Automatic norm continuity of weak* homeomorphisms

A UTOMA TIC NORM CONTINUITY OF WEAK ∗ HOMEOM O RPHISMS ANTONIO A VIL ´ ES Abstract. W e pro ve that i n a certain class E of nonseparable Banac h spaces the norm topology of the dual ball is definable in terms of its weak ∗ topology . Th us, if X, Y ∈ E and f : B X ∗ − → B Y ∗ is a we ak ∗ -to-wea k ∗ homeomorphism, then f is automatically norm-to-norm contin uous. 1. Introduction The aim of this no te is to prov e the automatic contin uit y in the norm top olo gy for the weak ∗ homeomorphisms of the dual ball of certain nonsepar able Banach spaces. W e start by observing that such a pro per t y never holds in the separable case. Prop ositio n 1. L et X b e a sep ar able infinite dimensional Banach sp ac e. Then, ther e exists a we ak ∗ -to-we ak ∗ home omorphism f : B X ∗ − → B X ∗ which is n ot norm- to-norm c ontinuous. Pro of: The space ( B X ∗ , w ∗ ) is a metriz a ble infinite-dimensional compact convex set, so it is ho meomorphic to the Hilb ert cub e [0 , 1] ω , b y Keller ’s Theorem. It is a known fact, cf. [4, p. 26 1], that the Hilb ert cub e is countable dense homog enous, which mea ns that if A and B are co un table weak ∗ dense s ubs ets of B X ∗ then there is a weak ∗ homeomorphism f : B X ∗ − → B X ∗ such that f ( A ) = B . Thus, it is eno ugh to find tw o such subsets A a nd B with so me different pro pe r ty relative to the no rm top ology . F or exa mple, A can be taken s o that its norm-clo sure is connected (b y choosing it to b e rationally conv ex) and B with disconnected norm-clo s ure (take B ′ a countable weak ∗ dense set which is not norm dens e and then a dd to B ′ a p oint out of its nor m-closure).  W e shall introduce tw o cla s ses E a nd E 0 of nonsepar able Banach spaces with the prop erties indicated in the tw o fo llowing theore ms . Theorem 2. L et X and Y b e sp ac es in the class E a nd let f : B X ∗ − → B Y ∗ b e a we ak ∗ -to-we ak ∗ home omorphism. Then f is norm-to-norm c ontinuous. 2000 Mathematics Subje ct Classific ation. 46B26. The author w as supported by a Mari e Curie Int ra-Europ ean F elloship MCEIF-CT2006-038768 (E.U.) and researc h pro jects M TM2005-08379 and S ´ eneca 00690/PI/04 (Spain). 1 2 ANTONIO A VIL ´ ES Theorem 3. L et X and Y b e sp ac es in the class E 0 and let f : B X ∗ − → B Y ∗ b e a we ak ∗ -to-we ak ∗ home omorphism. Then, if ( x ∗ n ) is a se quenc e in B X ∗ which we ak ∗ c onver ges to x ∗ and k x ∗ n k − → k x ∗ k , then k f ( x ∗ n ) k − → k f ( x ∗ ) k . The definition of these classes r equires some prelimina ry work a nd will b e g iven later but we can alr eady indicate s o me examples o f spaces which we know that belo ng in there. The most significant repr e sent atives in E ∩ E 0 are the spaces c 0 (Γ) and ℓ p (Γ) for 1 < p < ∞ , Γ b eing an uncountable set o f indices. More generally , any uncountable c 0 -sum or ℓ p -sum (1 < p < ∞ ) of s eparable spa ces b elongs to E 0 , and any space from E 0 with the dual Kadec - Klee prop erty b elo ngs to E . Recall that X has the dual Kadec- Klee pr op erty if whenever we have a weak ∗ conv ergent sequence ( x ∗ n ) in the dual such that the sequence of norms ( k x ∗ n k ) co n verges to the norm o f the limit, then ac tually the sequence is nor m- conv ergent. In addition, the class E is c lo sed under finite ℓ 1 -sums. On the other hand, one can show tha t ℓ 1 (Γ) 6∈ E ∪ E 0 , indeed: Prop ositio n 4. F or any infinite set Γ , ther e is a we ak ∗ -to-we ak ∗ home omorphism f : B ℓ 1 (Γ) ∗ − → B ℓ 1 (Γ) ∗ which is n ot norm-t o-n orm c ont inuous and do es not pr e- serve the limit of norms of we ak ∗ c onver gent se quenc es. Pro of: W e know that ℓ 1 (Γ) ∗ = ℓ ∞ (Γ) and hence B ℓ 1 (Γ) ∗ ≈ [ − 1 , 1] Γ where the weak ∗ top ology is identified with the p oint wise top ology . Consider elements e 1 = (1 , 0 , 0 , . . . ), e 2 = (0 , 1 , 0 , 0 , . . . ), etc. in [ − 1 , 1] Γ . W e can eas ily define co o r- dinatewise a homeomorphism f : [ − 1 , 1] Γ − → [ − 1 , 1] Γ taking f ( 1 2 n e n ) = 1 2 e n and f (0) = 0. The sequences ( 1 2 n e n ) and its image ( 1 2 e n ) weak ∗ conv erge to 0, but the fir st is no r m co nv ergent and the other is not, not even the sequence of no r ms conv erges to 0.  Apart from this extreme example, in fact it happ ens that these prop erties ar e sensitive to renor mings: spaces like c 0 ⊕ ℓ 1 c 0 (Γ) and ℓ p ⊕ ℓ 1 ℓ p (Γ) fail this automatic norm-contin uity prop e rty despite the fact that they ar e iso morphic to c 0 (Γ) and ℓ p (Γ), 1 < p < ∞ . Prop ositio n 5. L et S b e an infinite dimensional sep ar able Banach sp ac e and Y b e a n onsep ar able Banach sp ac e. L et X = S ⊕ ℓ 1 Y . Then, t her e exists a we ak ∗ - to-we ak ∗ home omorphism f : B X ∗ − → B X ∗ which is not norm-to-norm c ont inuous. Pro of: Notice that B X ∗ = B S ∗ × B Y ∗ and then apply P rop osition 1.  Let us p oint o ut wher e the difficulties a ppea r in proving Theor e m 2 for s ay the space X = c 0 (Γ). Suppos e that f : B X ∗ − → B X ∗ is a weak ∗ homeomorphism. It is a well known fact that the G δ po int s of B X ∗ are exactly the p oints of the spher e S X ∗ , hence f ( S X ∗ ) = S X ∗ . W e know in additio n that at the p oints o f the sphere the no r m and weak ∗ top ology coincide , so we co nclude that f is norm- contin uous at all the p oints of the sphere. And this is all that the usual sta ndard functional- analytic techniques can say to us. In order to get norm-c ontin uity at all p oints AUT OMA TIC NORM CONTINUITY OF WEAK ∗ HOMEOMORPHISM S 3 we shall a need a more p ow erful to o l co ming fr o m top ology: Shchepin’s s p ectr a l theory . This will allow us to define a certa in notion of conv ergence of a sequence in a compact space of unco un table weigh t, that we called fib er-co n vergence. Of course, fiber co n vergent sequences are resp e cted by homeo morphisms, though not by gene r al co nt inuous functions. Banach spa c es in class E are those for which the fiber conv erge n t sequences in ( B X ∗ , w ∗ ) are exactly the nor m c onv ergent sequences, while the cla ss E 0 consists of those spaces in which the fiber conv ergent s e quences of the dual ball are those weak ∗ conv ergent sequences whose s equence of nor ms conv erges to the no rm o f the limit. 2. Spectral theor y In this section, we summa r ize in a self-contained wa y what we need ab out s p ec- tral theor y , in the same wa y as it is exp os ed in our joint work with K alenda [1], which at the same time is a refor mulation of the idea s fro m [2] and [3] in a suitable language. Although this preliminary mater ial app ears alr eady in [1] with more de- tails, we found it conv enient to repr o duce it here. Let K b e a compact space. W e denote by Q ( K ) the set of all Hausdorff quotient spaces of K , that is the set of a ll Ha usdorff compact s paces of the form K /E en- dow ed with the quotient top o logy , for E an equiv a le nc e re la tion on K . An element of Q ( K ) can b e r epresented either b y the equiv a le nce relation E , o r by the quotient space L = K /E toge ther with the canonica l pro jection p L : K − → L . On the se t Q ( K ) there is a natura l order relation. In terms of eq uiv alence r e- lations E ≤ E ′ if a nd only if E ′ ⊂ E . Equiv a lently , in terms o f the quotient spaces, L ≤ L ′ if and only if ther e is a co ntin uous sur jection q : L ′ − → L such tha t q p L ′ = p L . T he set Q ( K ) endow ed with this o r der relation is a complete lattice, that is, every subset has a least upp er b ound or supr em um: if F is a family o f equiv a le nce rela tions o f Q ( K ), its leas t upp er b ound is the relation given by xE 0 y iff xE y for all E ∈ E , in other words E 0 = sup F = T F . It is easy to c heck that E 0 gives a Hausdor ff quotient if each element of F do es. Let Q ω ( K ) ⊂ Q ( K ) b e the family of a ll quo tien ts o f K which ha ve countable weigh t. Notice tha t sup A ∈ Q ω ( K ) for every countable subset A ⊂ Q ω ( K ) and also that sup Q ω ( K ) = K . A family S ⊂ Q ω ( K ) is called cofinal if for every L ∈ Q ω ( K ) there exists L ′ ∈ S such that L ≤ L ′ . The family S is ca lled a σ -semilattice if for every co un table subset A ⊂ S , the lea st upp er b ound of A b elong s to S . Theorem 6 (A version of Shchepin’s s p ectr a l theorem) . L et K b e a c omp act sp ac e of unc ountable weight and let S and S ′ two c ofinal σ -semilattic es in Q ω ( K ) . Then S ∩ S ′ is also a c ofinal σ - semilattic e in Q ω ( K ) . It is not so obvious to check whether a given σ -s emilattice is co final, so this theorem m ust b e applied tog ether with the following criterio n: 4 ANTONIO A VIL ´ ES Lemma 7. L et K b e a c omp act sp ac e of unc ountable weight and S a σ - semilattic e in Q ω ( K ) . Then, S is c ofinal if and only if sup S = K . The imp orta nce of this machinery is that it allows o ne to study a compact s pa ce of uncountable weight through the study of a cofinal σ -s emilattice of metr izable quotients and, in particular, thro ugh the natural pro jections betw een elements of the σ -semila ttice. In this way , the study of compa ct spaces of uncountable weigh t is related to the study o f contin uous surjections b etw een compa ct spaces of co un table weigh t. The following lang uage will b e useful: Definition 8. Let K b e a compact space of uncountable weigh t and le t P b e a prop erty . W e say that the σ -typical s ur jection o f K satisfies pro pe r ty P if there exists a cofinal σ -semila ttice S ⊂ Q ω ( K ) such that for every L ≤ L ′ elements of S , the natural pro jection p : L ′ − → L satisfies prop erty P . The sp ectral theorem has the following conse q uence: In order to chec k whether the σ -typical surjection of K ha s a certain pr op erty , it is e no ugh to do it on any given cofina l σ -semila ttice, na mely: Theorem 9. L et K b e a c omp act sp ac e of unc ountable weight, let P b e a pr op erty, and let S b e a fixe d c ofinal σ -semilattic e in Q ω ( K ) . Then the σ -typic al surje ction of K has pr op erty P if and only if ther e exists a c ofinal σ -semilattic e S ′ ⊂ S s u ch that for every L ≤ L ′ elements of S ′ , the natu r al pr oje ction p : L ′ − → L satisfies pr op erty P . When the compact space we are dealing with is the dual unit ball B X ∗ of a non- separable Bana ch X in the weak ∗ top ology , then a cofinal σ -semilattice in Q ω ( B X ∗ ) can b e obtained from a suitable family of sepa rable subspa c es of X . Prop ositio n 10 . L et F b e a family of sep ar able subsp ac es of X such that (1) span ( S F ) = X , and (2) if F ′ ⊂ F is a c ount able subfamily, then span ( S F ′ ) ∈ F . Then the family { B Y ∗ : Y ∈ F } is a c ofinal σ - semilattic e in Q ω ( B X ∗ ) . Notice that we view B Y ∗ as a quotient of B X ∗ for Y ⊂ X thro ugh the natural restriction map. The pro o f o f the prop osition is stra ightf orward. In particular, cofinality follows fro m the first condition and Lemma 7. 3. Fiber convergence Definition 11. Let π : K − → L b e a co n tinuous sur jection, and let ( x n ) b e a se - quence of elemen ts o f L co nv erging to x ∈ L . W e s ay that x n is π -fib er conv ergent if for every y ∈ π − 1 ( x ) there exist elements y n ∈ π − 1 ( x n ) s uch that y n conv erges to y . AUT OMA TIC NORM CONTINUITY OF WEAK ∗ HOMEOMORPHISM S 5 Definition 12. Let K b e a compact space o f uncountable weight and let ( x n ) b e a seque nce in K that con verges to x ∈ K . W e say tha t the sequence ( x n ) is fib er conv ergent if for the σ -typical sur jection π : L − → L ′ , the image of the sequence in L ′ , ( π L ′ ( x n )) n<ω , is π -fib er c o nv ergent. Definition 13. A nonseparable Ba nach space b elo ngs to the class E if the fibe r conv ergent sequences of ( B X ∗ , w ∗ ) a re e x actly the nor m convergen t seq uences. Definition 14 . A nonsepar a ble B anach s pace b elongs to the class E 0 if the fiber conv ergent sequences of ( B X ∗ , w ∗ ) a re e x actly thos e sequence s ( x ∗ n ) weak ∗ conv er- gent to a p oint x ∗ ∈ B X ∗ such that k x ∗ n k − → k x ∗ k . Notice that Theorems 2 and 3 are immediate cons e q uence of the definitions, b e- cause the notion of a fib er- conv ergent sequence is an intrinsic top olo g ical notion and hence it is preserved under homeomo r phisms. Lemma 15. L et X and Z b e Banach s p ac es and 1 ≤ p ≤ ∞ . Set Y = X ⊕ ℓ p Z and let π : B Y ∗ − → B X ∗ b e the r estriction map dual t o the n atur al inclusion X ⊂ Y . • If p = 1 t hen every we ak ∗ c onver gent se quenc e in B X ∗ is π -fi b er c onver gent. • Su pp ose that 1 < p ≤ ∞ and ( x ∗ n ) is a se quenc e in B X ∗ that we ak ∗ c on- ver ges to x ∗ 0 . Then ( x ∗ n ) is π -fi b er c onver gent if and only if the se quenc e of norms ( k x ∗ n k ) c onver ges to k x ∗ 0 k . Pro of: Let ( x ∗ n ) b e a seq uence in B X ∗ that weak ∗ conv erges to x ∗ 0 , and let y ∗ 0 = x ∗ 0 + z ∗ 0 ∈ π − 1 ( x ∗ 0 ). If p = 1, then k x ∗ + z ∗ k = ma x ( k x ∗ k , k z ∗ k ) for every x ∗ ∈ X ∗ and z ∗ ∈ Z ∗ , hence it is enough to take y ∗ n = x ∗ n + z ∗ 0 to re alize that ( x ∗ n ) is ac tua lly π -fib er conv ergent. If 1 < p ≤ ∞ , then no rms in the dual are computed as k x ∗ + z ∗ k = ( k x ∗ k q + k z ∗ k q ) 1 q , p − 1 + q − 1 = 1 Suppo se that the sequence of norms ( k x ∗ n k ) co n verges to k x ∗ 0 k a nd let y ∗ 0 = x ∗ 0 + z ∗ 0 be an ar bitrary elemen t of the fib er of x ∗ 0 . W e will find elements y ∗ n ∈ π − 1 ( x ∗ n ) such that y ∗ n − → y ∗ 0 . If z ∗ 0 = 0, then we can simply take y ∗ n = x ∗ n . Th us, we supp ose that z ∗ 0 6 = 0 a nd we define λ n = max { λ ∈ [0 , 1] : k x ∗ n + λz ∗ 0 k ≤ 1 } and y ∗ n = x ∗ n + λ n z ∗ 0 . W e hav e to chec k that λ n − → 1. Suppos e on the contrary that for s o me subseq ue nce and some µ < 1 we hav e λ n k < µ . Then ( k x ∗ n k k q + µ q k z ∗ 0 k q ) 1 q = k x ∗ n k + µz ∗ 0 k > 1 so pa ssing to the limit k x ∗ 0 + z ∗ 0 k = ( k x ∗ 0 k q + k z ∗ 0 k q ) 1 q > ( k x ∗ 0 k q + µ q k z ∗ 0 k q ) 1 q ≥ 1 which is a contradiction. 6 ANTONIO A VIL ´ ES Conv ersely , ass ume now that the seque nc e of norms k x ∗ n k does not conv erge to k x ∗ 0 k . Passing to a subseque nc e , we can supp ose without lo s s of gener ality that there is a num ber µ such that k x ∗ 0 k < µ ≤ k x ∗ n k for every n . Let ξ ∈ [0 , 1] b e such that k x ∗ 0 k q + ξ q = 1, a nd let z ∗ 0 be any vector of Z ∗ of nor m ξ . W e claim that there is no sequence ( x ∗ n + z ∗ n ) ⊂ B Y ∗ that conv erges to x ∗ 0 + z ∗ 0 . If it were the ca se, then z ∗ n − → z ∗ 0 , so sup {k z ∗ n k : n ∈ ω } ≥ k z ∗ 0 k = ξ , s o sup {k x ∗ n + z ∗ n k : n ∈ ω } ≥ ( µ q + ξ q ) 1 q > ( k x ∗ 0 k q + ξ q ) 1 q = 1 which co ntradicts that x ∗ n + z ∗ n ∈ B Y ∗ for every n .  Theorem 16 . L et { X α : α ∈ A } b e an unc ountable family of sep ar able Banach sp ac es. (1) The c 0 -sum L c 0 { X α : α ∈ A } b elongs to E 0 , and if 1 < p < ∞ then also L ℓ p { X α : α ∈ A } b elongs to E 0 . (2) Al l the we ak ∗ c onver gent se quenc es of the dual b al l of L ℓ 1 { X α : α ∈ A } ar e fib er c onver gent. Pro of: Let X 0 = L α ∈ A X α , where the t yp e of direct sum is the suitable one in each case. Consider I the family of all countable subs ets of A and set X i = L α ∈ i X α for i ∈ I , and F = { X i } i ∈ I . This family satisfies conditions (1) and (2) in Prop o- sition 10, and therefore S = { B Y ∗ : Y ∈ F } is a cofina l σ -semilattice in Q ω ( B X ∗ 0 ). W e notice that all the natural pro jections b etw een elements of S corr esp ond to the dual restric tion map of an inclusio n of Banach spaces of type X ⊂ X ⊕ ℓ p Z , so that Lemma 15 indicates which ar e exactly the π -fiber convergen t seq uences in all thos e cases. Le t us fo cus on pa r t (1). Let ( x ∗ n ) ⊂ B X ∗ be a sequence w eak ∗ conv ergent to x ∗ 0 . F or i ∈ I we denote by π i : B X ∗ 0 − → B X ∗ i the natural surjection dua l to the inclusion X i ⊂ X 0 . Let k b e a countable subset of A such that k x ∗ n k = k π k ( x ∗ n ) k for all n . W e hav e then that k x ∗ n k − → k x ∗ 0 k if and only if k π j ( x ∗ n ) k − → k π j ( x ∗ 0 ) k for a ll j ⊃ k , and by L e mma 15 if and only if ( x ∗ n ) is fib er convergen t.  Let K K ∗ denote the cla ss o f Banach spaces w ith the dual Ka dec-Klee pro pe rty . Then, no tice that E 0 ∩ E = E 0 ∩ K K ∗ . Since c 0 (Γ) and ℓ p (Γ) (1 < p < ∞ ) hav e K K ∗ , we g o t that these spaces b elong to E and satisfy Theorem 2. Lemma 17 . L et K = Q K i b e a finite or c ountable pr o duct of c omp act sp ac es of unc ountable weight and ( x n ) a c onver gent se quenc e in K . Then, t his se quenc e is fib er c onver gent in K if and only if e ach of the c o or dinate se quenc es ( x n ( i )) n<ω is fib er c onver gent in K i . Pro of: First of all it is straightforw ar d to chec k that if { π i : X i − → Y i } is an y family of contin uous s ur jections, then a sequence ( y n ) ⊂ Q Y i is Q π i -fib er conv er- gent if a nd o nly if each c o ordinate seq uence is π i -fib er conv ergent. Now, go ing back to the statement of the lemma, supp ose that every co ordinate sequence ( x n ( i )) n<ω is fib er conv erg en t. Let S i be a cofinal σ -s e milattice in Q ω ( K i ) such that in all surjections π inside S i , the pro jectio n of the s equence ( x n ( i )) is π -fib er co n vergent. Let S be the cofinal σ -semila ttice in Q ω ( K ) formed by all quo tien ts of the form AUT OMA TIC NORM CONTINUITY OF WEAK ∗ HOMEOMORPHISM S 7 Q π i : Q K i − → Q L i where L i ∈ S i . Then, in all surjections π inside S the pro- jection of the sequence ( x n ) is π -fib er co nvergen t. Conv ersely , supp ose that ( x n ) is fiber conv ergent and consider S the cofinal σ -semila ttice in Q ω ( K ) formed by all quotients which ar e pro ducts of quotients in each coo rdinate. There exists a cofinal σ -semilattice T ⊂ S such that in all surjections π inside T the pro jection of the sequence ( x n ) is π -fib er co n vergent. F or every i , we consider T i to b e the set of all quotients L of K i such that there is some quotient Q j L j with L i = L . Then T i is a cofina l σ -semilattice of Q ω ( K i ) and, b y the observ ation at the b eginning of this pro of, in all surjections π inside T i the pro jection of ( x n ( i )) is π -fib er conv ergent.  Prop ositio n 18. L et { X n : n < ω } b e a c oun t ably infi n ite family of n onsep ar able Banach sp ac es. Then L ℓ 1 { X n : n < ω } b elongs neither to E 0 nor t o E . Pro of: Notice that B X ∗ = Q n<ω B X ∗ n . F or x n an elemen t of the sphere of X ∗ n , the sequence ( x 0 , 0 , 0 , . . . ), (0 , x 1 , 0 , . . . ), . . . is fib er co n vergent to 0 but no t norm conv ergent.  Prop ositio n 19 . If X , Y ∈ E , then X ⊕ ℓ 1 Y ∈ E . Pro of: Let Z = X ⊕ ℓ 1 Y . Notice that B Z ∗ = B X ∗ × B Y ∗ . A seq uence in this pro duct is fib er-co n vergent if a nd only if b oth co ordina tes are fiber -conv ergent a nd the same ha pp ens for norm-convergence.  W e can provide a couple of extra examples. In b oth cases 1 < p < ∞ and c 0 can be s ubstituted for ℓ p : • ℓ p (Γ) ⊕ ℓ 1 ℓ p (Γ) ∈ E \ E 0 . This space b elong s to E by Pro p os ition 19, but it is not in E 0 bec ause it is in E but it fails the dual Kadec -Klee pro p er t y: If ( x ∗ n ) is a sequence inside the unit s phere of ℓ p (Γ) ∗ which weak ∗ conv erges to 0, then the sequence ( x ∗ 0 , x ∗ n ) s hows that K K ∗ do es not ho ld. • An uncountable ℓ p -sum of copies of ℓ p ( ω ) ⊕ ℓ 1 ℓ p ( ω ) b elongs to E 0 \ E . This space b elongs to E 0 by Theo rem 1 6 but ag ain it is not in E since it is in E 0 but it fails K K ∗ for s imilar r easons as in the previous case. References [1] A. Avil´ es and O. F. K. Kalenda, Fib er or ders and c omp act sp ac es of unc ountable weight , T o appear in F undamen ta M ath. [2] E. V. Shc hepin, T op olo gy of limit sp ac es of unc ountable inverse sp e ctr a . R uss . Math. Surv. 31 (1976), No. 5, 155–191. [3] E. V. Shc hepin, F unctors and unc ountable p owers of c omp acta . Russ. Math. Surv. 36 (1981), No 3, 1–71; [4] J. v an Mi ll, Infinite- dimensional top olo gy. Pr er e quisites and intr o duction. North-Holl and Mathematical Library , 43. North-Holl and, A msterdam. University of P a ris 7, Equipe de Logique Ma th ´ ema tique, UFR de Ma th ´ ema tiques, 2 place Jussieu, 7 5251 P aris, France E-mail addr ess : avileslo@um.es