Groups of quasi-invariance and the Pontryagin duality

Groups of quasi-in v ariance and the P o n try agin dualit y ✩ S.S. Gabriy ely an Dep artment of Mathematics, Ben-Gurion University of the Ne gev, Be er-Sheva, P.O. 653, Isr ael Abstract A P olish group G is called a group of quasi-inv ariance o r a QI-group, if there e xist a lo cally compact group X and a probabilit y measure µ on X suc h that 1) there exists a contin uous monomorphism of G to X , and 2) for eac h g ∈ X either g ∈ G and the shift µ g is equiv alent to µ o r g 6∈ G and µ g is orthogo nal to µ . It is pro v ed that G is a σ -compact subset of X . W e sho w that there exists a quotien t group T H 2 of ℓ 2 mo dulo a discrete subgroup which is a Polish monothetic non lo cally quasi-con v ex (and hence nonreflexiv e) pathw ise connected QI-gro up, a nd such that the bidual of T H 2 is not a QI- group. It is prov ed also that the bidual group of a QI-gro up may b e not a saturated subgroup of X . Key wor ds: Group of quasi-in v ariance, Pon t ry agin duality theorem, dual group, P olish group, quasi-con v ex g roup, T -sequence 2008 MSC : 22A10, 22A35, 43A05, 43A40 1. In tro duction Let X b e a P olish group a nd B the family o f its Bo rel sets. Let E ∈ B . The image and the in v erse image of E are denoted b y g · E and g − 1 E resp ectiv ely . Let µ and ν b e probabilit y measures o n X . W e write µ ≪ ν ( µ ∼ ν, µ ⊥ ν ) if µ is absolutely con tin uous relativ e to ν (resp ectiv ely: equiv alen t, m utually singular). F or g ∈ X w e denote b y µ g the measure determined by the relation µ g ( E ) := µ ( g − 1 E ), ∀ E . The set of all g suc h ✩ The author was partially supp orted by Isra el Ministry of Immigra nt Absorption Email addr ess: sa ak@mat h.bgu .ac.il (S.S. Gabriyely an) Pr eprint submitt e d t o Elsevier Octob er 25, 2018 that µ g ∼ µ is denoted b y E ( µ ). The Mac k ey-W eil The orem asserts that X = E ( µ ) for some µ iff X is lo cally compact. Some algebraic and t op ological prop erties of E ( µ ) are considered in [9] and [1 0]. In particular, it w as prov ed that E ( µ ) alwa ys admits a P olish group top o logy and, as a subgroup of X , is a G δσ δ -set. The P olish gro up top ology is defined b y the strong op erator top ology in the followin g wa y . If g , h ∈ E ( µ ), and { µ n } is a coun table dense subset in L 1 ( µ ) (with µ 1 = µ ) , then t he follo wing metric on E ( µ ) d ( h, g ) = ∞ X n =1 1 2 n k µ n g − µ n h k 1 + k µ n g − µ n h k + k µ n g − 1 − µ n h − 1 k 1 + k µ n g − 1 − µ n h − 1 k ! . defines the Polish gr o up top ology whic h is finer than the t o p ology induced from X . Note that althoug h the metric d dep ends on the chosen sequenc e { µ n } , the P olish group top o logy is unique a nd do es not dep end on d . F or a to p ological gr o up G , the group G ∧ of con tin uous homo mo r phisms (c haracters) in to the torus T = { z ∈ C : | z | = 1 } endo w ed with the compact- op en top olo gy is called the char a c ter gr oup of G and G is named Pontryagin r eflexive or r eflexive if the canonical homomorphism α G : G → G ∧∧ , g 7→ ( χ 7→ ( χ, g ) ) is a top ological isomorphism. A subset A of G is called quasi- c onv e x if for ev ery g ∈ G \ A , there is some χ ∈ A ⊲ := { χ ∈ G ∧ : R e ( χ, h ) > 0 , ∀ h ∈ A } , suc h tha t Re( χ, g ) < 0 , [23 ]. An Ab elian top o logical group G is called lo c al ly quasi-c onvex if it has a neigh b orho o d basis of the neutral elemen t e G , giv en b y quasi-conv ex sets. The dual G ∧ of an y top ological Ab elian group G is lo cally quasi-conv ex [2 3]. In fact, the sets K ⊲ , where K runs through the compact subsets of G , constitute a neigh b orho o d basis of e G ∧ for the compact op en top olog y . F ollowing E.G.Zelen yuk and I.V.Protaso v [24], w e say that a sequence u = { u n } in a g roup G is a T - se quenc e if there is a Hausdorff group top ology on G for whic h u n con v erges to zero. The group G equipp ed with the finest group top ology with this prop erty is denoted by ( G, u ). Set Z ∞ b = { n = ( n 1 , . . . , n k , n k +1 , . . . ) | n i ∈ Z and | n | b := sup {| n i |} < ∞} , Z ∞ 0 = { n = ( n 1 , . . . , n k , 0 , . . . ) | n j ∈ Z } . W e will consider the spaces l p and c 0 . F or our conv enience we set l 0 := c 0 . Eviden tly , Z ∞ 0 is a closed discrete subgroup of l p for an y 0 ≤ p < ∞ . 2 The follow ing groups pla y a crucial role in our consideration T H p := ( ω = ( z n ) ∈ T ∞ | ∞ X n =1 | 1 − z n | p < ∞ ) , 0 < p < ∞ , T H 0 := { ω = ( z n ) ∈ T ∞ | z n → 1 } . It is easy to prov e that T H p are P olish groups with p oin t wise m ultiplication and the t op ology generated by the metric d p ( ω 1 , ω 2 ) = ∞ X n =1 | z 1 n − z 2 n | p ! min(1 , 1 p ) , if 0 < p < ∞ , and d 0 ( ω 1 , ω 2 ) = sup ( | z 1 n − z 2 n | , n = 1 , 2 , . . . ) , if p = 0 . W e also need more complicated groups, whic h a re defined in [2]. Supp ose that a n ≥ 2 are in tegers suc h that P ∞ n =1 1 a n < ∞ and set γ (1) = 1 , γ ( n + 1) = Π n k =1 a k ( n ≥ 1). If 1 ≤ p < ∞ and z ∈ T , w e set k z k p = ∞ X n =1 | 1 − z γ ( n ) | p ! 1 /p and . Put G p = { z ∈ T : k z k p < ∞} and Q = { z ∈ T ; z γ ( n ) = 1 fo r some n } . Then ( G p , k . k p ) is a P olish group and Q is its dense subgroup [2]. It is clear that all G p are totally disconnected. F or p = 0 a nd z ∈ T w e set G 0 = { z ∈ T : z γ ( n ) → 1 } and k z k 0 = sup {| 1 − z γ ( k ) | , k = 1 , 2 . . . } . Then ( G 0 , k . k 0 ) is a P olish group [11]. This article w as inspired b y the follo wing old and imp ortan t problem in abstract harmo nic analysis and top ological algebra: find the “righ t” gener- alization of the class of lo cally compact groups. The comm utativ e harmonic analysis giv es us one of the b est indicator for the “r ig h t” generalization - the P on tryagin duality theorem. The P ontry agin theorem is known to b e true for sev eral classes of non lo cally compact groups: the additiv e group of a Ba nac h space, pro ducts of lo cally compact groups, complete metrizable n uclear groups [4], [15] [22]. Thes e examples suggest searc hing for p ossible generalizations not o nly in the direction of duality theory . The existence of the Haa r measure plays a crucial role in harmonic analysis. As it w a s men tioned ab ov e, existence of a left (quasi)inv a rian t measure is equiv alen t to 3 the lo cal compactness of a group. Therefore w e can use some similar notion only . Groups of the for m E ( µ ) are natural candidates. On the ot her hand, if a probabilit y measure µ on T is ergo dic under E ( µ ), then for eac h g ∈ T either µ g ∼ µ or µ g ⊥ µ [14]. Therefore, taking in to consideration ergo dic decomp osition, w e prop o se the follo wing generalization. Definition 1. A Pol i s h gr oup G is c a l le d a g r oup of quasi - i nvarianc e or a QI-gr oup, if ther e exist a lo c al c omp act gr oup X and a pr ob ability me as ur e µ on X such that G is c ontinuously emb e dde d in X , E ( µ ) = G and µ g ⊥ µ for al l g 6∈ E ( µ ) . W e will say that G is represen ted in X b y µ and denote it b y E µ . It is clear that, if G is Ab elian, t hen w e can a ssume that X is compact. In the general case w e can define Q I-groups in the following wa y (taking in to accoun t Theorems 5.14 and 8.7 in [12]): a top olo gical g roup G is called a QI-group if there exists a compact normal subgroup Y suc h t ha t G/ Y is a P olish QI-g r o up. In the a rticle w e are restricted to the Ab elian P olish case only . By the definition, it is clear that a QI-group has enough con tin uous c haracters. Eviden tly , an y lo cally compact Polish g roup is a QI-group. Let H b e a separable real Hilb ert space. Then H is a QI-g roup, since it is E ( µ ) for a Gaussian measure on R ∞ [21] and w e can con tin uously em b ed R ∞ in to ( T 2 ) ∞ = T ∞ in the usual w a y . Moreo v er, an y l p , 0 < p ≤ 2 , is a QI-gro up [7]. In [2], the autho rs prov ed that G 1 and G 2 are QI-g r o ups (see also [17]). W e will give a simple straigh tforward pro of that T H 1 and T H 2 are QI-groups to o (cf. [20 ] for T H 2 , see also [13]). If a probabilit y measure µ on T is ergo dic under E ( µ ), then E ( µ ) = E µ is a QI-group [14]. Hence, in the category of P olish groups, the set G QI of all gr o ups o f quasi-in v ar ia nce is wider t ha n the class of lo cally compact groups. Cho osing of suc h groups is motiv ated not only by the ab ov e-mentioned. Let µ o n T b e ergo dic under E ( µ ) = E µ . J.Aar onson and M.Nadk arni [2] sho w ed that E µ is the eigen v alue group of some non-singular transformation and illustrated a basic interaction b etw een eigen v alue groups and L 2 sp ec- tra. Moreo v er, they computed the Hausdorff dimension of some E µ , whic h is imp ortant in connection with the dissipative prop erties o f a non-singular transformation [1]. A deep pro p ert y of the eigen v a lues of the action of E µ giv es us a k ey prop ert y f o r solving subtle problems ab out sp ectra o f mea- sures a r ound the Wiener-Pitt phenomenon. Thes e and other applications to harmonic analysis are giv en in [1 4] and [16 ]. Below w e prov e t ha t a Q I-group is ev en a σ -compact subgroup o f some lo cally compact group. Hence, on the 4 one hand, QI-groups play a very imp ortant role in non- singular dynamics and harmonic analysis and, on the other hand, they are not “v ery big” (as, for example, the unitary group U ( H ) of the separable Hilb ert space or the infinite symmetric g roups S ∞ ). These arguments allo w us to consider the notion “to b e a Q I- group” as a p ossible generalization of the notion “to b e a lo cally compact group” and explain our in terest in suc h gr o ups. The main goal of the article is to consider some g eneral problems of the Pon try agin duality theory for groups of quasi-in v a r ia nce. The following question is natural: Question 1. Ar e al l gr oups of quasi-invarianc e Pontryagin r eflexive ? It is clear that lo cally compact Polish groups and l p , 1 ≤ p ≤ 2 , are reflexiv e. W e pro v e that T H 1 is reflexiv e to o. On the other hand, an y group l p , 0 < p < 1 , is no t ev en lo cally quasi-conv ex (8.2 7 ,[3]) a nd, hence, no t reflexiv e. Since the bidual group of a P olish group is alw a ys lo cally quasi-conv ex and P olish [6], w e can ask the follo wing. Question 2. Is the bidual G ∧∧ of a QI - g r oup G a QI-gr oup ? The answ er on question 2 is also negative . W e pro v e that the bidual group of T H 2 is not a QI-group. W e do not know the answ er on the next question. Question 3. L et G b e a lo c al ly quasi-c o nvex QI -g r oup. Is G r eflexive ? In [14], the authors pro v ed tha t eac h Q I-group is saturated. Since the bidual of a QI-group G is no t alwa ys a QI-group, w e can ask whether the bidual group is saturated. It is prov ed that G ∧∧ 2 = G 0 (and, hence, G 2 is not reflexiv e). Since G 0 is not a saturated subgroup of T [14], this sho ws that the bidual group of a QI-group ma y b e not saturated. On the other hand, the groups T H p are inte resting from the general p oint of view of the P on try agin duality theory . W e prov e the follo wing. • If 1 < p < ∞ , then T H p is a monothetic non lo cally quasi-conv ex Polish group and, hence, non reflexiv e. V. Pe stov [18] ask ed whether ev ery ˇ Cec h-complete gr oup G with sufficien tly man y characters is a reflexiv e group. Hence T H p giv es another negative answ er on this question (in 11.15 [3] an even stronger coun terexample is give n). • If p = 0 or p = 1, then T H p is a mono thetic r eflexive P olish non lo cally compact group. 5 • T H p is top olog ically isomorphic to l p / Z ∞ 0 . Since l p is P on try agin reflexiv e and Z ∞ 0 is its closed discrete (and hence) lo cally compact subgroup, w e see that their quotien t T H p is no t lo cally quasi-con v ex. Th us the answ er on question 14 [5] is negativ e. •  T H p  ∧∧ is top ologically isomorphic t o c 0 / Z ∞ 0 and reflexiv e. Hence nei- ther “ to b e dual” nor “to b e Pon tryagin r eflexiv e” is not a three space prop ert y . • If 1 < p < ∞ , then  T H p  ∧ =  T H 0  ∧ = Z ∞ 0 and is r eflexive . Thus there exists a con tinual chain (under inclusion) of Polish mono thetic non lo- cally quasi-conv ex pathw ise connected groups with t he same c ountable r eflexive dual. In fact,  T H 0  ∧ is a Graev free top ological ab elian group o v er the con v ergen t sequence . Analogous prop erties hold for t he family of groups G p . Let us not e only that, actually , w e can consider the group G p as a dually closed and dually em b edded (totally disconnected) subgroup o f T H p . 2. Main Results As it w as men tioned ab ov e, E ( µ ) is a G δσ δ -subset of X . F or a QI-g roup w e can pro v e the following. Prop osition 1. If a QI -gr oup G is r epr esen te d in X , then it is σ - c om p act in X . Pro of. Since the P olish group top ology τ on G is unique, we can consider G as ( E ( µ ) , d ) for some probability measure µ on X . Since τ is finer than the top olog y on X , w e can c ho ose ε 0 > 0 suc h that the ε 0 -neigh b orho o d U ε 0 of the unit e is con tained in a compact neigh b orho o d of e in X . Th us Cl X U ε is compact in X for any ε < ε 0 . If { h n } is a dense countable subset o f ( E ( µ ) , d ), then E ( µ ) = ∪ n h n U ε for ev ery ε > 0. Therefore, if w e will prov e that Cl X U ε ⊂ E ( µ ) for an enough small ε < ε 0 , then E ( µ ) = ∪ n h n Cl X U ε is a σ -compact subset of X . Set ε = min( ε 0 , 0 . 1). W e will pro v e that Cl X U ε ⊂ E ( µ ). Let g n → t in X , g n ∈ U ε . Assume the con v erse and t 6∈ E ( µ ), i.e. µ t ⊥ µ . Cho ose a compact K suc h that µ ( K ) > 0 , 9 and µ t ( K ) = 0. Cho o se a neighborho o d V δ ( K ) o f K suc h that µ t ( V δ ( K )) < 0 , 1. Then there exists an in teger N suc h that g − 1 n · K ⊂ t − 1 V δ ( K ) , ∀ n > N , and µ g n ( K ) = µ ( g − 1 n K ) < 0 , 1 , ∀ n > N . (1) 6 On the other hand, since d ( g , e ) < ε , then, b y the definition of d , k µ g − µ k < 0 , 2. But for an y g ∈ U ε w e ha v e 0 , 2 > k µ g − µ k ≥ k µ g | K − µ | K k ≥ | µ ( g − 1 K ) − µ ( K ) | , and µ ( g − 1 K ) = µ ( K ) + ( µ ( g − 1 K ) − µ ( K )) ≥ 0 , 9 − 0 , 2 = 0 , 7 . In particular, µ g n ( K ) > 0 , 7. This inequalit y contradicts to ( 1).  Prop osition 2. T H 1 and T H 2 ar e QI -gr oups. Pro of. W e can capture the idea of how to construct of examples as follo ws. Let µ b e absolutely con tin uous relativ e to the Haa r measure m R and assume that its densit y f ( x ) is smo oth. Then P ( ϕ ) := Z p f ( x ) f ( x + ϕ ) dx = Z f ( x ) s 1 + 1 f ( x ) ( f ( x + ϕ ) − f ( x )) dx = 1 + ϕ 2 Z f ′ ( x ) dx − ϕ 2 8 Z ( f ′ ) 2 − 2 f f ′′ f dx + ϕ 3 48 Z f 2 · f ′′′ − 6 f f ′ f ′′ + 3( f ′ ) 3 f 2 dx + O ( ϕ 4 ) . Hence w e can expect: if f is linear, then P ( ϕ ) ∼ 1 + cϕ ; and if R f ′ dx = 0, then P ( ϕ ) ∼ 1 − cϕ 2 . W e iden tify T with [ − 1 2 ; 1 2 ) , t 7→ e 2 iπ t . Since α/ 2 < sin α < α, α ∈ (0 , π / 2), then for ϕ ∈ [ − 1 2 ; 1 2 ) , z = e 2 π iϕ , 0 < p < ∞ , w e ha v e 2 p π p | ϕ | p ≥ | 1 − z | p = | 1 − e 2 π iϕ | p = 2 p | sin π ϕ | p ≥ π p | ϕ | p , p > 0 . (2) 1) L et us pr ove that T H 1 is a QI -gr oups. Let f ( x ) = x + 1. F or ϕ ∈ [0; 1 2 ) w e get f ( x − ϕ ) = x + 1 − ϕ, if x ∈ [ − 1 2 + ϕ ; 1 2 ) , and f ( x − ϕ ) = x + 3 2 − ϕ, if x ∈ [ − 1 2 ; − 1 2 + ϕ ] . Then the routine computations giv e us the following P ( ϕ ) ∼ 1 − 8 + 5 √ 2 6 + 4 √ 2 ϕ + O ( ϕ 2 ) 7 and max { P ( ϕ ) } = P (0) = 1 only a t 0. Hence P ( ϕ ) → 1 iff ϕ → 0. Consider the probability measures µ n = f ( x ) m T on T . Set µ = Q n µ n . Let ω = ( z n ) = (e 2 iπ ϕ n ). Then, b y the Kakutani Theorem [9], µ ω 6⊥ µ iff ω ∈ E ( µ ) iff Y n P n ( ϕ n ) < ∞ ⇔ X n ln P n ( ϕ n ) < ∞ ⇔ X n | ϕ n | < ∞ . Since ϕ n → 0 , then, b y (2), | ϕ n | ∼ | 1 − z n | · 2 π . Hence ω ∈ T H 1 . 2) L et us pr ove that T H 2 is a QI -gr oups. Let f c ( x ) = 1 a e − c | x | , where a = 2 c (1 − e − c/ 2 ). F or ϕ ∈ [0; 1 2 ) w e get f c ( x + ϕ ) = 1 a e − c | x + ϕ | , if x ∈ [ − 1 2 ; 1 2 − ϕ ) , and f c ( x + ϕ ) = 1 a e − c | x + ϕ − 1 | , if x ∈ [ 1 2 − ϕ ; 1 2 ) . Then the simple computations giv e us P c ( ϕ ) = Z 1 2 − 1 2 p f c ( x ) f c ( x + ϕ ) dx = 1 2 sh − 1 c 4  2sh c 4 (1 − 2 ϕ ) + cϕ c h c 4 (1 − 2 ϕ )  . It is easy to prov e that 1 − 1 8 ( cϕ ) 2 ≤ P c ( ϕ ) ≤ 1 − 1 32 ( cϕ ) 2 , ∀ c ∈ [0; 1] , ϕ ∈ [ − 1 2 ; 1 2 ) . (3) Consider the probability measures µ n = f c n ( x ) m T on T . Set µ = Q n µ n . Let ω = ( z n ) = (e 2 iπ ϕ n ). Then, by the Kakutani Theorem [9] and (3) , µ ω 6⊥ µ iff ω ∈ E ( µ ) iff Y n P c n ( ϕ n ) < ∞ ⇔ X n ln P c n ( ϕ n ) < ∞ ⇔ X n ( c n ϕ n ) 2 < ∞ . In particular, if c n = 1, then ϕ n → 0 . Therefore, b y (2), ϕ 2 n ∼ | 1 − z n | 2 · 4 π 2 . Hence ω ∈ T H 2 .  W e do not kno w an y general characterization of QI-v ector spaces . In particular, w e do not know the answ er on the following question (taking in to accoun t Theorem 1 [13]) Question 4. Is ther e p > 2 such that l p is a QI - g r oup ? 8 The next pro p osition allows to prov e the reflexivit y of the dual group of a group G . Prop osition 3. L et G b e a Polish g r oup. Set H = Cl( α G ( G )) . 1. If H = G ∧∧ , then G ∧ (and G ∧∧ ) is r eflexive. 2. If G is lo c al ly quasi-c o n vex and H = G ∧∧ , then G is r eflexive. Pro of. 1) Since G is P olish and G ∧ is a k -space [6 ], then α G and α G ∧ are con tin uous (corollary 5.1 2 [3]). Since H = G ∧∧ , α G ∧ is a contin uous isomorphism. Let α ∗ G b e the dual homomorphism of α G . Then it is the con v erse to α G ∧ , since ( α ∗ G ◦ α G ∧ ( χ ) , x ) = ( α G ∧ ( χ ) , α G ( x )) = ( χ, α G ( x )) = ( χ, x ) , ∀ χ ∈ G ∧ , ∀ x ∈ G. Th us α G ∧ is a top olog ical isomorphism and G ∧ is reflexiv e. 2) Let G b e lo cally quasi-conv ex. Then α G ( G ) is an em b edding with the closed image (Prop o sition 6.12 [3]). Th us, if H = G ∧∧ , t hen G = G ∧∧ is reflexiv e [6].  Prop osition 4. L et 0 ≤ p < ∞ . Then T H p is a monothetic Polish gr oup which is top olo gic al ly isomorph ic to l p / Z ∞ 0 . Pro of. 1 ) Let π p : l p → l p / Z ∞ 0 b e the canonical map 0 ≤ p < ∞ . Denote b y x the class of equiv alence of ( x n ), i.e. x = ( x n ) + Z ∞ 0 . T hen π p ( x n ) = ( x n (mo d1)). Indeed, ( x n ) + Z ∞ 0 = ( y n ) + Z ∞ 0 iff there exists an in teger N suc h that y n = x n + m n , m n ∈ Z , n = 1 , . . . , N , a nd y n = x n for n > N . Since x n and y n tend to zero, this is equiv alent to y n = x n (mo d1). Set s n = ( y n − x n )(mo d1) ∈ [ − 1 2 , 1 2 ). Then the metric on l p / Z ∞ 0 is defined as d ∗ ( x , y ) = inf { d (( x ′ n ) , ( y ′ n )) , ( x ′ n ) ∈ ( x n ) + Z ∞ 0 , ( y ′ n ) ∈ ( y n ) + Z ∞ 0 } , and so d ∗ ( x , y ) =  X | s n | p  min(1 , 1 p ) . (4) Let r : l p / Z ∞ 0 → T H p , r ( x ) =  e 2 π i ( x n (mod 1))  . Eviden tly that r is injectiv e. If ( z n ) = ( e 2 iπ ϕ n ) ∈ l p , ϕ n ∈ [ − 1 2 ; 1 2 ), then, b y (2), w e hav e π p ∞ X n =1 | ϕ n | p ≤ ∞ X n =1 | 1 − z n | p ≤ 2 p π p ∞ X n =1 | ϕ n | p . (5) Then d p ( p ( x ) , p ( y )) =  X | e 2 π is n − 1 | p  min(1 , 1 p ) =  X 2 p | sin π s n | p  min(1 , 1 p ) . (6) 9 Equations (4)-(6) show that π d ∗ ≤ d ≤ 2 π d ∗ and r is surjectiv e. Hence r is a top ological isomorphism. Analogously , we can consider the case p = 0. 2) S. Rolewicz [19] prov ed that T H 0 is monothetic. The case 0 < p < ∞ is considered analogically . W e follo w S. Rolewicz [19] and 1.5.4 [8]. Let 0 < a 1 < 1 / 2 b e an ir rational num ber. F or every n > 1 w e c ho o se an irrational n um b er a n suc h that a) a 1 > a 2 > · · · > a n > 0 are ra tionally indep enden t. b) a n < 1 2 n k n , where k n is the smallest nat ur a l n um b er suc h that for ev ery ( n − 1)- tuple ( y 1 , . . . , y n − 1 ) o f r eals there exist integers m 1 , . . . , m n − 1 and a natural n um b er k with k ≤ k n and | k a s − y s − m s | < 1 2 n for all s = 1 , . . . n − 1 (the existence of suc h k follows fro m t he Kronec k er Theorem). No w we set ω 0 = ( z 0 n ), where z 0 n = e 2 π ia n . Eviden tly ω 0 ∈ T H p for ev ery 0 ≤ p < ∞ . Let us prov e that the group h ω 0 i g enerated b y ω 0 is dense in T H p . Since the case p = 0 w as prov ed b y S. Ro lewicz [19], we assume that p > 0 . Let ε > 0 and ω = ( z n ) ∈ T H p , where z n = e 2 π iy n , y n ∈ [ − 1 2 ; 1 2 ). Set q = max( p, 1) and c ho ose n suc h that ∞ X s = n | 1 − z s | p + ( n − 1)2 p π p 2 pn + 1 2 pn 4 p π p 2 p − 1 < ε q . (7) By the definition of ω 0 , w e can c ho ose k ≤ k n and in tegers m 1 , . . . , m n − 1 suc h that | k a s − y s − m s | < 1 2 n for all s = 1 , . . . , n − 1 . It is remained to pro v e that d p ( ω , k ω 0 ) < ε . F or s = 1 , . . . , n − 1, b y (2), we ha v e | z s − ( z 0 s ) k | p = | 1 − exp { 2 π i ( k a s − y s ) }| p < 2 p π p | k a s − y s − m s | p < 2 p π p 2 pn . (8) F or s ≥ n w e hav e ( z 0 s ) k = e 2 π ik a s . Since k ≤ k n , b y (2), w e obtain | z s − ( z 0 s ) k | p ≤ | 1 − z s | p + | 1 − ( z 0 s ) k | p < | 1 − z s | p + 2 p π p 2 ps . (9) 10 Then, b y (7)-(9), w e hav e d q p ( ω , k ω 0 ) ≤ n − 1 X s =1 | z s − ( z 0 s ) k | p + ∞ X s = n  | 1 − z s | p + 2 p π p 2 ps  < ( n − 1)2 p π p 2 pn + ∞ X s = n | 1 − z n | p + 1 2 pn 4 p π p 2 p − 1 < ε d . Th us d p ( ω , k ω 0 ) < ε and h ω 0 i is dense.  By inequalit y (5), we will consider t he groups T H p , p = 0 or 1 ≤ p , under the following metrics: if ω j = ( z j n ) = (e 2 π iϕ j n ), where ϕ j n ∈ [ − 1 2 ; 1 2 ) , j = 1 , 2, then ρ p ( ω 1 , ω 2 ) = ∞ X n =1 | ϕ 1 n − ϕ 2 n | p ! 1 /p , if 1 ≤ p, and ρ 0 ( ω 1 , ω 2 ) = sup  | ϕ 1 n − ϕ 2 n | , n = 1 , 2 , . . .  , if p = 0 . In the sequel w e need some notat ions. F or p ≥ 1, nonnegative in tegers k and m and χ = ( n 1 , . . . , n k , 0 , . . . ) ∈ Z ∞ 0 , we set - | χ | p := p p | n 1 | p + · · · + | n k | p ; - l ( χ ) is t he n um b er of nonzero co o rdinates of χ ; - A ( k , m ) = { χ = (0 , . . . , 0 , n m +1 , . . . , n s , 0 , . . . ) ∈ Z ∞ 0 : | χ | 1 ≤ k + 1 } ; - the in tegral part of a real n um b er x is denoted by [ x ]. W e need the follow ing lemma. Lemma 1. F or any p ≥ 1 and 0 < ε < 1 / 4 we set A ε,q = { χ ∈ Z ∞ 0 : 4 ε | χ | q ≤ 1 } . Put a =  1 4 ε  − 1 an d b =  1 4 ε  q  . The n for every p > 1 we have A ( a, 0) ⊂ A ε,p ⊂ A ( b, 0) . Pro of. Let χ = ( n k ) ∈ Z ∞ 0 . Since q ≥ 1 and n k ∈ Z , w e ha v e | χ | 1 = X | n k | ≤ X | n k | q = | χ | q q ≤  X | n k |  q = | χ | q 1 11 and the a ssertion follow s.  Let us consider the sequenc e e = { e n } ∈ Z ∞ 0 , where e 1 = (1 , 0 , 0 , . . . ) , e 2 = (0 , 1 , 0 , . . . ) , . . . . Denote b y ( Z ∞ 0 , e ) t he group Z ∞ 0 equipped with the finest Hausdorff group top ology for whic h e n con v erges to zero (in fact, it is a Graev free top ological ab elian group ov er the conv ergent sequence e ∪ { 0 } ). Theorem 1. 1. T H p is not lo c al ly quasi-c onvex and, so, not r e flexive fo r any 1 < p < ∞ . 2. T H 1 is r eflexive and, henc e, lo c al ly quasi-c onvex. 3. T H 0 is r eflexive and, henc e, lo c al ly quasi-c onvex. I t i s not a QI -gr o up. 4.  T H p  ∧ is top olo gic al ly isomorphic to  T H 0  ∧ and, henc e, r eflexive for any 1 < p < ∞ . 5.  T H 0  ∧ = ( Z ∞ 0 , e ) . 6.  T H 1  ∧ is algeb r aic al ly isomorp hic to Z ∞ b . Pro of. 1. Ev iden tly , T n , n ≥ 1 , a r e close d subgroups of T H p . Let χ ∈  T H p  ∧ , 1 ≤ p < ∞ . Then χ | T n is a c haracter of T n . Hence χ | T n = ( m 1 , . . . , m n ) , m n ∈ Z . a) L et us pr ove that  T H p  ∧ is algebr aic al ly isomorphic to Z ∞ 0 for every 1 < p < ∞ . It is clear that Z ∞ 0 ⊂  T H p  ∧ . F or t he con v erse inclusion it is remained to pro v e that only finite num b er of integers m n are nonzero. Assume the con v erse and m s l 6 = 0 , l = 1 , 2 , . . . . W e can assume that m s l > 0. Set a 1 = 1 2 π , a 2 = − 1 2 π · 2 ln 2 , a 3 = − 1 2 π · 3 ln 3 , . . . , a k 1 = − 1 2 π · k 1 ln k 1 , where k 1 is the first n um b er suc h tha t P k 1 k =1 a k < − 1 2 π , a k 1 +1 = 1 2 π · ( k 1 + 1) ln( k 1 + 1) , . . . , a k 2 = 1 2 π · k 2 ln k 2 , where k 2 is the first n um b er suc h t ha t P k 2 k =1 a k > 1 2 π , a nd etc. Put z s l = exp(2 π i a l m s l ) and z n = 1 for the remainder n . Ob viously , ω = ( z n ) ∈ T H p . Since ∪ n T n is dense in T H p and χ is con tin uous, there exists ( χ, ω ) = lim l exp(2 π i P k l n =1 a n ). But Im exp( i P k l n =1 a n ) > sin 1 , for eve n l , a nd < − sin 1, for o dd l . It is a contradiction. b) L et us pr ove that  T H 1  ∧ is algeb r aic al ly isomorp hic to Z ∞ b . 12 F or the inclusion  T H 1  ∧ ⊂ Z ∞ b w e need to prov e that { m k } is b ounded. Assuming the con v erse, w e can c ho o se a subsequen ce k l suc h that | m k l | > l 2 , l = 1 , 2 , . . . . Set ω = ( z n ), where z n = exp(2 π i/ 3 m k l ), if n = k l , and z n = 1 otherwise. It is clear that ω ∈ T H 1 and ( χ, ω ) do es not exist. On the other hand, if | n | ∞ < ∞ , then χ = n is a con tin uous c haracter of T H 1 . Th us Z ∞ b ⊂  T H 1  ∧ . 2. Set U ε is the ε -neigh b orho o d of the unit in T H p , 1 ≤ p < ∞ . a) L et 1 < p < ∞ . We wil l pr ove that A ε,q = U ⊲ ε for any 0 < ε < 1 . By Lemma 1, this sho ws that the sets A ( k , 0 ) form a decreasing family of precompact sets suc h tha t an y compact set in the hemicompact group  T H p  ∧ is con tained in some A ( k , 0). Let χ ∈ U ⊲ ε . If χ = ( n 1 , . . . , n k , 0 , . . . ) and ω = ( z j ) = (e 2 iπ ϕ j ), where ϕ j ∈ [ − 1 2 ; 1 2 ), then ( χ, ω ) = z n 1 1 . . . z n k k = exp(2 iπ ( n 1 ϕ 1 + · · · + n k ϕ k )) . Since U ε is path wise connected, then Re( χ, ω ) ≥ 0 , ∀ ω ∈ U ε , iff − 1 4 ≤ n 1 ϕ 1 + · · · + n k ϕ k ≤ 1 4 , ∀ ω ∈ U ε , (10) and, in particular, for all ω k = ( z 1 , . . . , z k ) ∈ T k suc h that k X j =1 | ϕ j | p < ε p . (11) By H¨ older’s inequalit y , w e hav e | n 1 ϕ 1 + · · · + n k ϕ k | ≤ | χ | q · p p | ϕ 1 | p + · · · + | ϕ k | p ≤ ε | χ | q . It is clear that the suprem um of the function f = n 1 ϕ 1 + · · · + n k ϕ k under the condition (11) is ach iev ed when ϕ i = ε · sign ( n i )  | n i | | χ | q  q /p and equals to ε | χ | q . By (1 0), w e ha v e U ⊲ ε =  χ ∈  T H p  ∧ : sup f ( ω k ) ≤ 1 4 , ω ∈ U ε  = { χ ∈ Z ∞ 0 : 4 ε | χ | q ≤ 1 } . b) L et p = 1 and 0 < ε < 1 2 . Set Z ε =  n ∈ Z ∞ b : | n | b ≤ 1 ε  . We wil l pr ove that Z 4 ε ⊂ U ⊲ ε ⊂ Z 2 ε . 13 This sho ws that t he sets Z ε form a decreasing family of precompact sets and eac h compact set in the hemicompact gro up  T H 1  ∧ is contained in some Z ε . If ω ∈ U ε , then analogously to case a), w e ha v e the fo llo wing: since U ε is path wise connected, then χ ∈ U ⊲ ε ⇔ | X n i ϕ i | ≤ 1 4 , ∀ ω ∈ U ε . (12) Since | X n i ϕ i | ≤ | n | b · X | ϕ i | = | n | b · | ω | 1 ≤ | n | b · ε, w e ha v e Z 4 ε =  n ∈  T H 1  ∧ : | n | b ≤ 1 4 ε  ⊂ U ⊲ ε . Let us prov e the second inclusion. If | n | b ≥ 1 2 ε , then | n j | ≥ 1 2 ε for some j . If ω = ( z n ), where z n = e 2 π iε 3 4 if n = j and z n = 1 otherwise, then ω ∈ U ε and | P n i ϕ i | = | n j | 3 4 ε ≥ 3 / 8. This con tradicts to (12). Th us U ⊲ ε ⊂ Z 2 ε . 3. a) L et 1 < p < ∞ . L et us show that for e ach neighb o rho o d W of χ = 0 in ( T H p ) ∧ and a p ositive in te ger k ther e exists m such that A ( k , m ) ⊂ W . (13) According to corollary 4.4 [3 ], there exis ts a sequence { a n } suc h that a n → 1 and { a n } ⊲ ⊂ W . Let us show t ha t A ( k , m ) ⊂ { a n } ⊲ for all large m . Set a n = ( z n k ) = ( e 2 π iϕ n k ). Let q b e suc h that 1 p + 1 q = 1. Set A = max { q p | n 1 | q + · · · + | n k +1 | q , | n j | ≤ k + 1 } . Th us, if χ ∈ A ( k , 0), then l ( χ ) ≤ k + 1 and | χ | q ≤ A . No w choose N 1 suc h that ρ p ( e, a n ) ≤ 1 4 A , ∀ n ≥ N 1 , a nd c ho ose M > N 1 suc h that ∞ X k = M +1 | ϕ n k | p ≤ 1 (4 A ) p , ∀ n ≤ N 1 . Then for a ll n w e ha v e P ∞ k = M +1 | ϕ n k | p < 1 (4 A ) p . Hence, for all χ ∈ A ( k , m ) , m ≥ M , and a n , by H¨ older’s inequalit y , w e ha v e | n m +1 ϕ n m +1 + · · · + n l ϕ n l | ≤ | χ | q · p q | ϕ n m +1 | p + · · · + | ϕ n l | p ≤ A · 1 4 A = 1 4 . Therefore A ( k , m ) ⊂ { a n } ⊲ for all m ≥ M . 14 b) L et p = 1 , ε > 0 and W b e an op en neigh b orho o d of the neutr al elemen t of  T H 1  ∧ . Set Z l ε :=  n = (0 , . . . , 0 , n l +1 , n l +2 , . . . ) : | n | b ≤ 1 ε  ⊂ Z ε . L et us show that Z l ε ⊂ W for al l lar ge l . Analogously , according to corollary 4.4 [3], there exists a sequence { a k } suc h that a k → 1 and { a k } ⊲ ⊂ W . It is enough to show tha t Z l ε ⊂ { a k } ⊲ for all large l . Cho ose N suc h that ρ 1 ( e, a k ) < ε 4 , ∀ k > N . Cho ose l 0 > N suc h that ∞ X i = l 0 +1 | ϕ k i | < ε 4 , for ev ery k = 1 , . . . , N . Then the last inequalit y is true for all k . Therefore for ev ery l ≥ l 0 , ev ery χ ∈ Z l ε and a k , w e ha v e | ∞ X i =1 n i ϕ k i | = | ∞ X i = l +1 n i ϕ k i | ≤ | n | b ∞ X i = l +1 | ϕ k i | < 1 ε · ε 4 = 1 4 . Hence n ∈ { a n } ⊲ . Thus Z l ε ⊂ { a n } ⊲ . 4. a) L et 1 < p < ∞ and ε < 0 , 01 . L et χ α → χ , wher e χ = ( n 1 , . . . , n s , 0 , . . . ) , χ α , χ ∈ U ⊲ ε . L et us pr ove that fo r every M ther e ex i s ts α 0 such that χ α = ( n 1 , . . . , n s , 0 , . . . , 0 M , n α M + 1 , . . . ) , ∀ α ≥ α 0 . By item 2a and Lemma 1, w e ha v e U ⊲ ε = A ε,q ⊂ A  1 4 ε  q  , 0  . Th us | n j | + | n α j | ≤ 2  1 4 ε  q  + 2 . (14) Set q = max n  2  1 4 ε  q  + 2  2 + 1 , 2 ε 2 o . Let a sequenc e { a n } b e suc h that a n → e a nd consists the follo wing elemen ts (exp  2 π i k 1 q  , . . . , exp  2 π i k M q  , 1 , . . . ) , where k i = 0 , ± 1 , . . . , ± q . 15 Since { a n } is compact, { a n } ⊲ is op en. Th us there exists α 0 suc h that χ − χ α ∈ { a n } ⊲ for α > α 0 . In pa r t icular, Re  exp  2 iπ ( n j − n α j ) k q  ≥ 0 , j = 1 , . . . , M , | k | ≤ q . (15) No w w e a ssume the conv erse and | n j − n α j | > 0 for some 0 < j ≤ M . Then, b y (14), 1 ≤ | n j − n α j | ≤ | n j | + | n α j | ≤ √ q . Since q > 2 ε 2 and ε < 0 , 0 1 , then 1 q ≤ | n j − n α j | q ≤ √ q q < ε < 0 , 01 . Hence there exists | k j | ≥ 1 suc h tha t 1 4 < ( n j − n α j ) k j q < 1 2 . Therefore f o r this k j the inequalit y (15) is wrong. b) L et us pr ove that A ( k , m ) is c omp act (1 < p < ∞ ) . Let a net { χ α } ⊂ A ( k , m ) is fundamen tal. Since, b y item 2, A ( k , m ) is precompact and is contained in some U ⊲ ε , then it con v erges to some χ = ( n 1 , . . . , n s , 0 , . . . ). Cho ose α 0 suc h that χ α = ( n 1 , . . . , n s , 0 , . . . , 0 M , n α M + 1 , . . . ) , ∀ α ≥ α 0 . Since | χ | 1 ≤ | χ α | 1 ≤ k + 1, w e obtain that χ ∈ A ( k , m ). 5. L et us pr ove that  T H p  ∧∧ = T ∞ 0 , 1 < p < ∞ . It is clear that  T H p  ∧∧ ⊂ (( Z ∞ 0 ) d ) ∧ = T ∞ . Let ω = ( z n ) = (e 2 iπ ϕ n ) ∈  T H p  ∧∧ . a) L et us show that  T H p  ∧∧ ⊂ T ∞ 0 , i.e. z n → 1 . Assume the con v erse and z n 6→ 1. Then there exists a subsequen ce n k suc h tha t ϕ n k → α 6 = 0. W e will sho w that ω is discon tin uous at 0. Let W b e a neigh b orho o d of the neutral elemen t of ( T H p ) ∧ . By (13), there exists m suc h that A (1 , m ) ⊂ W . In particular, if n k > m , then χ k = (0 , . . . , 0 , 1 , 0 , . . . ), where 1 o ccupies p osition n k , b elongs to W . Then ( ω , χ k ) = exp ( 2 iπ ϕ n k ) → exp(2 iπ α ) 6 = 1 . Th us ω is discon tin uous. Hence  T H p  ∧∧ ⊂ T ∞ 0 . b) L et us pr ove the c onverse in clusion:  T H p  ∧∧ ⊃ T ∞ 0 , i.e. if z n → 1 , then ω is a c ontinuous cha r acter of  T H p  ∧ . Since ( T H p ) ∧ is a hemicompact k - space [6], b y item 2a, it is enough to pro v e that ω is con tin uous on A ( k , 0 ). Let ε > 0 and χ α → χ , where χ α , χ ∈ A ( k , 0). Let q b e suc h that 1 p + 1 q = 1. Set A = max { q p | n 1 | q + · · · + | n k +1 | q , | n j | ≤ k + 1 } . Th us, if η ∈ A ( k , 0), then | η | q ≤ A . 16 Cho ose M suc h that p p P k =1 | ϕ M + k | p < ε 2 π A . By item 4a, for M w e can c ho ose α 0 suc h that χ α = ( n 1 , . . . , n s , 0 , . . . , 0 M , n α M + 1 , . . . ) , where χ = ( n 1 , . . . , n s , 0 , . . . ) , ∀ α > α 0 . Then ( ω , χ α − χ ) = exp  2 iπ P ∞ k =1 n α M + k ϕ M + k  . By H¨ older’s inequalit y , w e ha v e | X n α M + k ϕ M + k | ≤ | χ α | q · p q X | ϕ M + k | p < A · ε 2 π A = ε 2 π . Therefore for α > α 0 , b y (2), w e obtain | ( ω , χ α ) − ( ω , χ ) | = | 1 − ( ω , χ α − χ ) | < ε and ω is con tin uous. c) L et us pr ov e that ( T H p ) ∧∧ is top olo gic a l ly isomorphic to T H 0 . Since T H p is P olish, then ( T H p ) ∧∧ is P olish to o [6]. Since ( T H p ) ∧∧ and T H 0 are the same Borel subgroup of T ∞ , they m ust coincide top o lo gically . 6. a) L et us pr ov e that Z ∞ 0 is dens e in  T H 1  ∧ = Z ∞ b . Let n 0 = ( n i ) ∈ Z ∞ b and W b e a neigh b orho o d of t he neutral elemen t. Let ε b e suc h that | n 0 | b ≤ 1 ε . By item 3b of the pro of, we can choo se l suc h that Z l ε ⊂ W . Set n = ( n 1 , . . . , n l , 0 , . . . ) ∈ Z ∞ 0 . Then n 0 − n ∈ Z l ε ⊂ W q.e.d. b) L et us pr ove that T H 1 is r eflexive. Set t : T H 1 → T ∞ is the natural con tinu ous monomorphism. Since the image of t is dense, t ∗ : ( T ∞ ) ∧ = Z ∞ 0 → ( T H 1 ) ∧ = Z ∞ b is injectiv e. As it w as prov ed in part a), t ∗ has the dense imag e. Hence t ∗∗ : ( T H 1 ) ∧∧ → T ∞ is injectiv e. By corollary 3 [6], it is enough to prov e that T H 1 = ( T H 1 ) ∧∧ algebraically . Let ω = ( z n ) = ( e 2 π iϕ n ) ∈ ( T H 1 ) ∧∧ . If P | ϕ n | = ∞ , then, in the standard wa y , we can construct n = ( ± 1) such that ( ω , n ) do es not exist. Th us ω mus t b e contained in T H 1 . 7. a) L et 1 < p < ∞ . L et us pr ove that 1. T H p is not lo c al ly quasi c onvex. 2. ( T H p ) ∧ is r eflexive. 3. ( T H p ) ∧ = ( T H 0 ) ∧ and, he nc e, ( T H p ) ∧ do es not dep end on p . 4. T H 0 is r eflexive. 17 Let α p : T H p → ( T H p ) ∧∧ = T H 0 b e the canonical ho momorphism. Then α p has the dense image. By Prop osition 3.1, ( T H p ) ∧ and T H 0 = ( T H p ) ∧∧ are reflexiv e. Thu s, ( T H p ) ∧ = ( T H 0 ) ∧ do es not dep end on p . By Prop osition 3.2, T H p is not lo cally quasi con v ex. b) L et us pr ove that T H 0 is not a QI-gr oup. Assume the con v erse and T H 0 is a QI-group. Assume that T H 0 is repre- sen ted in T ∞ . Set U d ε = { h ∈ T H 0 : d ( e, h ) < ε } , U ε = { h ∈ T H 0 : ρ 0 ( e, h ) < ε } . By Prop osition 1, there exis ts ε 0 > 0 such that Cl T ∞ ( U d ε 0 ) is compact in T ∞ and it is contained in T H 0 . Since the P olish group top olo gy is unique, there exists ε > 0, suc h t hat U ε ⊂ U d ε 0 . Set z = exp  2 π i ε 2 π  . Then ω k = ( z , . . . , z , 1 k +1 , 1 , . . . ) ∈ Cl T ∞ ( U ε ) ⊂ T H 0 for ev ery k . But ω k con v erges in T ∞ to ω = ( z ) 6∈ T H 0 . Hence T H 0 can not b e represen ted in T ∞ . If T H 0 is represen ted in ano t her lo cally compact gro up X , then, b y 25.3 1 (b) [12], X = T ∞ × X 1 and T H 0 condensates to T ∞ . Hence T H 0 is not a QI-group. 8. L et us pr ove that  T H p  ∧ and ( Z ∞ 0 , e ) ar e top olo gic al ly isomo rphic. Since e n ∈ A (0 , n ) for ev ery n , e n con v erges to zero in  T H p  ∧ b y (13). Th us, b y definition, id : ( Z ∞ 0 , e ) 7→  T H p  ∧ is con tin uous. Let us prov e the conv erse, i.e. if W is an op en neigh b orho o d o f zero in ( Z ∞ 0 , e ), then W is op en in  T H p  ∧ . At first w e note tha t since  T H p  ∧ is a hemicompact k -space, a set Y is closed if and only if Y ∩ A ( k , 0) is closed in A ( k , 0). By the construction of ( Z ∞ 0 , e ) [24], we ma y assume that W has t he form W = ∪ ∞ k =1 ( A ∗ i 1 + A ∗ i 2 + · · · + A ∗ i k ) , where 1 ≤ i 1 < i 2 < . . . , A ∗ n = { 0 , ± e m : m ≥ n } . No w assume the con v erse and W is not op en in  T H p  ∧ . Then there exists k 0 suc h that A ( k 0 , 0) \ W is not closed in A ( k 0 , 0). In particular, A ( k 0 , 0) \ W 6 = ∅ . Let a net { χ α } ⊂ A ( k 0 , 0) \ W b e suc h that χ α con v erges t o χ 0 = ( n 1 , . . . , n s , 0 , . . . ) ∈ A ( k 0 , 0) ∩ W . Since ( Z ∞ 0 , e ) is a top ological gro up, w e can find a neigh b orho o d W 1 of zero such that χ 0 + W 1 ⊂ W , where W 1 = ∪ ∞ k =1 ( A ∗ j 1 + A ∗ j 2 + · · · + A ∗ j k ) , 1 ≤ j 1 < j 2 < . . . By the construction of W 1 , w e hav e A ( k 0 , j k 0 +1 ) ⊂ W 1 . By 4a , w e can c ho ose α 0 suc h that χ α = ( n 1 , . . . , n s , 0 , . . . , 0 j k 0 +1 , n α j k 0 +1 +1 , . . . ) ∈ A ( k 0 , 0) , ∀ α ≥ α 0 . 18 Th us χ α − χ 0 ∈ A ( k 0 , j k 0 +1 ) for every α ≥ α 0 . So χ α = χ 0 + ( χ α − χ 0 ) ∈ χ 0 + W 1 ⊂ W , ∀ α ≥ α 0 . It is imp ossible since χ α 6∈ W . Th us W is op en in  T H p  ∧ . The theorem is pro v ed.  Remark 1. Let G b e the group Z ∞ 0 with the discrete top ology and H = ( Z ∞ 0 , e ). Then G a nd H are reflexiv e groups and id : G → H is a contin uous isomorphism, but id ∗ : H ∧ = T H 0 → G ∧ = T ∞ is only a con tin uous injection.  Remark 2. W e can prov e that ( T H 0 ) ∧ = Z ∞ 0 algebraically using the follo wing observ atio n. L et X , G, and Y b e a top olo gic al ab elian gr oups such that ther e exi s t c on- tinuous monomo rphisms i : X → G a n d j : G → Y with the den se imag e. If i ∗ ◦ j ∗ is bije ctive, then i ∗ and j ∗ ar e bije ctive to o. (Since i has the dense ima g e, i ∗ is injectiv e. Since i ∗ ◦ j ∗ is bijectiv e, then i ∗ m ust b e surjectiv e. Henc e i ∗ is bijectiv e. Th us j ∗ = ( i ∗ ) − 1 ( i ∗ ◦ j ∗ ) is bijectiv e.) No w, since T H p ⊂ T H 0 ⊂ T ∞ and ( T H p ) ∧ = ( T ∞ ) ∧ = Z ∞ 0 , then, by the observ ation, ( T H 0 ) ∧ = Z ∞ 0 algebraically .  No w we consider the gro up G p , 1 < p < ∞ . Since Q is dense in G p , G ∧ p ⊂ Q ∧ d = ∆ a , where a = ( a 1 , a 2 , . . . ) (here Q d denotes the group Q with discrete top ology). By section 25.2 [12], w e hav e ( χ, z ) = z P ∞ k =1 ω k γ ( k ) , ∀ z ∈ Q, where χ = ( ω k ) ∈ ∆ a , ω k ∈ { 0 , 1 , . . . , a k − 1 } . W e need the follow ing lemma. Lemma 2. The gr oup Q is de n se in G 0 . Pro of. W e iden tify T with [0 , 1) and denote by h x i the distance of x from the nearest in teger. By (2), we can consider the followin g equiv alen t metric on G 0 r 0 ( x 1 , x 2 ) = sup {h γ ( n )( x 1 − x 2 ) i , n ∈ N } . If x ∈ [0 , 1), w e can write x = ∞ X k =1 ε k ( x ) a 1 a 2 . . . a k , where ε k ( x ) = 0 , . . . , a k − 1 , and for ev ery M > 0 there exists k > M suc h that ε k ( x ) < a k − 1. Th us γ ( n ) x (mo d1) = ∞ X k = n ε k ( x ) a n . . . a k = ε n ( x ) a n + θ n a n , 19 where 0 ≤ θ n < 1, since θ n = ε n +1 a n +1 + ε n +2 a n +1 a n +2 + ε n +3 a n +1 a n +2 a n +3 + · · · < a n +1 − 1 a n +1 + a n +2 − 1 a n +1 a n +2 + a n +3 − 1 a n +1 a n +2 a n +3 + . . . =  1 − 1 a n +1  +  1 a n +1 − 1 a n +1 a n +2  +  1 a n +1 a n +2 − 1 a n +1 a n +2 a n +3  + · · · = 1 . By the definition of G 0 , w e ha v e h γ ( n ) x i = h ε n ( x ) + θ n a n i → 0 . (16) No w w e set x N = P N − 1 k =1 ε k ( x ) a 1 a 2 ...a k ∈ Q . Then x − x N = ∞ X k = N ε k ( x ) a 1 a 2 . . . a k = 1 a 1 a 2 . . . a N − 1 · ε N ( x ) + θ N a N and γ ( n )( x − x N ) = ( γ ( n )( x )(mo d1 ) , for N ≤ n 1 a n ...a N − 1 · ε N ( x )+ θ N a N , for 1 ≤ n < N (17) Let ε > 0. Since a n → ∞ , b y (1 6), w e can c ho ose N suc h that h γ ( n ) x i < ε for all n ≥ N and a N − 1 > 1 /ε . Then, b y (17), w e obtain r 0 ( x, x N ) < ε . Th us Q is dense in G 0 .  F or 1 < p < ∞ and p = 0 w e consider the following homomorphisms S p : G p 7→ T × T H p ≈ T H p , S p ( z ) = ( z , z γ (2) , . . . , z γ ( k ) , . . . ) . It is clear that S p is a top ological isomorphism of G p on to the followin g closed subgroup of T H p { ω ∈ T H p : ω = ( z γ (1) , z γ (2) , . . . , z γ ( n ) , . . . ) } . W e will iden tify G p with this subgroup. Set γ = { γ n } . Then, by Lemma 2, γ is a T B -sequence . Denote b y ( Z , γ ) the group of in tegers equipped with the finest Hausdorff group top olog y in whic h γ n con v erges to zero. Theorem 2. L et 1 < p < ∞ . Th e n 1. G p is not lo c al ly quasi-c onvex and, henc e, not r eflexive . 2. G 0 is r eflexive and, henc e, lo c al ly quasi-c onvex. 20 3. ( G p ) ∧ = ( G 0 ) ∧ = ( Z , γ ) and, henc e, r eflexive. Pro of. 1. L et us pr ove that G p , 1 < p < ∞ or p = 0 , is dual ly close d in T H p . Let ω = ( z γ ( n ) ) a nd ω 0 = ( z 1 , z 2 , . . . ) 6∈ G p . Then there exists the minimal i > 1 suc h that z i 6 = z γ ( i ) 1 . Set H i = { ( z , z γ (2) , . . . , z γ ( i ) ) , z ∈ T } . Then H i is closed in T i and ω ′ 0 = ( z 1 , . . . , z i ) 6∈ H i . Let π i b e the natura l pro jection from T H p to T i . It is clear that π i ( G p ) ⊂ H i . If n ′ = ( n 1 , . . . , n i ) ∈ H ⊥ i is suc h that ( n ′ , ω ′ 0 ) 6 = 1, then n = ( n 1 , . . . , n i , 0 , . . . ) ∈ G ⊥ p and ( n , ω 0 ) = ( n ′ , ω ′ 0 ) 6 = 1. Hence G p is dually closed. 2. L et us pr ove that G p , 1 < p < ∞ , is dual ly emb e dde d in T H p and G ∧ p is algebr aic al ly isomorph ic to Z . As it w as prov ed in [2], χ = ( ω k ) ∈ G ∧ p iff either ω k = 0 for all large k or ω k = a k − 1 for all la rge k . Hence w e can iden tify χ = ( ω 1 , . . . , ω m , 0 , . . . ) ∈ G ∧ p with n ∈ Z in the following w a y (10.3, [12]): if n = ω 1 + ω 2 γ (2) + · · · + ω m γ ( m ) > 0, then n 7→ χ = ( ω 1 , . . . , ω m , 0 , . . . ) and − n 7→ χ = ( a 1 − ω 1 , a 2 − ω 2 − 1 , . . . , a m − ω m − 1 , a m +1 − 1 , a m +2 − 1 , . . . ) . Therefore G ∧ p = Z = T ∧ and ( n, z ) = z n , ∀ n ∈ Z , z ∈ G p . Hence w e can extend ev ery n ∈ G ∧ p to a c haracter of ( T H p ) ∧ , for example, in the follo wing w ay n 7→ n = ( n, 0 , . . . ) and ( n , ω ) = z n 1 , where ω = ( z 1 , z 2 , . . . ) . Hence G p is a dually em b edded subgroup of T H p . 3. L et us c ompute G ⊥ p . By definition, w e hav e G ⊥ p = { n = ( n 1 , n 2 , . . . , n s , 0 , . . . ) : z P s k =1 n k γ ( k ) = 1 , ∀ z ∈ G p } . Since Q is dense in G p and T , w e hav e n ∈ G ⊥ p if and only if s X k =1 n k γ ( k ) = 0 . 4. L et us pr ove that ( T H p ) ∧ /G ⊥ p = ( Z , γ ) . 21 Denote b y π 0 : ( T H p ) ∧ 7→ ( T H p ) ∧ /G ⊥ p the natural homomorphism. By item 3, if χ = ( n 1 , n 2 , . . . , n s , 0 , . . . ) ∈ ( T H p ) ∧ , then χ + G ⊥ p = χ ′ + G ⊥ p , where χ ′ = ( n, 0 , . . . ) with n = P s k =1 n k γ ( k ). Thus π 0 ( e s ) = γ s . Hence, by L emma 3 [11], ( T H p ) ∧ /G ⊥ p = ( Z , γ ). In what follo ws w e need some notations. Set A ( k , m ) = ( n 1 γ ( r 1 ) + · · · + n s γ ( r s ) | m ≤ r 1 < · · · < r s , s X i =1 | n i | ≤ k ) , A H ( k , m ) = ( n 1 e r 1 + · · · + n s e r s | m ≤ r 1 < · · · < r s , n i ∈ Z , s X i =1 | n i | ≤ k ) . Then A ( k , m ) is a subset of ( T H p ) ∧ /G ⊥ p , and A H ( k , m ) is a subset of  T H 0  ∧ . Let us consider the em b edding S p : G p → T H p . By item 2, G p is dually em b edded. Thus S ∗ p : ( T H p ) ∧ → G ∧ p is surjectiv e. So S ∗∗ p : G ∧∧ p → ( T H p ) ∧∧ = T H 0 is a con tin uous monomo r phism and φ p : ( T H p ) ∧ /G ⊥ p 7→ G ∧ p is a con tin uous isomorphism. 5. L et us pr ove that for ev ery δ > 0 ther e e x i s ts k > 0 such that U ⊲ δ ⊂ φ p ( A ( k , 0)) . Let m ∈ U ⊲ δ . If m > 0, then there exists the follo wing unique decomp osi- tion of m m = ω 1 γ (1) + · · · + ω s γ ( s ) , where 0 ≤ ω k < a k . (18) a) L e t us pr ove that ther e exist C 1 = C 1 ( δ ) > 0 such that for al l m ∈ U ⊲ δ with de c omp osition (18) we have ω k < C 1 . Indeed, a ssume the con v erse and t here exist m k ∈ U ⊲ δ , m k = P n ω k n γ ( n ), and an index n k suc h that ω k n k → ∞ . W e will a ssume that ω k n k > 2 and iden tify T with [0; 1). Set x k = ε n k a 1 a 2 . . . a n k , where ε n k =  3 a n k 2 ω k n k  . Then γ ( n ) x k (mo d1) = ε n k a n a n +1 ...a n k if n ≤ n k , and γ ( n ) x k (mo d1) = 0 if n > n k . Therefore, b y (2), w e hav e k x k k p p ≤ n k X n =1 2 p π p ε p n k ( a n a n +1 . . . a n k ) p < 2 p +1 π p ε p n k a p n k < 2 · 3 p π p 1 ( ω k n k ) p < δ p 22 if ( ω k n k ) p > 2 · 3 p π p δ p . Hence for some k 0 , w e ha v e x k ∈ U δ , ∀ k ≥ k 0 . Since 1 2 π i Arg( m k , x k )(mo d1) = n k X n =1 ω k n ε n k a n a n +1 . . . a n k = ε n k a n k ω k n k + n k − 1 X n =1 ω k n a n . . . a n k − 1 ! , then ω k n k ε n k a n k < 1 2 π i Arg ( m k , x k )(mo d1) < ( ω k n k + 2) ε n k a n k and 1 2 π i Arg( m k , x k )(mo d1) → 3 2 (mo d1) = 1 2 . Hence m k 6∈ U ⊲ δ . It is a contradiction. b) L et us pr ove that ther e exists an inte ger M = M ( δ ) such that at most M c o efficien ts ω k ar e not e qual to 0. Denote b y M ( n ) the nu mber of all nonzero co efficien ts ω i of n in the decomp osition (18). Denote b y Z A ( n ) the set of all co efficien ts ω i of n whic h are equal t o A . Set M A ( n ) is the cardinality of Z A ( n ). By a), w e need to pro v e that M A ( n ) is b ounded on U ⊲ δ for ev ery A . Let us assume the conv erse and there exists a sequence { m k } ⊂ U ⊲ δ suc h that M A ( m k ) → ∞ . Cho ose k ′ suc h that (w e remaind that a k → ∞ ) 2 2 p +1 π p C 1 ∞ X k = k ′ 1 k p < δ p and a k > 100 , ∀ k ≥ k ′ . (19) Cho ose a sequen ce 2 p C 1 k ′ < T 0 < T 1 . . . suc h that 2 5 < T i +1 X l = T i +1 A l < 3 5 , ∀ i = 0 , 1 , . . . (20) and c ho ose k 0 > 10 k ′ suc h that a k > 10 T 1 for all k ≥ k 0 . Cho ose m 1 ∈ U ⊲ δ suc h that M A ( m 1 ) > T 1 + k 0 and set k ′ 1 > k 0 is the maximal index of nonzero ω i in the decomp o sition (18) of m 1 . Let us denote b y k 0 < l 1 < · · · < l T 1 the first T 1 indexes suc h that ω l i = A . Cho ose k 1 > k ′ 1 suc h that a k > 10 2 T 2 for all k ≥ k 1 . Cho ose m 2 ∈ U ⊲ δ , m 2 6 = m 1 , such that M A ( m 2 ) > T 2 + k 1 and set k ′ 2 > k 1 is the maximal index of nonzero ω i in the decomp o sition (18) o f m 2 . Let us 23 denote b y k 1 < l T 1 +1 < · · · < l T 2 the first T 2 − T 1 indexes suc h that ω l i = A . Cho ose k 2 > k ′ 2 suc h that a k > 10 3 T 3 for all k ≥ k 2 . And so on. R emark that, b y our c ho osing of l k , a l T k − 1 + n > 10 k T k , k = 1 , 2 . . . , 1 ≤ n ≤ T k − T k − 1 . (21) Set x k = T k − T k − 1 X n =1  a l T k − 1 + n T k − 1 + n  · 1 a 1 a 2 . . . a l T k − 1 + n , k = 1 , 2 . . . Then w e ha v e: for 1 ≤ s ≤ l T k − 1 +1 γ ( s ) x k (mo d1) = 1 a s a s +1 . . . a l T k − 1 +1 − 1 T k − T k − 1 X n =1  a l T k − 1 + n T k − 1 + n  · 1 a l T k − 1 +1 . . . a l T k − 1 + n = 1 a s a s +1 . . . a l T k − 1 +1 − 1  a l T k − 1 +1 T k − 1 + 1  + θ k s  · 1 a l T k − 1 +1 , (22) where 0 < θ k s < 1; for l T k − 1 + r < s ≤ l T k − 1 + r +1 , 0 < r < T k − T k − 1 , γ ( s ) x k (mo d1) = 1 a s a s +1 . . . a l T k − 1 + r +1 − 1 T k − T k − 1 X n = r +1  a l T k − 1 + n T k − 1 + n  · 1 a l T k − 1 + r +1 . . . a l T k − 1 + n = 1 a s a s +1 . . . a l T k − 1 + r +1 − 1  a l T k − 1 + r +1 T k − 1 + r + 1  + θ k s  · 1 a l T k − 1 + r +1 , (23) where 0 < θ k s < 1; and γ ( s ) x k (mo d1) = 0 for l T k < s . Since a k ≥ 2 and p > 1, for ev ery q ≥ 1, we hav e q X s =1 1 ( a s a s +1 . . . a q ) p ≤ q X s =1 1 a s a s +1 . . . a q < 2 . (24) Since a k ≥ 2 , p > 1 and T 0 > k ′ , b y (2) and (19)-(24), w e hav e k x k k p p < 2 p π p T k − T k − 1 X n =1 2  2  a l T k − 1 + n T k − 1 + n  p · 1 a p l T k − 1 + n < 24 2 2 p +1 π p T k − T k − 1 X n =1  1 T k − 1 + n  p < δ p . Hence x k ∈ U δ for all k . F or m k = P n ω k n γ ( n ) w e denote b y - B k 0 is the set o f indexes 1 < s < l T k − 1 suc h that ω k s 6 = 0. - B k r , 0 < r < T k − T k − 1 , is the set of indexes l T k − 1 + r < s < l T k − 1 + r +1 suc h that ω k s 6 = 0. - B k T k − T k − 1 is the set of indexes l T k < s suc h that ω k s 6 = 0. Then w e can represen t m k in the form m k = T k − T k − 1 X n =1 Aγ  l T k − 1 + n  + T k − T k − 1 X r =0 X s ∈ B k r ω k s γ ( s ) . Then, b y (22) and (23), w e ha v e (0 < θ k s < 1) Aγ  l T k − 1 + n  x k (mo d1) = A T k − 1 + n + A a l T k − 1 + n  a l T k − 1 + n T k − 1 + n  − a l T k − 1 + n T k − 1 + n + θ k s  . (25) for ev ery 1 ≤ n ≤ T k − T k − 1 . F o r ev ery 0 ≤ r < T k − T k − 1 , by (2 2)-(24), we ha v e X s ∈ B k r ω k s γ ( s ) x k (mo d1) < l T k − 1 + r +1 − 1 X s =1 ω k s a s a s +1 . . . a l T k − 1 + r +1 − 2 · 1 a l T k − 1 + r +1 − 1 < 2 C 1 a l T k − 1 + r +1 − 1 , (26) and X s ∈ B k T k − T k − 1 ω k s γ ( s ) x k (mo d1) = 0 (27) Th us, b y (25)-(2 7), we hav e ( mo d1)      1 2 π i Arg( m k , x k ) − T k − T k − 1 X n =1 A T k − 1 + n      < 25 T k − T k − 1 X n =1 A a l T k − 1 + n + T k − T k − 1 X r =0 2 C 1 a l T k − 1 + r +1 − 1 < 4 C 1 ( T k − T k − 1 ) 10 k T k → 0 . Hence, b y (21), m k 6∈ U ⊲ δ . It is a con tradiction. Since U ⊲ δ is symmetric, w e prov ed the following: for each δ > 0 t here exists a constant C = C ( δ ) suc h that if n ∈ U ⊲ δ and | n | = ω 1 + ω 2 γ (2) + · · · + ω m γ ( m ) , then ω 1 + · · · + ω m < C . Hence U ⊲ δ ⊂ φ p ( A ( k , 0)) for some in teger k > 0. 6. L et us pr ove that φ p : ( T H p ) ∧ /G ⊥ p 7→ G ∧ p is a top olo gic al iso m orphism. Let δ > 0. Since A H ( k , 0) is compact in  T H 0  ∧ (see pro of of Theorem 1) and A ( k , 0) = π 0 ( A H ( k , 0)), then A ( k , 0) is compact. Therefore φ p is a homeomorphism on A ( k , 0) ⊃ φ − 1 p ( U ⊲ δ ). Since G ∧ p is hemicompact and any compact set in G ∧ p is con tained in some U ⊲ δ , φ p is a t o p ological isomorphism. 7. L et us pr ove that G ∧∧ p = G 0 . By items 4 and 6, we hav e G ∧ p = ( T H p ) ∧ /G ⊥ p = ( Z , γ ). Th us the assertion follo ws from Theorem 3 [11]. In particular, G ∧ p do es not dep end on p . 8. Let α p : G p 7→ G ∧∧ p = G 0 b e t he canonical ho mo mo r phism. Since Q is dense in G p [2] and G 0 (Lemma 2), then α p ( G p ) is dense in G 0 . Th us, by Prop osition 3.1, G ∧ p , G ∧ 0 and G 0 are reflexiv e. In particular, G ∧ p = G ∧ 0 . By Prop osition 3.2, G p is not lo cally quasi-con v ex.  Note that t he groups G p , 1 < p < ∞ , form a con tin ual c hain (under inclusion) o f Polish non lo cally quasi-conv ex to tally disconnected groups with the same c ountable r eflexive dual. References [1] J. Aaronson, The eigen v alues of non-singular t r a nsformations, Israel J. Math. 4 5 (1983), 297-312 . [2] J. Aaronson, M. Nadk arni, L ∞ eigen v alues and L 2 sp ectra of non-singular transformations, Pro c. London Math. So c. (3) 55 (19 8 7), 53 8 -570. [3] L. Außenhofer, Con tributions to the dualit y theory of Ab elian top olo gical groups and to the theory of n uclear g r o ups, Dissertation, T ¨ ubingen, 1998 , Dissertationes Math. (Rozpra wy Mat.) 384 (1999) 113. [4] W. Banaszczyk, Additiv e subgroups of top olog ical v ector spaces, LNM 1466, Berlin-Heidelberg- New Y o rk 1 9 91. 26 [5] M. Bruguera, Some prop erties of lo cally quasi-conv ex gr o ups, T op ology Appl. 77 (1997), 87-94. [6] M.J. Chasco M.J., P on try agin duality for metrizable groups, Arch. Math. 70 (1998) 22-28. [7] S.D. Chatterji, V. Mandrek ar, Quasi-in v aria nce of measures under trans- lation, Math. Z. 154(1) (1977), 19-29. [8] Dikranjan D ., Pro da nov I., Sto janov L., T op olo g ical groups. Characters, dualities, and minimal gro up t o p ologies, Marcel Dekk er, Inc., New-Y ork- Basel, 1990. [9] S.S. Gabriyely an, Admis sible transformations of measures, J. Math. Ph ysics, Analisys, Geometry , 1( 2 ) (2005) 155-181 . (Russian) [10] S.S. Gabriy ely an, T op ological prop erties of the set of admissible transfor- mations of measures, J. Math. Ph ysics, Analisys, Geometry , 2( 1 ) (20 0 6) 9-39. [11] S.S. Gabriy ely an, On T -sequences and ch aracterized subgroups, arXiv:math.GN/0902.07 23. [12] E. Hewitt, K.A. R oss, Abstract Harmonic Analysis, V ol. I, 2nd ed. Springer-V erlag , Berlin, 1 979. [13] A. Hora, Quasi-inv arian t measures on comm utativ e Lie groups of infinite pro duct t yp e, Math. Z. 206( 2 ) (1991), 1 6 9-192. [14] B. Host, J.- F. Mela, F. P arreau, Non singular transformat io ns and sp ec- tral analysis of measures, Bull. So c. Math. F rance 1 19 (1 9 91), 33-90. [15] S. K aplan, Extensions of the Pon try agin dualit y , I: Infinite pro ducts, Duk e Math. 15(1948 ) 649- 658. [16] F. Parreau, Measures with r eal sp ectra, Inv ent. Math. (2) 98 (1989), 311-330 . [17] F. Parreau, Ergo dicite et purete des pro duits de Riesz, Ann. Inst. F ourier (2) 40 (1990), 3 91-405. 27 [18] V. P esto v, F ree a b elian top ological g r o ups and the P ontry agin- v an Kam- p en dualit y , Bull. Austral. Math. So c. 52(199 5), 297 -311. [19] S. Rolewicz, Some remarks o n monothetic g roup, Collo quium Math. (1 ) XI I I (1964), 27-28 . [20] G . Sadasue, On quasi-in v a riance of infinite pro duct measures, J. Theor. Probab. 21 ( 2 008) 571-58 5 . [21] G .E. ˇ Silo v, F an Dyk Tin’, In tegral, measure and deriv ative o n linear spaces, Izdat. ”Nauk a”, Mosco w, 196 7. [22] M.F. Smith, The P ontrjagin dualit y theorem in linear spaces, Ann. Math. 5 6(2) (1952) 248-2 53. [23] N. Y a. Vilenkin, The theory of c haracters of top ological Ab elian g roups with b oundedness g iv en, Izv. Ak ad. Nauk SSSR, Ser. Math. 15 ( 5 ) (19 5 1) 439-462 . (Russian) [24] E.G. Z eleny uk, I.V. Protasov, T op ologies on ab elian groups, Math. USSR Izv. 37 ( 1 991) 445- 460. Russian original: Izv. Ak a d. Nauk SSSR 54 (1990) 1090-1107 . 28