Strong reflexivity of Abelian groups

STRONG REFLEXIVITY OF ABELIAN GROUPS Montserrat Bruguera Dept. de Matematica Aplicada I Universidad Politecnica de Catalu~na e-mail: [email protected]c.es Mara Jes us Chasco Dept. de Fsica y Matematica Aplicada Universidad de Navarra e-mail: mjchasco@sica.unav.es Abstract A reexive top ological group G is called strongly reexiveifeach closed subgroup and each Hausdor quotient of the group G and of its dual group is re- exive. In this pap er we establish the adequate concept of strong reexivity for convergence groups and weprove that the pro duct of countable many lo cally compact top ological groups and complete metrizable nuclear groups are BB-strongly reexive. 1991 Mathematics Sub ject Classication: Primary 22A05. Secondary 46A16. Key words and phrases : Pontryagin duality theorem, dual group, convergence group, continuous convergence, reexive group, strong reexive group, k-space,  Cech complete group, k-group . The second author was partially supported by D.G.I.C.Y.T. PB 96-0651-C03-03 . 1 Intro duction Along this pap er we deal with strong reexivity of top ological groups and convergence groups. All the groups considered will be Ab elian. For an Ab elian top ological group G ,the symbol ; G denotes the set of continuous characters (i.e., continuous homomorphisms from G into T , the multiplicative group of complex numb ers with mo dulus 1). The set ; G with multiplication dened pointwise and endowed with the compact op en top ology is a Hausdor top ological Ab elian group which is called the dual group of G and is denoted by G ^ . The bidual group of G , G ^^ is dened as ( G ^ ) ^ and  G : G ! G ^^ stands for the canonical embedding. A top ological Ab elian group is said to be reexive if  G is a top ological isomorphism. The Pontryagin duality theorem states that every lo cally compact Ab elian group is reexive. This yields, in an obvious way, that also closed subgroups and Hausdor quotients of lo cally compact Ab elian groups are reexive. This is not the case for other reexive groups, whichmayhave non reexive closed subgroups or non reexive quotients. For instance, Leptin proved in 11] the existence of apro duct of discrete groups with a non reexive closed subgroup. Thus, it is natural to intro duce a new class of reexive groups stable for those op erations. This is done in 1], where such groups are called strongly reexive. On the other hand, in a set of pap ers by Beattie, Binz, Butzmann, Muller and several others, a new concept of reexivity of top ological groups is given using the continuous convergence structure to dene the dual of a top ological Ab elian group. Since the continuous convergence structure do es not derive in general (unless the departure group is lo cally compact) from a top ology, the dual group is only a \convergence group". However this incursion into convergence groups is only an auxiliary to ol: the bidual is again top ological. Thus, if duals are endowed with the continuous convergence structure instead of the compact op en top ology, a new kind of reexivity is obtained by 2 requiring that the canonical emb edding into the bidual b e a bicontinuous isomorphism. In 7] such groups are called BB-reexive and it is proved that this new notion of re- exivity is indep endent of the classical notion of Pontryagin reexivity. It is an op en question to determine the class of groups for whichthey coincide in 6] it is proved that it contains metrizable groups, and as Corollary 2.5 states, it also contains  Cech complete groups and direct sums of lo cally compact groups. The imp ortance of this new concept is mainly related to completeness prop erties of the group. For example, a BB-reexive top ological group must be complete (Prop osition 2.2). On the other hand a top ological vector space is BB-reexive if and only if it is lo cally convex and complete 2]. The class of BB-reexive groups is more likely to b e stable for the op eration of taking closed subgroups. So, in this resp ect BB-reexivity b ehaves b etter than Pontryagin reexivity. It makes sense to dene BB-strongly reexive groups, as those BB-reexive groups such that the Hausdor quotients of them and of their duals are also BB-reexive. As Theorem 3.4 states, in the class of BB-strongly reexive groups, the general corresp ondences between duals of closed subgroups and the whole character groups mo dulo annihilators characteristic for Pontryagin duality, are also valid. 1 Preliminary background For the denitions of convergence structure and convergence space we refer the reader to 9] and 2]. Top ological notions such as continuity, cluster point, closed, op en or compact sets, etc, can be stated in terms of convergence of lters, therefore they have corresp onding denitions for convergence spaces. A top ology denes in a natural way a convergence structure, namely, the one given by its convergent lters or nets. However, not every convergence structure comes from a top ology on the supp orting set. A convergence structure onaset X is said to b e topological if it is given by the 3 convergent lters of some top ology. Aset A  X is open if it b elongs to every lter whichconverges to a p ointof A . The family of all op en sets in the convergence space ( X ) fullls the axioms of a top ology   , called the associated topology to the convergence structure . The convergence in   has more convergent lters than  and they coincide if the convergence structure  is top ological. One may prove that a map f dened on a convergence space ( X ) with values in a top ological space Y is continuous i f :( X   ) ! Y is continuous. If H is a subspace of aconvergence space ( X ),   j H is ner than the asso ciated top ology to the convergence structure  j H , and they coincide for compact convergence subspaces, as stated in the following Lemma. Lemma 1.1 If ( X ) is a convergencespaceand H  X is compact, then   j H =   j H Pro of. Since the convergence structure  is ner than   , H is compact in   . Then,   j H and   j H are comparable compact top ologies therefore, they coincide. A Hausdor top ological space X is a k-space if its closed sets are characterized by the following fact: F  X is closed in X if and only if F \ K is closed in K , for every compact subset K of X .This condition means that the top ology of a k-space is the nest top ology with the same compact sets and it is equivalent to the following one: A function dened on X with values in a top ological space Y is continuous i its restriction to any compact subset is continuous. It is well known that for a top ological Ab elian group the nest top ology with the same compact subsets is not in general a group top ology. By this reason Noble intro duced in 13] the notion of k-group as the appropriate analogue to k-space for Hausdor top ological groups. A top ological group is a k-group if its top ology is the nest group top ology with the same compact subsets (equivalently: each homomorphism from G into another top ological group is 4 continuous if its restriction to each compact is continuous). This notion has some better p ermanence prop erties than the one of k-space. Quotient groups and pro ducts of k-groups are k-groups. Obviously, every k-space is also a k-group. In the framework of convergence spaces we have the following result. Prop osition 1.2 Let ( X ) a convergencespace such that the associatedtopological space ( X   ) is Hausdor. Then the fol lowing conditions are equivalent. a) F  X is closed in X if and only if F \ K is closed in K ,for every compact subset K of X . b) A function f dened on X with values in a topological space Y is continuous if and only if its restriction to any compact subset of X is continuous. Pro of. a ) ) b )Let f : ( X ) ! ( Y  )be such that f j K is continuous for every -compact K. In order to prove that f is continuous, it is enough to see that f ; 1 ( C ) \ K is -closed in K for each closed subset C of Y , but this yields from the equality f ; 1 ( C ) \ K =( f j K ) ; 1 ( C )andthe continuityof f j K . b ) ) a ) Consider the family H = f H  X such that K \ H is -closed for all -compact, K g : This family fullls the axioms of closed sets for a top ology  H which is ner than   and coincides with it on the -compact subsets of X .Hence the identitymap from ( X   )to ( X  H ) is bicontinuous and that means that H is the family of closed subsets of ( X ). 5 Remark. Observe that each one of the ab ove equivalent conditions implies, by Lemma 1.1, that the asso ciated top ological space ( X   ) is a k-space. We will call k-convergence spaces those convergence spaces satisfying one of them. Local ly compact convergence spaces are convergence spaces for whichevery convergent lter has a compact memb er. They are k-convergence spaces as can be shown in the following Prop osition. Prop osition 1.3 Let ( X ) be a local ly compact convergence space such that ( X   ) is Hausdor. Then, a function f denedonXwith values in a convergencespace Y is continuous i its restriction to any compact subset is continuous. Pro of. Let f : ( X ) ! ( Y  0 )be suchthat f j K is continuous for every -compact K. Let F be a lter in X convergent to x .Since X is lo cally compact, the lter F has a compact member K . The trace of F in K is a lter whichconverges to x in K , so its image by the continuous function f j K is a lter in Y whichconverges to f ( x ). Therefore, the lter f ( F )converges to f ( x ). 2 BB-reexive convergence groups Fischer dened the convergence groups, as groups endowed with a convergence structure compatible with the group op eration. All the convergence groups considered in this pap er will b e Hausdor, that is, a lter converges to at most one p oint. If G isaconvergence group, we use the symbol ; G to denote the set of all continuous homomorphisms from G into T . The continuous convergence structure  c in ; G is dened in the following way: A lter F in ; G converges in  c to an element  2 ; G if for every x 2 G and every lter 6 H in G that converges to x , ! ( FH )converges to  ( x )in T (here, FH denotes the lter generated by the pro ducts F  H , where F 2F , H 2H and ! ( FH )denotes the lter generated by ! ( F  H ):= f f ( x ) f 2 F x 2 H g ). It can be said that  c is the coarsest convergence structure in ; G for which the evaluation mapping ! :; G  G ! T is continuous (; G  G has the natural pro duct structure). The dual group ; G of a convergence group ( G ), endowed with the convergence structure  c ,isaconvergence group which is denoted by; c G and is called the convergence dual group of G . Aconvergence group is called BB-reexive if the canonical homomorphism  G : G ! ; c ; c G is a bicontinuous isomorphism (here ; c ; c G has the obvious meaning). Observe that, due to the continuityof ! :; G  G ! T ,  G is always continuous. For lo cally compact Ab elian top ological groups, the compact op en top ology and the continuous convergence structure in the dual group, have the same convergent lters. This fact characterizes the lo cally compact groups in the class of top ological reexive groups 12]. Prop osition 2.1 Let G be a local ly compact convergence group then: a) The continuous convergence structure on the dual group is topological and it coincides with the compact open topology. b) If compact subsets of G are topological, ; c G is complete. Pro of. a) Let F be a lter  co -convergent to the neutral element of ; G and let H be a convergent lter in G . Since G is lo cally compact the lter H has a compact 7 member H . If W 2 B T (1), the set ( H W ) is a neighb orho o d of the neutral element of ; G therefore, it contains some F 2 F . As ! ( F  H )  W , we conclude that F is  c -convergent. The converse holds without any restrictions. b) We are going to see that ; c G is complete in the uniformity of uniform convergence on compact sets. If ( f  ) is a Cauchy net in this uniformity, for all x 2 G ,( f  ( x )) is also Cauchy in T . Let f be the homomorphism dened by f ( x ) = lim( f  ( x )). Since for each compact K  G ,( f  j K )isin C ( K T ) and this top ological space is complete, wehavethat f j K is continuous for each compact K  G and therefore, by Prop osition 1.3, it is continuous on G . It is also clear that the convergence of ( f  )to f is uniform on compact sets. We collect in the next Prop osition some prop erties of the continuous convergence structure on the dual of a top ological group G . Prop osition 2.2 Let G be a topological Abelian group, then: a) ; c G is a local ly compact convergencegroup. b) (; G   c ) is a k-space. c) If A  ; G is equicontinuous, the continuous convergence structureon A coincides with the topology of pointwise convergenceand with the compact open topology. d) Compact subsets of ; c G are equicontinuous. e) Compact subsets of ; c G are topological. f) ; c ; c G is topological and complete. 8 g) If  G is continuous, G ^ and ; c G have the same compact subsets, and therefore G ^^ is a topological subgroup of ; c ; c G . h) In case  G continuous, G ^ is a k-space if and only if  co =   c . Pro of. a) This is Prop osition 1 of 7]. We observe that the requirement that  G b e continuous can be dropp ed in the pro of. b) follows from a) and Prop osition 1.3. c) it is proved in Lemma 1 of 8]. d) it is proved in Theorem 7 of 8]. e) follows from c) and d). f) follows from a) and Prop osition 2.1. b). g) it is proved in 7], Remark 1 and Theorem 1. h) follows from b) and g). Theorem 2.3 For a topological group G , the fol lowing assertions are equivalent: a) The topological groups ; c ; c G and G ^^ coincide. b)  G is continuous and every homomorphism : G ^ ! T such that  j K is continuous for al l compact subsets K , is continuous. 9 Lemma 3.5 If ( G ) isaconvergencegroup and H  G a closedsubgroup, then   =H is the associated topology to  =H . Pro of. We must show that   =H =   =H .Let p : G ! G=H be the canonical pro jection.  ) Take O 2   =H and let F be a lter in G=H ,  =H -convergent to  z ] 2 O . If L is a lter in G such that L  ! x 2 p ; 1  z ] and p ( L )  F , then p ; 1 ( O ) 2L since p ; 1 ( O ) is   -op en and x 2 p ; 1 ( O ). Thus O = p ( p ; 1 ( O )) 2 p ( L )  F and so, O 2   =H .  )Let U 2   =H and let L be a lter in G such that L  ! t 2 p ; 1 ( U ). Then p ( L ) !  t ] 2 U in G=H and, since U is   =H -op en, U 2 p ( L ). Thus p ; 1 ( U ) 2L , and so U 2   =H . Theorem 3.6 Let f G n g n 2 N be a sequence of local ly compact Abelian groups. Then, G = Q G n is BB-strongly reexive. Pro of. In 1](17.3), it is proved that G is Pontryagin strongly reexive, then for each closed subgroup H of G , the quotient G=H is  Cech complete and Pontryagin reexive, thus it is BB-reexive. Let L be a closed subgroup of ; c G .Since G ^ is a k-space,  co is the asso ciated top ology to the continuous convergence structure and L is a closed subgroup of G ^ . Being G Pontryagin strongly reexive, L is dually closed so, there exists a closed subgroup H of G such that H o = L (1](14.2)). We are going to see that ; c G=H o is 16 reexive. The convergence group ; c G is lo cally compact and the same happ ens with the quotient ; c G=H o . Thus, ; c (; c G=H o ) is top ological and it carries the compact op en top ology. On the other hand, the top ological group asso ciated to ; c G=H o is, by Lemma 3.5, G ^ =H o . Therefore ;(; c G=H o )=;( G ^ =H o ). The group ; c (; c G=H o )isbicontinuously isomorphic to H oo =  G ( H )which is isomorphic to H and  : ; c G=H o ! ; c H is a continuous isomorphism, since H is dually emb edded. So, taking into account the commutativityof the diagram ; c G=H o  ; c G=H o ;;;;;! ; c ; c (; c G=H o )  ? ? y x ? ? ; ' H o ; c H ;;; ;  G j H ; c H oo and due to the fact that ; c H is lo cally compact, using Prop osition 1.3, we only need to prove that the restrictions of  ; 1 to the compact subsets of ; c H are continuous. Let C b e a compact subset of ; c H , i : H ! G the inclusion and p :; c G ! ; c G=H o the canonical pro jection C is top ological and equicontinuous. As it is proved in 1](8.2), there exists an equicontinuous set E in ; G suchthat ; i ( E ) = C . Let ! E the  co - clausure of E  ! E is closed and equicontinuous and therefore compact in G ^ .Being  G continuous, ! E is also compact in ; c G . Consequently p ( ! E )is compact in ; c G=H o . Since  ; 1 ( C )   ; 1 (; i ( ! E )) = p ( ! E ) and  ; 1 ( C )is closed in ; c G=H o , we have that  ; 1 ( C ) is compact in ; c G=H o . We are going to see now that  ; 1 ( C )istop ological: The set K = ! E is top ological and compact in G ^ = P G ^ n  so, there exists some n 2 N such that K  G ^ 1 + G ^ 2 + ::: + G ^ n =: G n and p ( K )  p ( G n )whichis top ologically isomorphic to G n =G n \ H o .Let us see that G n =G n \ H o inherits from ; c G=H o the natural top ology. Let F be a lter in G n =G n \ H o convergent to  x ]in the natural top ology, q : G n ! G n =G n \ H o the canonical pro jection and H a lter in G n convergent 17 to x 2 q ; 1 ( x ]) such that q ( H )  F .If L converges to y in G = Q G n , since H is in G n = G ^ 1 + G ^ 2 + ::: + G ^ n , H ( L ) ! x ( y ) therefore H! x in ; c G and then F !  x ] in ; c G=H o . For each compact C of ; c H ,wehave seen that  ; 1 ( C ) is compact and top ological. The map  :  ; 1 ( C ) ! C , surjective and continuous, is in fact a top ological isomorphism, and consequently the restriction of  ; 1 to the compact set C is continuous. For the class of nuclear groups intro duced by Banaszczyk in 1]wehave the following result. Theorem 3.7 Every complete metrizable nuclear group is BB-strongly reexive. Pro of. By 1](17.3) every complete metrizable nuclear group G is Pontryagin strongly reexive. Then, for every closed subgroup H of G , H and G=H are Pontryagin reexive. So, b eing H and G=H metrizable, are also BB-reexive (6]). Dual convergence groups of BB-reexive groups are also BB-reexive, therefore ; c G and ; c H are BB-reexive. Let L be a closed subgroup of ; c G . Being  co the asso ciated top ology to the continuous convergence structure, L is a closed subgroup of G ^ .As in the pro of of Theorem 3.6, there exists a closed subgroup H of G suchthat H o = L .Using now that ; c G=H o is bicontinuosly isomorphic to ; c H (5]), we obtain ; c G=H o is BB-reexive. Acknowledgement. The authors are indebted to Professor E. Mart#$n-Peinador for very helpful suggestions. 18 References 1] Banaszczyk, W. 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