ∈ A ; (ii) The indexing set A is κ - complete; (iii) S X is κ -contin uous, i.e. for eac h ch ain { α i : i ∈ I } ⊆ A with | I | ≤ κ and α = sup { α i : i ∈ I } , the diagonal pro duct △{ p α α i : i ∈ I } : X α → lim { X α i , p α j α i , I } is a homeomorphism. One of the main results concerning κ - sp ectra is the fo llowing result of ˇ S ˇ cepin (kno wn as the ˇ S ˇ cepin’s Sp ectral Theorem, see [2, Theorem 1.3.4]). Theorem 2.2. L et S X = { X α , p β α , A } and S Y = { Y α , q β α , A } b e two κ -sp e ctr a. Then for every map f : lim S X → lim S Y ther e exist a c ofinal and κ -close d subse t B ⊆ A and map s f α : X α → Y α , α ∈ B , such that f = lim { f α : α ∈ B } . If f is a home omorphism, then we ma y assume that e ach f α , α ∈ B , is also a home omorphism. 2.3. Σ -pro duct s. If C ⊆ B ⊆ A , t hen π B : R A → R B and π B C : R B → R C denote the corresp onding pro jections. Similarly b y i B : R B → R A and i B C : R C → R B w e denote sections of π B and π B C defined as follo ws: i B ( { x t : t ∈ B } ) = ( { x t : t ∈ B } , { 0 t : t ∈ A \ B } ) and i B C ( { x t : t ∈ C } ) = ( { x t : t ∈ C } , { 0 t : t ∈ B \ C } ) . Belo w we consider the Σ-pro duct of real lines , whic h is the subspace Σ( A ) =  { x t : t ∈ A } ∈ R A : |{ t ∈ A : x t 6 = 0 }| ≤ ω }  of the pro duct R A . F or eac h B ⊆ A , Σ( B ) is iden tified with the subspace Σ( A ) ∩ i B ( R B ) of Σ( A ). Note t hat Σ( A ) = S { i B ( R B ) : B ∈ exp ω A } . Retractions r B = i B π B : Σ( A ) → Σ( B ), B ⊆ A , play an imp ortan t role b elo w. W e refer the reader to [5] f or relev an t definitions and results. F o r a given closed subset F ⊆ Σ( A ) w e are in terested in finding subsets B ⊆ A suc h t hat r B ( F ) ⊆ Remarks on V al divia compact spaces 3 F or, equiv alen tly , F ∩ Σ( B ) = r B ( F ). Note that if Y = cl R A F is compact, then r B ( F ) ⊆ F if a nd only if r B ( Y ) ⊆ Y or, equiv alen tly , cl( Y ∩ Σ( B )) = r B ( Y ). In this case B is called an F -go o d (or Y - go o d) subset of A . It turns out (see [4, Lemma 1], [1, Lemma 1.2]) that there are many Y -go o d subs ets . Precise statemen t we use b elow is recorded in the fo llo wing lemma. Lemma 2.3. L et ω ≤ κ < τ , | A | = τ and Y = cl R A ( Y ∩ Σ( A )) b e c omp act. Then (a) the set A V κ of Y -go o d subsets of A of c ar dinality ≤ κ is c ofin a l and κ -close d in exp κ A ; (b) union of an incr e asing c ol le ction of Y -go o d subsets of A is again a Y - go o d subset o f A ; (c) if B is a Y -go o d subset of A , then r B | Y : Y → Y B is a r etr action. 3. Proo f of the M ain Resul t In this section w e pro v e our main result - Theorem 3.3. First w e need the follo wing statement. Lemma 3.1. L et ω ≤ κ < τ , | A | = τ and a c omp act gr oup X b e emb e dde d into R A . Then the set exp κ A c ontains a c ofinal and κ -close d subset A G κ such that π B ( X ) is a top olo gic al gr oup and the pr oje ction π B | X : X → π B ( X ) is a homomorphism for e ach B ∈ A G κ . Pr o of. Let λ : X × X → X and µ : X → X denote group op erations in X , i.e. λ ( x, y ) = x · y and µ ( x ) = x − 1 , x, y ∈ X . Let also X B = π B ( X ), p B = π B | X and p B C = π B C | X B for an y C , B ⊆ A with C ⊆ B . Consider the standard κ -sp ectrum S X = { X B , p B C , e xp κ A } and note that X = lim S X . By Theorem 2.2, a pplied to the map µ : X → X and the κ -sp ectrum S X , there exist a cofinal and κ -closed subset A µ κ of exp κ A and maps µ B : X B → X B , B ∈ A µ κ , such that p B µ = µ B p B for eac h B ∈ A µ κ and µ = lim { µ B : B ∈ A µ κ } . Next consider the κ - sp ectrum S X × S X = { X B × X B , p B C × p B C , e xp κ A } . Ob vi- ously , X × X = lim S X × S X . Applying Theorem 2.2 to the map λ : X × X → X and to the sp ectra S X × S X and S X , w e conclude that there exist a cofinal a nd κ -closed subset A λ κ of exp κ A and maps λ B : X B × X B → X B , B ∈ A λ κ suc h that p B λ = λ B ( p B × p B ) for each B ∈ A λ κ and λ = lim { λ B : B ∈ A λ κ } By Proposition 2.1, the in tersection A G κ = A µ κ ∩ A λ κ is still cofinal and κ -closed in exp κ A . Note that for eac h B ∈ A G κ w e ha v e t wo maps λ B : X B × X B → X B and µ B : X B → X B . These maps define a group structure on X B as follows: x B · y B = λ B ( x B , y B ) and x − 1 B = µ B ( x B ), x B , y B ∈ X B . The unit elemen t in X B is defined as follo ws: e B = p B ( e ), where e is the unit in X . It is easy to see that the pro jection p B : X → X B b ecomes a group ho mo mo r phism. Indee d, for x, y ∈ X w e hav e p B ( x · y ) = p B λ ( x, y ) = λ B ( p B × p B )( x, y ) = λ B ( p B ( x ) , p B ( y )) = p B ( x ) · p B ( y ) . 4 A. Chigogidze  Lemma 3.2. L et X b e a top olo gic al gr oup which is a r etr act of a V aldivia c om- p act. Then ther e ex ists a wel l-or der e d c on tinuous sp e ctrum S X = { X α , p α +1 α , τ } such that (1) τ = w ( X ) ; (2) w ( X α ) ≤ | ω | · | α | , α < τ ; (3) X α is a top olo gic al gr oup and a r etr act o f a V aldivi a c omp act, α < τ ; (4) T he limit pr oje ction p α : X → X α is a top olo gic al ho momorphism and a r etr action, α < τ . Pr o of. Let Y b e a V aldivia compact of suitably embedded into R A with | A | = τ > ω . Supp ose also that s : Y → X is a retraction and X is a top ological group. Without loss of g eneralit y we may assume that w ( X ) = τ . Let ω ≤ κ < τ . By Lemma 2.3(a), the collection S κ Y = { Y B , q B C , C, B ∈ A V κ } , where q B C = r B C | Y B , whenev er C , B ∈ A V κ and C ⊆ B , forms a κ -spectrum. Also consider the κ -sp ectrum S κ X = { X B , p B C , C, B ∈ A V κ } , where X B = r B ( X ) and p B C = r B C | X B , whenev er C , B ∈ A V κ and C ⊆ B . Clearly , Y = lim S κ Y and X = lim S κ X . Note that pro jections q B C : Y B → Y C and q B : Y → Y B of the sp ectrum S κ Y are retractions (L emma 2.3(c)). Theorem 2.2, applied to the κ -sp ectra S κ Y and S κ X and to the retraction s : Y → X , guaran tees existence o f a κ -closed and c ofinal subset A κ s of A V κ suc h t hat for eac h B ∈ A s there is a map s B : Y B → X B satisfying the equality p B s = s B q B . Next observ e that s B : Y B → X B , B ∈ A κ s , is a retraction. Indeed, let x B ∈ X B and x ∈ X is suc h that p B ( x ) = x B . Then w e hav e s B ( x B ) = s B ( p B ( x )) = s B ( q B ( x ) = p B ( s ( x )) = p B ( x ) = x B . Next note tha t since in the diagram ( B ∈ A κ s ) X p B   Y s o o q B   X B Y B s B o o maps s , s B and q B are retractions, it follows that p B is also a retraction. By Lemma 3.1, we may assume without loss of generality that there exists a κ - closed and cofinal subset A κ of A κ s ∩ A G κ suc h that for eac h B ∈ A κ compactum X B is a top ological group and the pro jection p B : X → X B is a topo logical homomorphism. Th us, f or an y κ with ω ≤ κ < τ w e ha ve a κ - closed and cofinal subset A κ of exp κ A suc h that the follo wing conditions are satisfied for eac h B ∈ A κ : (i) Y B is a V a ldivia compact (ii) q B : Y → Y B is a retraction Remarks on V al divia compact spaces 5 (iii) There exis ts a retraction s B : Y B → X B suc h that p B s = s B q B . (iv) p B : X → X B is a retraction (v) X B is a top o logical group (vi) p B : X → X B is a top ological homomorphism. Let us no w define subsets A α , α < τ , of A . Fix a w ell-ordering { B α : α < τ } of the set A κ , let A 0 = B 0 , s 0 = s A 0 and a ssume that the sets A β and retra ctio ns s β : Y A β → X A β ha ve b een constructed for eac h β < α , where α < τ , in suc h a w ay that the follow ing conditions are satisfied: (a) β | A β | ≤ | ω | · | β | (b) β A γ ⊆ A β whenev er γ ≤ β < α (c) β A β = ∪{ A γ : γ < β } whe nev er β is a limit ordinal. (d) β A β +1 ⊇ A β ∪ B β +1 , whenev er β + 1 < α . (e) β There exists a retra ctio n s β : Y A β → X A β suc h that p A β s = s β q A β (f ) β X A β is a top o logical gr o up (g) β p A β : X → X A β is a retraction and a top ological homomorphism If α is a limit ordinal, then let A α = ∪{ A β : β < α } . Note that for eac h γ , β with γ < β < α the follo wing diagram X A β p A β A γ   Y A β s β o o q A β A γ   X A γ Y A γ s γ o o is comm utative . Note also t ha t X A α = lim { X A β , p A β A γ , α } and Y A α = lim { Y A β , q A β A γ , α } . Consequen tly , collection { s β : β < α } defines a map s α : Y A α → X A α suc h tha t p A β s α = s β q A β for eac h β < α . Since s β is a re- traction, for any x ∈ X A α w e hav e p A β ( s α ( x )) = s β ( q A β ( x )) = s β ( p A β ( x )) = p A β ( x ) , whic h sho ws t ha t s α ( x ) = x . This v erifies condition (e) α . By (b) β and Lemma 2.3(b), A α is a Y -go o d subset of A . Consequen tly , by Lemma 2.3 (c), q A α : Y → Y A α is a retraction. But then it is easy to conclude that p A α : X → X A α is also a retraction. By conditions (f ) β and (g) β , it fo llo ws that f or β < α , X A α = lim { X A β , p A β A γ , α } is also a top ological group with naturally defined group op erations and the pro jection p A α : X → X A α is a top ological homomorphism. Next consider the case α = β + 1. Since, b y condition (a) β , | A β | ≤ | ω | · | β | and since A κ (with κ = | ω | · | β | ) is cofinal in ex p κ A , there exis ts A α ∈ A κ suc h that A α ⊇ A β ∪ B α . Then there exists a retraction s α : Y A α → X A α suc h that 6 A. Chigogidze p A α s = s α q A α , p A α : X → X A α is a retraction, X A α is a topo logical group and p A α is a top o logical homomorphism. This completes the construction. Let X α = X A α and p α = p A α for eac h α < τ . It only remains to note that X = lim { X α , p α +1 α , τ } and that all the required conditions (1)–(4) are satisfied.  Theorem 3.3. L et X b e a c omp act g r oup. Then the fol low i n g c onditions ar e e quivalent: (1) X is a V aldivia c omp act; (2) X is a r etr act of a V al d ivia c omp act; (3) X is home omorphic to a pr o duct of m e trizable c omp acta; Pr o of. Implications (1 )= ⇒ (2) and (3) = ⇒ (1) are trivial. Let us prov e im- lication (2)= ⇒ (3) . W e proceed b y induction. F or metrizable groups whic h are retracts of V a ldivia compacta there is nothing to pro ve. Let X denote a top ological group whic h is a retract of a V alidivia compact. Suppo se that the statemen t is true in cases when w ( X ) < τ , τ > ω , and consider X of w eight w ( X ) = τ . Let S X = { X α , p α +1 α , ω 0 ≤ α < τ } b e a sp ectrum supplie d b y Lemma 3.2. Note that by prop erty (4) in Lemma 3.2 eac h limit pro jec- tion p α : X → X α is a retraction and a group homomorphism. This implies that eac h short pro jection p α +1 α : X α +1 → X α is a gro up homomorphism (and ob viously a retraction). Consequen tly , X α +1 is homeomorphic to the pr o duct X α × k er ( p α +1 α ) (see [3, Lemma 0.2 ]) (corresp onding homeomorphism is pro- vided b y the formula h ( x ) = ( p α +1 α ( x ) , x · ( p α +1 α ( x )) − 1 ), x ∈ X α +1 ). Thu s X is homeomorphic to the pro duct X ω 0 × Q { k er( p α +1 α ) : ω 0 ≤ α < τ } . Finally note that k er( p α +1 α ) is a retract of X α +1 and X α +1 , b y Lemma 3.2(3 ) , is a retract of a V aldivia compact. Therefore k er( p α +1 α ) is a retract o f a V aldivia compact. By Lemma 3.2(2), w (k er( p α +1 α )) ≤ w ( X α +1 ) ≤ | ω | · | α + 1 | < τ . Then the inductiv e assumption guarantee s that eac h k er( p α +1 α ) is homeomorphic to a pro duct of metrizable compact groups. Pro of is completed.  Reference s [1] A. Argyro s , S. Mercoura kis, A. Negrep ontis, F unctional-analytic pr op erties of Corson- c omp act sp ac es , Studia Ma th. 89 (1988), 197–2 29. [2] A. Chig ogidze, Inverse Sp e ctra , North Holland, Amsterdam, 1996. [3] A. Chigogidze , Nonmetrizable ANRs admitting a gr oup st ructur e ar e manifolds , T o p ology Appl. 153 (2006), 107 9 -108 3 [4] S. P . Gul’ko, On pr op erties of su bsets of Σ - pr o ducts , Soviet Math. Dokl. 18 (1977 ), 1438– 1442. [5] S. P . Gul’ko, Semilattic es of re tr actions and the pr op erties of c ontinuous function sp ac es of p art ial maps , Recen t progr ess in function spaces , 93– 155, Qua d. Ma t., 3 , Dept. Math., Seconda Univ. Nap oli, Case r ta, 1998. [6] O . K alenda, V aldivia c omp act sp ac es in top olo gy and Banach sp ac e t he ory , E xtracta Math. 15 (2000), 1–8 5. Remarks on V al divia compact spaces 7 [7] O . K alenda, Natur al ex amples of V aldivi a c omp act sp ac es , J. Math. Anal. Appl., to app ear, av ailable o nline at http:/ /dx.do i.org/10.1016/j.jmaa.2007.07.069 . [8] W. Kubi ´ s, V ald ivia c omp act A b elian gr oups , preprint; av ailable online at http:/ /xxx. lanl. gov/abs/0706.3772 . [9] W. Kubi ´ s, V. Usp enskij, A c omp act gr oup which is not V aldivi a c omp act , Pro c. Amer. Math. So c. 133 (2005), 248 3–248 7. Dep ar tment of Ma thema tics and S t a tistics, U niversity of Nor th Car olina a t G reensbor o, 3 83 Br y an Bldg, Greensbor o, N C, 2740 2, USA E-mail addr ess : chigo gidze @uncg .edu