Ambitable topological groups

Am bitable top ological groups Jan P ac hl Fields I nstitu te T or on t o, Ontario, Can ada Ma y 25, 2009 (v ersion 4) Abstract A top ological group is said to b e ambi table if eac h uniformly bounded u n iformly equicontin uous set of functions on the group with its righ t uniformity is con tained in an am b it. F or n = 0 , 1 , 2 , . . . , every lo cally ℵ n -b ounded top ological group is either precompact or am bitable. In the familia r semigroups constructed o ver am bitable groups, topological centres hav e an effectiv e characteriza tion. 1 Ov erview A top o logical group G may be naturally e m b edded in la rger spa ces, algebraic ally and top olog- ically . Two s uch spa ces of par ticular int e rest in a bstract har monic analysis are • the norm dua l of the space o f b ounded rig h t uniformly co ntin uous functions on G , denoted here M ( r G ) (also k nown as LUC( G ) ∗ ); and • the uniform compa c tification of G with its right unifor mit y , deno ted here r G (also known as G LUC or G LC ). It is customary to study these “right” versions o f the t wo spaces; the pr o p erties of the corr e - sp onding “left” versions are obtained by symmetry . Both M ( r G ) and r G are right top olog ical semigro ups . In inv estigating their structure it is very helpful to hav e a tractable characterizatio n of their top olog ical centres. F easible candidates for such characterizations are the space of uniform measures M u ( r G ) and the completion c r G of the right uniformity on G . When G is lo cally compact, M u ( r G ) is the space of finite Radon measure s on G , and c r G is G itself. In this case, La u [8] and Lau a nd Pym [9] pr ov ed that M u ( r G ) and c r G = G are the top olog ical centres of M ( r G ) a nd r G . These characteriza tions generalized a num b er o f previous results for sp ecia l classes of lo cally compa ct groups. More recently , Neufang [10] applied his factorization metho d to simplify the pro of of Lau’s result. Then F erri and Neufang [5] used a v aria nt of the facto r ization method to prov e that M u ( r G ) and c r G are the topo logical centres of M ( r G ) a nd r G for ℵ 0 -b ounded (not necessar ily lo cally compa ct) top o logical gr o ups. 1 This pap er deals with another v ariant o f the factoriza tion metho d, similar to that used by F erri and Neufang. By definition, am bitable topolo gical g roups are those in whic h a suitable factorization theo rem holds ; equiv a lently , in the language of top olo g ical dynamics, those in which ev er y uniformly b ounded uniformly equicontin uous set of functions is contained in an ambit. In such gr oups M u ( r G ) and c r G are the top olog ical ce n tr es o f M ( r G ) and r G . Several classes of top o logical groups ar e shown to b e ambit a ble. In particular , if n is a p ositive integer then every lo ca lly ℵ n -b ounded gro up is either precompact or ambitable, which y ie lds a common generaliza tion o f the afor ementioned res ults by Lau, La u and P ym, and F erri and Neufang. This is a n up dated prelimina ry version of the pap er. The fina l version will b e published in T op ology and its Applications in 200 9. 2 Basic definitions All top olog ic al groups considered in this pap er are assumed to b e Ha usdorff, and a ll linear spaces to b e ov er the field R of reals. Most of the notation used her e is defined in [12]. In particula r , if p is a mapping from X × Y to W then \ x p ( x, y ) is the mapping x 7→ p ( x, y ) from X to W and \ y p ( x, y ) is the mapping y 7→ p ( x, y ) fro m Y to W . Let G b e a gr o up, f a real-v alued function o n G a nd x ∈ G . Define ρ x ( f ) (the right tr anslation o f f by x ) to be the function \ z f ( z x ). The set orb( f ) = { ρ x ( f ) | x ∈ G } is the (right) orbit o f f . Denote by orb( f ) the clos ure of orb( f ) in the pro duct spa ce R G (the set o f real-v alued functions o n G with the to po logy of p oint wise conv erg ence). When ∆ is a pseudometric on G , define BLip + (∆) = { f : G → R | 0 ≤ f ( x ) ≤ 1 a nd | f ( x ) − f ( y ) | ≤ ∆( x, y ) for all x, y ∈ G } . Then B L ip + (∆) is a compact subset of the pro duct s pa ce R G ; w e alw ays consider BLip + (∆) with this compact top olog y . When G is a top olog ical gro up, deno te by RP ( G ) the s e t of all con tinuous rig ht - inv ariant pseudometrics on G . The right uniformity on G is the uniformity g enerated b y RP ( G ). The set G with the right uniformity is denoted r G , a nd the space of all b o unded uniformly contin uous real-v alued functions on r G is denoted U b ( r G ). The group G is said to b e pr e c omp act if the uniform spa ce r G is precompa ct. The following lemma summarizes several prop er ties of ρ x ( f ) needed in this pap er. De V ries ([4], sec. IV.5 ) provides a compre hensive tre atment of the r ole of ρ x ( f ) in top ologic a l dynamics. Lemma 2.1 L et G b e any t op olo gic al gr oup and ∆ ∈ RP ( G ) . 1. If f is a r e al-value d function on G and x, y ∈ G then ρ xy ( f ) = ρ x ( ρ y ( f )) . 2. If f ∈ BLip + (∆) and x ∈ G then ρ x ( f ) ∈ BLip + (∆) . 3. The mapping ( x, f ) 7→ ρ x ( f ) is c ont inuous fr om G × BLip + (∆) to BLip + (∆) . Pro of. 1. ρ xy ( f )( z ) = f ( z xy ) = ρ y ( f )( z x ) = ρ x ( ρ y ( f ))( z ). 2. If f ∈ BLip + (∆) then | ρ x ( f )( z ) − ρ x ( f )( z ′ ) | = | f ( z x ) − f ( z ′ x ) | ≤ ∆( z x, z ′ x ) = ∆( z , z ′ ). 2 3. T o prov e that the mapping ( x, f ) 7→ ρ x ( f ) to B L ip + (∆) is co nt inuous, it is sufficient to prov e that the mapping ( x, f ) 7→ ρ x ( f )( z ) to R is contin uous for each z ∈ G . T ake any z ∈ G , ( x 0 , f 0 ) ∈ G × BLip + (∆) and ε > 0. The set U = { ( x, f ) ∈ G × BLip + (∆) | ∆( z x, z x 0 ) < ε and | f ( z x 0 ) − f 0 ( z x 0 ) | < ε } is a neighbourho o d of ( x 0 , f 0 ) in G × BLip + (∆). F or ( x, f ) ∈ U we hav e | ρ x ( f )( z ) − ρ x 0 ( f 0 )( z ) | = | f ( z x ) − f 0 ( z x 0 ) | ≤ | f ( z x ) − f ( z x 0 ) | + | f ( z x 0 ) − f 0 ( z x 0 ) | < 2 ε. Thu s the mapping ( x, f ) 7→ ρ x ( f )( z ) is co ntin uous at ( x 0 , f 0 ).  When G and ∆ ar e as in the lemma, BLip + (∆) with the a ction ( x, f ) 7→ ρ x ( f ) is a compact G -flow, in the terminolo gy o f top ologica l dynamics [4 ]. If f ∈ U b ( r G ) then there exis ts ∆ ∈ RP ( G ) such that sf + t ∈ BLip + (∆) for some s, t ∈ R . Thus s · orb( f ) + t = orb( sf + t ) ⊆ BLip + (∆) and therefore the set orb( f ) is compact in the topolog y of point wise conv erg ence, and orb( f ) with the action ( x, f ) 7→ ρ x ( f ) is also a compact G -flow. Recall that a compact G -flow is an ambit if it con ta ins an elemen t with dense orbit ([4], IV.4.1). F or a fix e d G , all a m bits can b e constructed from those of the form orb( f ), wher e f ∈ U b ( r G ) ([4], IV.5 .8 ). Say that a top olog ical gr oup G is ambitable if every B Lip + (∆), where ∆ ∈ RP ( G ), is contained in an a mb it within U b ( r G ). In o ther w o rds, G is ambitable iff for each ∆ ∈ RP ( G ) there exists f ∈ U b ( r G ) such that BLip + (∆) ⊆ orb( f ). Theorem 2.2 No pr e c omp act top olo gic al gr oup is ambitable. Pro of. Let G b e preco mpa ct. Fix a ny f ∈ U b ( r G ), a nd define ∆( x, x ′ ) = sup y ∈ G | f ( xy ) − f ( x ′ y ) | for x, x ′ ∈ G . Then ∆ ∈ RP ( G ), and there is a finite set F ⊆ G such that ∆( x, F ) < 1 3 for every x ∈ G . Consider the constant functions 0 and 1. If 0 ∈ orb( f ) then there is y ∈ G suc h that ρ y ( f )( x ) < 1 3 for ev ery x ∈ F , hence ρ y ( f )( x ) < 2 3 for every x ∈ G . Thus f ( x ) < 2 3 for ev ery x ∈ G , and 1 6∈ orb( f ). This prov e s that ther e is no f ∈ U b ( r G ) for which 0 , 1 ∈ orb( f ).  Question 1 Is every top olo gic al gr ou p eithe r pr e c omp act or ambitable? This question is motiv ated by in vestigations of top ologica l centres in certain semigr oups that ar ise in functional analysis. The c o nnection is explained in s ection 5 b elow. 3 Cardinal functions The reade r is referr ed to Jech [7] for de finitio ns r egarding car dina ls. The cardina lit y o f a set X is | X | . The cardinal successor of a cardinal κ is κ + . The sma lle s t infinite cardinal is ℵ 0 , a nd ℵ n +1 = ℵ + n . The smalles t cardinal la rger than ℵ n for n = 0 , 1 , 2 , . . . is ℵ ω . Let G b e a group and ∆ a pseudometric on G . Sufficient conditions in the next section ar e expressed in ter ms of thr e e car dinal functions: 3 • d (∆), the ∆ -density of G (the smallest ca rdinality of a ∆-dense subset of G ); • η ♯ (∆), the smallest car dinality of a set P ⊆ G s uch that G = [ p ∈ P { x ∈ G | ∆( p, x ) ≤ 1 } ; • η (∆), the smallest cardinalit y of a set P ⊆ G for which there exists a finite set Q ⊆ G such that G = [ q ∈ Q [ p ∈ P { x ∈ G | ∆( p, qx ) ≤ 1 } . The following lemma collec ts ba sic facts ab out these three functions. Pro o fs follow directly from the definition. Lemma 3.1 L et ∆ b e a pseudometric on a gr oup G . L et B = { x ∈ G | ∆( e, x ) ≤ 1 } , wher e e is the identity element of G . 1. η (∆) ≤ η ♯ (∆) ≤ d (∆) . 2. d (∆) = lim k →∞ η ♯ ( k ∆) . 3. If ∆ ′ is another pseudometric on G such that ∆ ≤ ∆ ′ then η (∆) ≤ η (∆ ′ ) , η ♯ (∆) ≤ η ♯ (∆ ′ ) and d (∆) ≤ d (∆ ′ ) . 4. If ∆ is left-invariant and η (∆) ≥ ℵ 0 then η (∆) = η ♯ (∆) . 5. If ∆ is right-invariant then η ♯ (∆) is the smal lest c ar dinality of a set P ⊆ G such that G = B P and η (∆) is the smal lest c ar dinality of a set P ⊆ G for which ther e exists a finite set Q su ch t hat G = QB P . It is easy to see that a top olo gical group G is precompact if and o nly if η ♯ (∆) is finite for each ∆ ∈ RP ( G ). Part 1 in Theor e m 3.3 below yields a stronger sta tement: G is precompa c t if and only if η (∆) is finite for e ach ∆ ∈ RP ( G ). This is equiv alent to the theor em of Usp enskij ([14], p. 338; [15], p. 1 581), for which a simple pro of was given b y Bouzia d and T roa llic [3]. F erri and Neufang [5] gave another pro of using a result of Protasov ([13], Th. 11 .5.1). The case of finite P in the next Lemma is due to B ouziad and T ro allic ([3 ], Lemma 4.1). The pro of below is a straightforward gener alization of their approa ch, which in turn w as adapted from Neumann [1 1]. Lemma 3.2 L et G b e a gr oup, P ⊆ G , and A k ⊆ G for 1 ≤ k ≤ n . If G = S n k =1 A k P then ther e ar e a set P ′ ⊆ G and j , 1 ≤ j ≤ n , su ch that G = A − 1 j A j P ′ and • if P i s finite t hen so is P ′ , and • if P i s infinite t hen | P ′ | ≤ | P | . 4 Pro of pro ceeds by induction in n . When n = 1 , the statement is tr ue with j = 1 and P ′ = P . F or the induction s tep, let m ≥ 1 a nd assume that the statemen t in the lemma is true for n = m . Let P ⊆ G and A 1 , A 2 , . . . , A m +1 ⊆ G b e such that G = S m +1 k =1 A k P . If G = A − 1 m +1 A m +1 P then set j = m + 1 and P ′ = P . On the other hand, if G 6 = A − 1 m +1 A m +1 P then take any x ∈ G \ A − 1 m +1 A m +1 P . It follows tha t A m +1 x ∩ A m +1 P = ∅ , and A m +1 ⊆ S m k =1 A k P x − 1 . Thus G = m [ k =1 A k ( P ∪ P x − 1 P ) . By the induction hypo thesis, there are P ′ ⊆ G and j , 1 ≤ j ≤ m , such that G = A − 1 j A j P ′ , P ′ is finite if P is, and | P ′ | ≤ | P | if P is infinite. Thu s in either c ase the statement holds for n = m + 1.  Theorem 3.3 L et G b e a t op olo gic al gr oup, and ∆ ∈ RP ( G ) . 1. If η (∆) is fi nite then η ♯ ( 1 2 ∆) is fi nite. 2. If η (∆) is infi n ite then η ♯ ( 1 2 ∆) ≤ η (∆) . Pro of. Let B = { x ∈ G | ∆( e, x ) ≤ 1 } , where e is the identit y elemen t of G . If y , z ∈ B then y − 1 z ∈ { x ∈ G | 1 2 ∆( e, x ) ≤ 1 } , b ecause ∆( e, y − 1 z ) = ∆( z − 1 , y − 1 ) ≤ ∆( z − 1 , e ) + ∆( y − 1 , e ) = ∆( e, z ) + ∆( e, y ) ≤ 2 . By part 5 of Lemma 3.1, there ar e sets P , Q ⊆ G such tha t Q is finite, | P | = η (∆) and G = QB P . By Lemma 3 .2, there are q ∈ Q a nd P ′ ⊆ G such that P ′ is finite if P is, | P ′ | ≤ | P | if P is infinite, and G = ( q B ) − 1 q B P ′ = B − 1 B P ′ ⊆ { x ∈ G | 1 2 ∆( e, x ) ≤ 1 } P ′ which sho ws that η ♯ ( 1 2 ∆) ≤ | P ′ | . If η (∆) is finite then | P | and | P ′ | are finite a nd therefore η ♯ ( 1 2 ∆) is finite. If η (∆) is infinite then η ♯ ( 1 2 ∆) ≤ | P ′ | ≤ | P | = η (∆).  Corollary 3.4 L et G b e a t op olo gic al gr oup, and ∆ ∈ RP ( G ) . If d (∆) > ℵ 0 then d (∆) = lim k →∞ η ( k ∆) . Pro of. Combine parts 1 and 2 of Lemma 3.1 with part 2 of Theo rem 3.3.  Let κ b e an infinite cardinal. F ollowing the terminolog y of Guran [6] (see a lso se c. 9 in [1]), say that a topolo gical gr oup G is κ -b ounde d if for ev e r y neighbourho o d U o f the identit y element in G there exists a set H ⊆ G such that | H | ≤ κ and U H = G . 5 Lemma 3.5 L et κ b e an i n finite c ar dinal. The fol lowing c onditions for a top olo gic al gr oup G ar e e quivalent: (i) G is κ -b oun de d; (ii) d (∆) ≤ κ for every ∆ ∈ RP ( G ) ; (iii) η ♯ (∆) ≤ κ for every ∆ ∈ RP ( G ) ; (iv) η (∆) ≤ κ for every ∆ ∈ RP ( G ) . Pro of. The family o f all sets { x ∈ G | ∆( e, x ) ≤ 1 } , where ∆ ∈ RP ( G ), is a ba sis of neighbour- ho o ds of the identit y elemen t e in G . Therefor e (i) ⇔ (iii), by pa rt 5 in Lemma 3.1. (ii) ⇒ (iii) ⇒ (iv) by par t 1 in Lemma 3.1, and (iv) ⇒ (ii) by Co rollary 3.4.  Say that a top olo gical g roup G is lo c al ly κ - b ounde d if its identit y element ha s a neighbour- ho o d U such that for each ∆ ∈ RP ( G ) ther e is a ∆-dense subset H of U , | H | ≤ κ . When κ is an infinite ca rdinal, every lo cally compa ct gr o up is lo cally κ - bo unded, and so is every κ -b ounded group. 4 Sufficien t conditions This sectio n contains s everal sufficient conditions for a top olog ical gr o up to b e a mbit a ble. F or each s uch condition w e pr ov e a sligh tly stronger prop erty; namely , that for ev er y ∆ ∈ RP ( G ) there exists ∆ ′ ∈ RP ( G ), ∆ ′ ≥ ∆ such that BLip + (∆ ′ ) is an a mb it. The key result is Lemma 4 .3, which is another form of the factorization theorems o f Neufang [10] and F er ri and Neufang [5]. Lemma 4.1 L et ∆ b e a pseudometric on a gr ou p G such that η (∆) ≥ ℵ 0 . L et A b e a set of c ar dinality η (∆) , and for e ach α ∈ A let F α b e a n on- empty finite subset of G . Then ther e ex ist elements x α ∈ G for α ∈ A such that ∆( F α x α , F β x β ) > 1 whenever α, β ∈ A , α 6 = β . Pro of. Without loss of gener ality , as sume that A is the set o f o rdinals smaller tha n the first ordinal of cardinality η (∆). The construction of x α pro ceeds by transfinite induction. F or γ ∈ A , let S ( γ ) b e the statement “there exist elements x α ∈ G for all α ≤ γ such that ∆( F α x α , F β x β ) > 1 whenever α < β ≤ γ .” An y c hoice o f x 0 ∈ G makes S (0) tr ue. Now assume that γ ∈ A , γ > 0, and S ( γ ′ ) is true for all γ ′ < γ . W e want to prov e S ( γ ). Since η (∆) ≥ ℵ 0 and the cardinalit y of γ is less tha n η (∆), from the definition of η (∆) we get G 6 = [ q ∈ F γ [ α<γ [ p ∈ F α x α { x ∈ G | ∆( p, qx ) ≤ 1 } . Thu s there exists x γ ∈ G such that ∆( p, q x γ ) > 1 for a ll q ∈ F γ and all p ∈ F α x α where α < γ . That means ∆( F α x α , F γ x γ ) > 1 for a ll α < γ .  Lemma 4.2 L et G b e a t op olo gic al gr oup, ∆ ∈ RP ( G ) and η (∆) ≥ ℵ 0 . If O is a c ol le ction of non-empty op en subsets of BLip + (∆) and |O| ≤ η (∆) , then ther e exists f ∈ BLip + (∆) such that orb( f ) interse cts every set in O . 6 Pro of. Without loss of g e nerality , a ssume that every set in O is a basic neighbour ho o d. Th us each U ∈ O is of the for m U = { f ∈ B Lip + (∆) | | f ( x ) − h U ( x ) | < ε U for x ∈ F U } where F U ⊆ G is a finite set, h U ∈ BLip + (∆), a nd ε U > 0. By Lemma 4.1 with O in place of A , there are x U ∈ G for U ∈ O such that ∆( F U x U , F V x V ) > 1 whenever U , V ∈ O , U 6 = V . Define the function f : G → R by f ( x ) = sup V ∈ O max y ∈ F V ( h V ( y ) − ∆( x, y x V ) ) + for x ∈ G. Each function \ x ( h V ( y ) − ∆( x, y x V ) ) + belo ngs to BLip + (∆), and thus f ∈ BLip + (∆). It remains to be pro ved that f ( xx U ) = h U ( x ) for ev er y U ∈ O and x ∈ F U . Once that is established, it will follow that ρ x U ( f ) ∈ U for every U ∈ O . T ake any U ∈ O and x ∈ F U . F rom the definition of f w e get f ( xx U ) ≥ h U ( x ). T o prov e the opp osite inequality , consider any V ∈ O and any y ∈ F V . Case I: V = U . F rom h U ( y ) − h U ( x ) ≤ | h U ( y ) − h U ( x ) | ≤ ∆( x, y ) ≤ ∆( xx U , y x U ) a nd h U = h V , x U = x V , we get ( h V ( y ) − ∆( xx U , y x V )) + ≤ h U ( x ). Case I I: V 6 = U . F r om ∆( xx U , y x V ) > 1 we get ( h V ( y ) − ∆( xx U , y x V )) + = 0 ≤ h U ( x ). Thu s ( h V ( y ) − ∆( xx U , y x V )) + ≤ h U ( x ) in bo th cases , and now f ( xx U ) ≤ h U ( x ) follows from the definition of f .  Lemma 4.3 L et G b e a top olo gic al gr oup and ∆ ∈ RP ( G ) . If d (∆) = η (∆) ≥ ℵ 0 then ther e exists f ∈ BLip + (∆) such that B L ip + (∆) = orb( f ) . Pro of. Let H b e a ∆-dense subset of G suc h that | H | = d (∆) = η (∆). On BLip + (∆), the top ology of point wise con vergence o n G co incides with the top ology of point wise con vergence on H . Th us BLip + (∆) is homeomor phic to a subset of the pro duct space R H , its to po logy has a base of cardinality at most η (∆), and by Le mma 4 .2 ther e is f ∈ BLip + (∆) whose orbit int er sects every nonempt y op en set in BLip + (∆).  In this pap er, Lemma 4.3 is the key for finding sufficient co nditions for ambitabilit y . If for every ∆ ∈ RP ( G ) there exists ∆ ′ ∈ RP ( G ) such that ∆ ′ ≥ ∆ and d (∆ ′ ) = η (∆ ′ ) ≥ ℵ 0 then the group G is ambitable. Theorem 4.4 L et κ b e an infin ite c ar dinal, and G a lo c al ly κ -b ounde d top olo gic al gr oup. If ther e exists ∆ 0 ∈ RP ( G ) such that η ♯ (∆ 0 ) ≥ κ then G is ambi t able. Pro of. T ake any ∆ ∈ RP ( G ). Let e b e the identit y element of G , and B = { x ∈ G | ∆( e, x ) ≤ 1 } . Without loss of generality , assume that η ♯ (∆) ≥ κ and B ha s a ∆-dense subset H such that | H | ≤ κ . (If ∆ do es not hav e these prop er ties then replace ∆ by a la rger pseudometric in RP ( G ) that do es.) By Lemma 3.1, there is P ⊆ G such that | P | = η ♯ (∆) a nd G = B P . The set H P is ∆-dense in G , therefor e d (2∆) = d (∆) ≤ κ · η ♯ (∆) = η ♯ (∆). Th us d (2∆) ≤ η (2∆) by Theo rem 3.3, and by Lemma 4.3 there is f ∈ BLip + (2∆) such that BLip + (2∆) = orb( f ).  7 Corollary 4.5 L et κ b e an infinite c ar dinal. Every top olo gic al gr oup that is lo c al ly κ + -b ounde d and not κ -b ou n de d is ambitable. Pro of. Let G b e lo ca lly κ + -b ounded and not κ -b ounded. By Lemma 3.5 there is ∆ ∈ RP ( G ) for which η ♯ (∆) ≥ κ + , and Theo rem 4.4 applies with κ + in place o f κ .  Theorem 4.6 When n is a p ositive inte ger, every lo c al ly ℵ n -b ounde d top olo gic al gr oup is either pr e c omp act or ambitable. Pro of. Let G b e lo cally ℵ n -b ounded for some n . Let m ≥ 0 b e the smalle s t int e g er for which G is lo cally ℵ m -b ounded. If m ≥ 1 then G is ambitable by Corollar y 4.5 with κ = ℵ m − 1 . If m = 0 and G is no t precompact then there exists ∆ ∈ RP ( G ) such that η ♯ (∆) ≥ ℵ 0 and G is ambitable by Theor em 4.4.  Corollary 4.7 Every lo c al ly c omp act top olo gic al gr oup is either c omp act or ambitable. Corollary 4.8 Every ℵ 0 -b ounde d top olo gic al gr oup is either p re c omp act or ambitable. Lemma 4.3 yields also o ther cla s ses of ambitable groups, s uch a s those in the next tw o theorems. Theorem 4.9 If G is the additive gr oup of an infinite norme d sp ac e with t he norm top olo gy then G is a m bitable. Pro of. Let k · k be the no rm, and ∆( x, y ) = k x − y k , x, y ∈ G . The top ology of G is defined by the metric ∆. Since η (∆) = η ( k ∆) ≥ ℵ 0 for k = 1 , 2 , . . . , it follows that d (∆) = η (∆) b y Lemma 3.1. Let ∆ ′ be any pseudometr ic in RP ( G ). Without loss of g enerality , assume that ∆ ′ ≥ ∆ (if not then replace ∆ ′ by ∆ + ∆ ′ ). Since η (∆ ′ ) ≥ η (∆) = d (∆) ≥ d (∆ ′ ), by Lemma 4 .3 there exists f ∈ BLip + (∆ ′ ) such that BLip + (∆ ′ ) = orb( f ).  Let κ b e an infinite cardina l. Define cf( κ ), the c ofinality o f κ , to b e the smallest cardinality of a set A of ca r dinals such that κ ′ < κ for ea ch κ ′ ∈ A and sup A = κ . Jech ([7], 1 .3) discusses cofinality in detail. Theorem 4.10 If G is a top olo gic al gr oup and for every ∆ ∈ RP ( G ) ther e exists ∆ ′ ∈ RP ( G ) such that ∆ ′ ≥ ∆ and cf( d (∆ ′ )) > ℵ 0 , then G is ambitable. Pro of. Let ∆ ′ ∈ RP ( G ) be such that cf( d (∆ ′ )) > ℵ 0 . Then d (∆ ′ ) > ℵ 0 , and therefore d (∆ ′ ) = lim k →∞ η ( k ∆ ′ ) by Co rollar y 3.4. Since cf( d (∆ ′ )) > ℵ 0 , it follows tha t d (∆ ′ ) = η ( k ∆ ′ ) for so me k . Thus d ( k ∆ ′ ) = η ( k ∆ ′ ) a nd b y Le mma 4.3 there exists f ∈ BLip + ( k ∆ ′ ) such that BLip + ( k ∆ ′ ) = orb( f ).  Note that cf( ℵ ω ) = ℵ 0 . It is an op en ques tion whe ther every ℵ ω -b ounded topo logical gr oup is either precompact or a m bita ble. 8 5 T op ological cent res Recall [12] tha t for a top ologica l g roup G , • M ( r G ) is the norm dua l of U b ( r G ); • the subspace M u ( r G ) o f M ( r G ) is defined as follows: µ ∈ M u ( r G ) iff µ is con tinuous on BLip + (∆) for ea ch ∆ ∈ RP ( G ); • for x ∈ G , δ x ∈ M ( r G ) is defined b y δ x ( f ) = f ( x ) for f ∈ U b ( r G ); • the mapping δ : x 7→ δ x is a top olo gical embedding of G to M ( r G ) with the weak ∗ top ology; • when µ, ν ∈ M ( r G ), the c onvolution of µ and ν is defined by µ ⋆ ν ( f ) = µ ( \ x ν ( \ y f ( xy ))) for f ∈ U b ( r G ); • r G , the uniform semigr oup c omp actific ation of G , is the w ea k ∗ closure of δ ( G ) in M ( r G ), with the weak ∗ top ology and the conv olution o pe ration ⋆ ; • c r G = r G ∩ M u ( r G ) is a completion of r G . A semig roup S with a to po logy is a right top olo gic al semigr oup if the mapping \ x xy from S to S is contin uous for ea ch y ∈ S ([2], 1.3 ). F or any r ight top ologica l semigroup S , define its top olo gic al c ent r e Λ( S ) = { x ∈ S | the mapping \ y xy is contin uous on S } . The spaces studied here ar e deno ted b y v ar ious s ymbols in the literatur e. Some of the more common notations are: in this pap er alternative notatio ns U b ( r G ) LUC( G ) or R U C ∗ ( G ) or U r ( G ) or LC ( G ) M ( r G ) LUC( G ) ∗ or U r ( G ) ∗ or LC ( G ) ∗ M u ( r G ) Leb( G ) r G G LUC or G LC Λ( M ( r G )) Z ( G ) or Z t ( G ) Let G b e a topo logical group. Then M ( r G ) with the ⋆ oper a tion and the weak ∗ top ology is a right top olo gical semigro up. This semigro up a nd its subs e mig roup r G hav e an imp ortant role in functional ana lysis on G . Significa n t r esearch efforts hav e b een dev o ted to characterizing their top ological cen tres . In the rest of this section w e will see ho w the kno wn results follow from results ab out am bitable groups. The same a pproach yields a characteriza tio n of to po logical centres not only in M ( r G ) and r G but also in an y in ter mediate semigroup b etw een M ( r G ) and r G . Research in abstract ha r monic analys is is often concerned with linear spa ces ov er C , the field of complex num b ers, r ather than the field R used here. How ever, it is an easy exer c ise to derive the C -version of any r esult in this pap er from its R -version. F or every top o lo gical group G w e have M u ( r G ) ⊆ Λ( M ( r G )); s e e Prop osition 4.2 in [5] or s ection 5 in [1 2]. If G is precompac t then M u ( r G ) = M ( r G ) and therefore M u ( r G ) = Λ( M ( r G )). 9 Question 2 Is M u ( r G ) = Λ( M ( r G )) for every top olo gic al gr oup G ? The p ositive answer w a s pr ov ed for lo cally c ompact g roups by Lau [8 ] and for ℵ 0 -b ounded groups by F erri and Neufang [5]. By Cor o llary 5.4 b elow, the answer is p ositive for every ambitable gr oup. The situation is s imila r for Λ( r G ). F or every top olo gical gro up G w e hav e c r G ⊆ Λ( r G ). If G is prec ompact then c r G = r G is a c o mpact gro up and therefore c r G = Λ( r G ). Question 3 Is c r G = Λ( r G ) for every top olo gic al gr oup G ? The p ositive answer was prov ed for lo cally co mpact gr oups by Lau and P y m [9] and for ℵ 0 -b ounded groups by F err i a nd Neufang [5]. Again the a nswer is p ositive for every ambitable group, by Corollar y 5.4. Lemma 5.1 L et G b e a t op olo gic al gr oup and f ∈ U b ( r G )) . 1. The mapping ϕ : ν 7→ \ x ν ( \ y f ( xy )) is c ontinuous fr om r G to the pr o duct sp ac e R G . 2. ϕ ( r G ) = or b( f ) . Pro of. 1. As no ted a bove, δ x ∈ Λ ( r G ) for eac h x ∈ G , and th us the mapping ν 7→ δ x ⋆ ν is weak ∗ contin uous from r G to itself. Since δ x ⋆ ν ( f ) = ν ( \ y f ( xy )), this means that the mapping ν 7→ ν ( \ y f ( xy )) fro m r G to R is co nt inuous for each x ∈ G , and ther efore the mapping ν 7→ \ x ν ( \ y f ( xy )) is contin uous fro m r G to R G . 2. ϕ ( δ x ) = ρ x ( f ) for all x ∈ G , a nd therefore ϕ ( δ ( G )) = orb( f ). The mapping ϕ is contin uous by pa r t 1, r G is compact, and δ ( G ) is de ns e in r G . It follows that ϕ ( r G ) = orb( f ).  Lemma 5.2 L et G b e any t op olo gic al gr oup, µ ∈ M ( r G ) and f ∈ U b ( r G ) . If the mapping ν 7→ µ ⋆ ν fr om r G to M ( r G ) is we ak ∗ c ontinuous then µ is c ontinu ous on orb( f ) . Pro of. As in Lemma 5 .1, define ϕ ( ν ) = \ x ν ( \ y f ( xy )) for ν ∈ r G . r G orb( f ) R ✲ ❄ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s ϕ µ ν 7→ µ ⋆ ν ( f ) By the definition of co nvolution, µ ⋆ ν ( f ) = µ ( \ x ν ( \ y f ( xy ))) = µ ( ϕ ( ν )). Thus µ ◦ ϕ is contin uous from r G to R . By Lemma 5.1, ϕ is co ntin uo us from r G to orb( f ), and ϕ ( r G ) = or b( f ). Since r G is compact, it fo llows that µ is contin uous on o r b( f ).  Theorem 5.3 If G is an ambitable t op olo gic al gr oup, S ⊆ M ( r G ) , and S with the ⋆ op er ation is a semigr oup such that r G ⊆ S , then Λ( S ) = M u ( r G ) ∩ S . 10 Pro of. As was noted ab ov e, M u ( r G ) ⊆ Λ( M ( r G ). Ther efore M u ( r G ) ∩ S ⊆ Λ( S ) for every semigroup S ⊆ M ( r G ). T o pr ov e the opp osite inclusion, ta ke any µ ∈ Λ( S ) and any ∆ ∈ RP ( G ). Since r G ⊆ S , the mapping ν 7→ µ ⋆ ν from r G to M ( r G ) is weak ∗ contin uous by the definition of Λ( S ). Since G is a mbit a ble, BLip + (∆) ⊆ orb( f ) for some f ∈ U b ( r G ). By 5.2, µ is contin uous on orb( f ) and therefore also on BL ip + (∆). Thus µ ∈ M u ( r G ).  Corollary 5.4 If G is an ambi t able top olo gic al gr oup then M u ( r G ) = Λ( M ( r G )) and c r G = Λ( r G ) . Pro of. Apply 5.3 with S = M ( r G ) and with S = r G .  Combined with T heo rem 4.6, Corollar y 5.4 gives p ositive answers to Questions 2 and 3 for lo cally ℵ 0 -b ounded groups, a nd in particular for lo cally co mpact and for ℵ 0 -b ounded groups. This ge ne r alizes the previous ly published r esults cited a bove. Note that Theorem 5 .3 a pplies not only to the semigro ups r G a nd M ( r G ) in Cor ollary 5 .4, but also to many other semigr oups b etw een r G and M ( r G ) — for ex ample, the semigroup of all p ositive element s in M ( r G ), or the semigr o up of all finite linea r combinations of e lement s of r G with integral co efficients. Corollary 5.5 No u niquely amenable top olo gic al gr oup is ambitable. Pro of. If a g r oup G is ambitable then M u ( r G ) = Λ( M ( r G )) by Corolla ry 5.4. If G were also uniquely amenable, then G would be prec o mpact by Theo r em 5.2 in [12], which would contradict The o rem 2.2 ab ove.  Ac kno wl edgement. 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