Completeness properties of locally quasi-convex groups

Completeness prop erties of lo cally quasi-convex groups. M. Bruguera Dept. de Matematica Aplicada I Universidad Politecnica de Catalu~na e-mail: [email protected]c.es M.J. Chasco  Dept. de Fsica y Matematica Aplicada Universidad de Navarra e-mail: mjchasco@sica.unav.es E. Martn-Peinador y Dept. de Geometra y Topologa Universidad Complutense e-mail: p[email protected] V. Tarieladze Muskhelishvili Institute of Comp. Math. Georgian Academy of Sciences e-mail address: [email protected] Abstract It is natural to extend the Grothendieck Theorem on completeness, valid for lo cally convex top ological vector spaces, to ab elian top ological groups. The adequate framework to do it seems to be the class of lo cally quasi-convex groups. However, in this pap er we present examples of metrizable lo cally quasi-convex groups for which the analogue to Grothendieck Theorem do es not hold. By means of the continuous convergence structure on the dual of a top ological group, we also state some weaker forms of Grothendieck Theorem valid for the class of lo cally quasi-convex groups. Finally,weprove that for the smaller class of nuclear groups, BB-reexivity is equivalent to completeness. Intro duction The character group ; G of an ab elian top ological group G is the set of all continuous homomorphisms from G into the torus T = f z 2 C : j z j = 1 g ,with pointwise multiplication. Homomorphisms from G into T are usually named characters. The dual group of G is de- ned as ; G , endowed with the compact-op en top ology  co . It will be denoted by G ^ , while G ^^ := ( G ^ ) ^ stands for the bidual. We will also use the notations Hom ( G T ) and Hom ( G R ) to denote homomorphisms, and CHom ( G T )(also denoted by ; G ) and CHom ( G R ) continuous homomorphisms. 1991 Mathematics Sub ject Classication: Primary 22A05. Secondary 46A16. Key words and phrases : Completeness, Grothendieck theorem, Pontryagin duality theorem, dual group, convergence group, continuous convergence, reexive group, k-space, k-group.  Partially supp orted by D.G.I.C.Y.T. PB 96-0651-C03-03. y Partially supp orted by D.G.I.C.Y.T. PB 96-0651-C03-03. 1 The canonical emb edding  G : G ! G ^^ is dened by  G ( g )(  )=  ( g )forevery g 2 G and every  2 G ^ .If  G is a top ological isomorphism, the top ological group G is called reexive. The Pontryagin-Van Kamp en theorem states that lo cally compact ab elian groups are reexive. However the class of reexive groups includes other typ es of groups, like complete metrizable lo cally convex spaces and reexive top ological vector spaces 22] (b oth classes considered as top ological groups, i.e. forgetting the linear structure), arbitrary pro ducts of reexive groups 17], complete metrizable nuclear groups 2], etc. Our aim in this pap er is to study completeness of a top ological ab elian group and also of its dual, and how these prop erties are related with reexivity. Since completeness of lo cally convex vector spaces is totally characterized by the Grothendieck theorem and its corollaries, it seems natural to center the question for lo cally quasi-convex groups and to start with the underlying group of a top ological vector space. For such an ob ject E , completeness is indep endent of the point of view, i.e. if it is lo oked at as a vector space or as a group. However the character group ; E is no longer a vector space, and is obviously dierent from the set of continuous linear forms L E , which roughly sp eaking is the natural dual of a vector space. Thus, if a theorem of Grothendieck-typ e is to be obtained for the dual group of alo cally convex vector space, some work must be done, even for this very particular class of top ological groups. On the other hand, the continuous convergence structure can be dened in the dual of a top ological vector space and some fundamental results in duality theory heavily rely on it, although it may not be explicitely stated. Continuous convergence was rst dened in the dual of aconvergence group by Binz and Butzmann giving rise to the notion of BB-reexive convergence groups 3]. In 8] it is proved that alo cally convex vector space is BB-reexive if and only if it is complete. In corollary 4.4 we see that this result is also valid for nuclear groups, a class of top ological ab elian groups intro duced by Banaszczyk in 2], which can b e considered as the class of groups generated by lo cally compact ab elian groups and nuclear lo cally convex vector spaces, by the op erations of taking subgroups, arbitrary pro ducts, quotients by closed subgroups and countable direct sums. 1 Preliminary background A top ology denes in a natural way a convergence structure, namely,the one given by its convergent lters or nets. Conversely, one can start declaring which nets (or lters) on a set X converge, and the corresp onding limit p oints, and this is a convergence structure for the set X . If some general conditions (convergence axioms) are satised so that there exists a top ology in X for which the convergent nets (or lters) are the given \a priori" 18], it can b e said that the convergence derives from a top ology,orsimplythatitistop ological. If the convergence structure do es not full all the requirements to b e derived from a top ology, then weonlyhave a convergence space. In the literature there is not an unanimous acceptance of which are the axioms that must dene this concept. We are interested just on the continuous convergence structure and we have followed the text of Binz 3], where the reader can nd a go o d account of information. We also take his notations. Top ological notions such as continuity, cluster p oint, closed, op en or compact sets, etc, can b e stated in terms of convergence of lters or nets, therefore they have corresp onding denitions for convergence spaces. Convergence 2 groups are groups endowed with a convergence structure compatible with the group op eration, 15]. If G is a convergence group, we also 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 H in G that converges to x , e ( FH ) converges to  ( x ) in T (here, e ( FH ) denotes the lter generated by the sets e ( F  H ):= f f ( x ) f 2 F x 2 H g , where F 2 F , H 2 H ). By means of nets, the denition should be as follows: A net ( f  )  2 D in ; G is  c -convergent to f 2 ; G if for every net ( x  )  2 E in G converging to x 2 G ,the net ( f  ( x  )) (  ) 2 D  E ( D  E has the pro duct direction) converges to f ( x )in T . It is well known that a top ology in ; G for whichtheevaluation e :; G  G ! T is continuous (; G  G has the natural pro duct structure) must be ner than the compact op en top ology  co , but  co itself very seldom makes e continuous. Therefore a convergence structure may be designed in ; G in order to obtain the continuity of the evaluation mapping e : ; G  G ! T as well as the prop erty of b eing the coarsest with this condition. This is the real motivation to intro duce the continuous convergence structure on a dual. The dual group ; G of a convergence group ( G ), endowed with the convergence structure  c ,is aconvergence group, denoted by ; c G and called the convergence dual 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 e : ; c G  G ! T ,  G is always continuous. Analogously, a convergence vector space E is BB-reexive as a space if the canonical emb edding  E : E ! L c L c E is a bicontinuous isomorphism. In the category of Hausdor top ological groups, BB-duality and Pontryagin duality are indep endent notions 12], but they coincide, for instance, in the family of metrizable top ological groups 11]. The compact op en top ology and the continuous convergence structure in the dual of a lo cally compact ab elian top ological group, have the same convergent lters. This fact characterizes the lo cally compact groups in the class of reexive top ological groups 19]. If E is a real top ological vector space, the dual group E ^ ,and the dual vector space E  (i.e. the set of all continuous linear forms endowed with the compact op en top ology) are related through the exp onential mapping f ! exp (2 if ), which in this case happ ens to b e a top ological isomorphism (see 2], (2.3)). Here the compact op en top ology plays some role it would not be a top ological isomorphism if the supp orting sets were endowed by the corresp onding weak top ologies. The duality theory for top ological vector spaces is usually restricted to lo cally convex vector spaces where the Hahn-Banach theorem works. In an arbitrary top ological group, the notion of convexity has no sense. Nevertheless, a similar notion, the so called quasi-convexity, was intro duced by Vilenkin in 24], where he also dened the lo cally quasi-convex groups. A subset A of a top ological group G is called quasi-convex if for every g 2 G n A , there is some  2 A o := f  2 ; G : Re ( z )  0  8 z 2 A g , such that Re ( g ) < 0. The quasi-convex hull of any subset H  G is dened as the intersection of all quasi-convex subsets of G containing H . An ab elian top ological group G is called lo cally quasi-convex if it has a neighb orho o d basis of the neutral element e G , given by quasi-convex sets. The dual G ^ of any top ological ab elian group G is lo cally quasi-convex. In fact, the sets K o , where K runs through the compact 3 subsets of G , constitute a neighb orho o d basis of e G ^ for the compact op en top ology. The additive group of a top ological vector space is lo cally quasi-convex if and only if the vector space itself is lo cally convex, 2]. Therefore it is natural to restrict the duality theory of top ological ab elian groups to the lo cally quasi-convex ones. Some of the well known results on lo cally convex spaces have analogic versions valid for lo cally quasi-convex groups. In particular a top ology on a group G is lo cally quasi-convex if and only if it is an S -top ology (uniform convergence top ology) for the family S of equicontinuous subsets of the dual G ^ (13] Prop osition 3.9). A duality theory for groups is extensively presented in 13]. Here we will only state what is needed for our aims. If G is a top ological group, the Bohr top ology on G is the weakest top ology that makes continuous all characters of ; G . We will denote it by ! ( G ; G ), and the p ointwise top ology on ; G will b e denoted by ! (; G G ). Very interesting results on the Bohr top ology of a lo cally compact ab elian group, from a top ological p oint of view, are obtained in 14]. The pap er is organized as follows: in section 2we present examples of complete metrizable lo cally quasi-convex groups which are not Pontryagin reexive. In doing so we are concerned with lifting of characters on a group G to homomorphisms from G into R . We use essentially a result of Nickolas. In section 3 we present the Grothendieck completeness theorem for the underlying group of a lo cally convex space and its dual group. In the last section we see that the most natural version of Grothendieck theorem for top ological groups do es not hold, even in the class of metrizable lo cally quasi-convex groups. The examples which prove it, are precisely the groups considered in section 2. We then study a weaker form of Grothendieck theorem valid for lo cally quasi-convex groups and prove that for the smaller class of nuclear groups, or of lo cally convex vector groups, the result can be improved. 2 A family of nonreexive complete metrizable lo cally quasi-convex groups The groups L p Z 0  1], for p > 1, have the ab ove conditions as proved by Aussenhofer in (1], p 50). We obtained this result indep endently,but she did it earlier, and her pro of includes also the description of the dual of such groups. For the sake of completeness we describ e here these groups. Let L p 0  1] or simply L p be the vector space of all classes of real measurable functions f such that k f k := ( R 1 0 j f ( t ) j p dt ) 1 =p < 1 .It is well known that the spaces L p , ( p> 1), endowed with the norm kk are Banach spaces. Now L p Z is the subset of L p of all the classes of integer valued functions, with the induced top ology. Evidently it is a complete metrizable lo cally quasi-convex top ological ab elian group, but it is not avector subspace. Nowwe summarize the steps, interesting for our aims, which lead to the pro of that L p Z is a 4 nonreexive group. Crucial to all of them is the following result of Nickolas: If an ab elian top ological group G is a k-space, then the path comp onent of the identityin G ^ is the union of all the one-parameter subgroups of G ^ . (  ) By a one-parameter subgroup of G it is commonly understo o d the image of R byacontinuous homomorphism from R into G . First we are concerned with lifting of characters to real valued characters. As we already mentioned, every continuous character dened in a top ological vector space can be lifted to a continuous linear form. The same assertion can be made for certain groups, as we exp ose in the next prop osition. Its pro of is essentially contained in the pro of of (*), given in 20]. Prop osition 2.1 Let G be a topological abelian group such that G is a k-space and its dual G ^ is pathwise connected. Then every continuous homomorphism ' : G ! T can be lifted to a continuous homomorphism ~ ' : G ! R such that p ~ ' = ' , where p : R ! T is the covering projection. Remark The assumption that G is a k-space is not a necessary condition. In 1] Corollary 8.12, an example of a lo cally convex vector space E , which is not a k-space is presented. Clearly,the lifting prop erty for E derives from the natural isomorphism b etween E  and E ^ . We can now state the following: Theorem 2.2 If G is a metrizable, reexive pathwise connected group, then: a) Every continuous character ' : G ^ ! T can be lifted to a real continuous character (i.e. there exists ~ ' : 2 CHom ( G ^  R ) such that p ~ ' = ' ) b) G is the union of its one-parameter subgroups c) G is divisible Pro of. a) If G is metrizable, G ^ is a k-space as shown in 11]. On the other hand ( G ^ ) ^ is top ologically isomorphic to G , therefore pathwise connected. By prop osition 2.1, every continuous character ' : G ^ ! T can be lifted to say ~ ' : G ^ ! R such that p ~ ' = ' . Furthermore the lifting is unique (see 23] pp.69, 2nd. paragraph), since any lifting to a continuous character ~ ~ ' must b e such that ~ ~ ' (  0 )=0 2 R , where  0 is the neutral element of G ^ . b) follows also from (*). c) In order to prove the last assertion, we express G as the union of its one-parameter subgroups, say G = f  ( R ) :  2 CHom ( R G ) g . For any x 2 G and any n 2 N , there exists  2 CHom ( R G )and r 2 R , such that  ( r ) = x .Now   r n  is suchthat n  r n  = x . 5 Remarks A top ological group which is the union of its one-parameter subgroups must be pathwise connected. Thus, the condition that G ^ b e pathwise connected cannot b e dropp ed in prop osition 2.1. It was known to Dixmier (see 16] pp.393) that for alo cally compact ab elian group G the condition that every character in G can b e lifted to a real character is equivalent to the fact that the dual G ^ is the union of its one-parameter subgroups. That this also holds for metrizable reexive groups can be deduced from the proofof(*)together with Theorem 2.2. Prop osition 2.3 The group G = L p Z 0  1] ( p> 1 )is not Pontryagin reexive. Pro of. The pro of follows easily from the fact that G is contractible, therefore pathwise connected. Since it is a metrizable group, if it were reexive, G would satisfy all the assumptions of Theorem 2.2, therefore it would b e divisible. But this is not the case obviously for the function f constant to one, there is no g 2 L p Z suchthat 2 g = f .The fact that L p Z is contractible can be seen in 1]. Nevertheless we sketch the pro of. Denote by  0 l ) the characteristic function of 0 l )in0  1]. The mapping F : L p Z  0  1] ! L p Z ( f t ) !  0  1 ; t )  f establishes a homotopybetween the identity mapping in L p Z and the constant to null mapping. It is therefore a contraction of L p Z . 3 The Grothendieck completeness theorem on the additive group of a lo cally convex vector space In this section we prove that the underlying group of a top ological vector space and its dual group satisfy an analogue to Grothendieck Theorem (GT). We rst give a few lemmas which will simplify our job. In the next prop ositions, E will denote a top ological vector space. We keep the standard notations E ^ , ; c E and ; S E for the character group of E ,endowed with the compact op en top ology, with the continuous convergence structure and with an S -top ology resp ectively. Also by E  , by L c E and by L S E we mean the set of continuous linear forms L E endowed with the compact op en top ology, with the continuous convergence structure and with an S -top ology resp ectively. Lemma 3.1 Let ( E  ) bealocal ly convex vector space and let S be a family of closedbounded convex and balanced sets covering E . i) Denote by  : Lin ( E R ) ! Hom ( E T ) the exponential mapping,  ( f )= exp (2 if ) , 8 f 2 Lin ( E R ) . A character ' belongs to Im (  ) if and only if ' j L is continuous for al l onedimensional vector subspace L  E . 6 ii) The fol lowing assertions are equivalent: (a) Every character with continuous restriction on al l S 2 S , is continuous. (b) Every linear form with continuous restriction on al l S 2 S , is continuous. Pro of. i) Supp ose ' =  ( f ) for some f 2 Lin ( E R ). If L  E is a one-dimensional vector subspace, f j L is continuous, therefore ' j L is continuous. Conversely, let ' 2 Hom ( E T ). Denote by a ] the subspace generated by a non null vector a 2 E . Since ' j  a ] is continuous, it can be considered as a continuous character dened on R , and consequently there is aunique real number t a suchthat ' ( ra )= exp (2 it a r ) for all r 2 R . It is easy to checkthat t a = t a and t a + b = t a + t b ,forany  2 R and any b 2 E . Therefore by sup erp osition of the one-dimensional linear forms f a ( ra ) = t a r we obtain a linear form f : E ! R . Clearly ' = exp (2 if ). ii) ( a ) ) ( b ) Let f : E ! R be a linear form continuous on all S 2 S . The corresp onding character exp (2 if )iscontinuous in all S 2 S , and bya)itiscontinuous. Therefore, by 2] (2.3), f is continuous. ( b ) ) ( a ) Let  : E ! T be a character with continuous restriction on each S 2 S . From this it is easily seen that the restriction of  to nite dimensional subspaces is continuous and, by i), there exists a linear form f : E ! R such that exp (2 if )=  . Now for S 2 S ,and  > 0, there is some balanced neighb orho o d of e such that j exp (2 if ( x )) ; 1 j <= 2, for all x 2 S \ U .Then j exp (2 itf ( x )) ; 1 j <= 2, for all j t j  1, and all x 2 S \ U . Consequently j f ( x ) j <  and the restriction of f to all elements of S is continuous. By b) f is continuous on E ,and so is  = exp (2 if ). Lemma 3.2 Let ( E  ) bealocal ly convex vector space and let S be a family of closedbounded convex and balanced sets covering E . The exponential mapping  : L S E ! ; S E is a topological isomorphism. Pro of. The continuityof  is straightforward, and holds without any conditions on the sets S 2 S . An argument similar to that of ( b ) ) ( a ) of the previous lemma proves the continuity of the inverse mapping. In fact only the prop erties that the sets S 2 S are balanced and cover E are used. Next we state that the continuous convergence restricted to equicontinuous subsets of ; G coincides with the pointwise convergence. The pro of is straightforward. Lemma 3.3 Let G bea topological group and let H bean equicontinuous subset of ; G . If (   ) is a net contained in H and  2 ; G , the fol lowing assertions are equivalent: 7 1. (   ) is  c -convergent to  2. (   ) is  co -convergent to  3. (   ) is ! (; G G ) -convergent to  . By the previous lemma equicontinuous subsets of ; c G are top ological. The family of closed equicontinuous subsets of ; G actually coincides with that of  c -compact subsets. If  G is continuous, then they also coincide with the family of  co -compact subsets. For complete metrizable groups, more can be said. The following statement is comparable to the uniform b oundedness principle. Since the latter is a signicant result in the theory of top ological vector spaces, one can reasonably exp ect that this sort of \equicontinuity principle" may have some imp ortance for ab elian top ological groups. The pro of of it can be seen in 13] (Theorem 1.5), where it is established in a more general setting. Lemma 3.4 If G is a complete metrizable topological abelian group, then every ! (; G G ) - compact subset of ; G is equicontinuous. The convergence dual of a top ological group is lo cally compact, and has prop erties similar to those of k-spaces. Lemma 3.5 Let G be a topological group. The fol lowing assertions hold: 1. ; c G is a local ly compact convergence group. 2. If a character  :; c G ! T is such that  j K is continuous for al l compact K  ; c G , then  is continuous. 3. ; c ; c G is topological and carries the compact open topology relative to the compact subsets of ; c G . Furthermore, it is complete. Pro of. It can be seen in 3] and 6], (3.2.2), (1.5.4) and (3.2.5). Lemma 3.6 If G is a Hausdor local ly quasi-convex group, then  G : G !  G ( G )  ; c ; c G is an embedding. Pro of. In order to prove that  G is op en and injective, take into account Lemma 3.5, 3 and follow the pro of of the same facts for  G , 2], (14.3). On the other hand  G is always continuous. Next we see that ! ( G ; G ), and ! (; G G ) are the natural analogues to the weak and to the weak* top ologies dened in a top ological vector space and in its dual. 8 Lemma 3.7 Let G be an abelian topological group. 1. The dual group of ( G ! ( G ; G )) is ; G . 2. If ; G separates points of G , then every continuous character on (; G ! (; G G )) is an evaluation at some point of G ,i.e. the dual group of (; G ! (; G G )) can be algebraical ly identied with G . Pro of. It can be seen in 13], Theorem 3.7. The identication of lemma 3.7(2) is even top ological for some classes of groups, as weprove now: Theorem 3.8 If G is a complete metrizable local ly quasi-convex group, then the dual group X =(; G ! (; G G )) ^ is topological ly isomorphic to G . Pro of. By lemma 3.7(2), X can be algebraically identied with G .If K  ; G is ! (; G G )- compact, then it is equicontinuous by lemma 3.4. This means that o K := f z 2 G : Re ( z )  0  8  2 K g is a 0-neighb orho o d in G . On the other hand o K can b e identied with K o , and so we have that every 0-neighborhood in X is a 0-neighb orho o d in G . Conversely, if V is a quasi-convex 0-neighborhood in G , V o is ! (; G G )-compact (2] 1.5), therefore V = o ( V o ) is a 0-neighb orho o d in X . For a lo cally convex vector space E ,we bring together the two view p oints, as a group and as a space, in the next two theorems. Theorem 3.9 Let ( E  ) be a local ly convex space and let S be a family of closed bounded convex and balanced sets covering E .The fol lowing statements are equivalent: (a) L E is complete under the S -topology. (b) Every linear form f on E which is  -continuous on each S 2 S ,is continuous on ( E  ) . (c) ( L E  S ) is BB-reexive, i.e. E is bicontinuously isomorphic to L c L c ( L E  S ) (d) The group (; E  S ) is complete. (e) Every character on E , which is continuous on each S 2 S ,is continuous on ( E  ) . (f ) (; E  S ) is BB-reexive, i.e. it is bicontinuously isomorphic to ; c ; c (; E  S ) . Pro of. The equivalence between a) and b) is prop erly Grothendieck Theorem. The pro of can b e seen in any classical treatise, for example 21]. In 8] itisproved that a lo cally convex vector space is complete if and only if it is BB-reexive as a vector space, thus a) , c). 9