The unit ball of the Hilbert space in its weak topology

THE UNIT BALL OF THE HILBER T SP A CE IN ITS WEAK TOPOLOGY ANTONIO A VIL ´ ES Abstract. W e show that the unit ball of ℓ p (Γ) i n its w eak topology is a con- tin uous image o f σ 1 (Γ) N and w e d educe some com binatorial properties of its lattice of open sets whic h are not shared b y the balls of other equiv alen t norms when Γ is uncoun table. F or a set Γ and a real n umber 1 < p < ∞ , the Banach s pace ℓ p (Γ) is a reflexive space, hence its unit ball is co mpact in the w eak top olo gy and in fact, it is homeo - morphic to the following closed subset of the Tychonoff cub e [ − 1 , 1] Γ : B (Γ) =    x ∈ [ − 1 , 1] Γ : X γ ∈ Γ | x γ | ≤ 1    . The homeomorphism h : B ℓ p (Γ) − → B (Γ) is g iven b y h ( x ) γ = sig n ( x γ ) · | x γ | p . The spaces homeomorphic to clo sed subsets of so me B (Γ) cons titute the class of uniform Ebe rlein co mpacta, intro duced b y Beny amini a nd Starbird [6]. The s pace σ k (Γ), the compact subset of { 0 , 1 } Γ which consists of the functions with at most k nonze ro co or dinates ( k a p ositive in teger) is an example of a uniform E be rlein compact. In fact, the following r esult was pr ov en in [5]: Theorem 1 (Benyamini, Rudin, W age) . Every uniform Eb erlein c omp act of weight κ is a c o ntinu ous image of a close d subset of σ 1 ( κ ) N . In the same pap er [5], it was p os ed the problem whether in fact, it was p ossi- ble to get any uniform Eb erlein c ompact as a contin uous ima ge o f the full σ 1 (Γ) N . This question was answ ered in the neg ative b y Bell [2], by considering the following prop erty: A co mpact spa ce K verifies pro per ty (Q ) if for every uncountable regula r cardinal λ and every family { U α , V α } α<λ of op en subsets of K with U α ⊂ V α one of the following tw o alternatives must ho ld: (1) either there exists a s et A ⊂ λ with | A | = λ such that U α ∩ U β = ∅ for every tw o different elements α and β in A , 2000 Mathematics Subje ct Classific ation. 46B50, 46B26, 46C05, 54B30, 54D15. Key wor ds and phr ases. Unifor m Eb erlein compact, p olyadic space, ℓ p spaces, Hilb ert space, we ak top ology , equiv alen t norm . Author supported by FPU grant of MEC of Spain. 1 2 ANTONIO A VIL ´ ES (2) or either there exists a set A ⊂ λ with | A | = λ such that V α ∩ V β 6 = ∅ for every tw o different elements α and β in A . Bell prov ed in [2] that prop erty (Q) is satisfied by a ll po lyadic spaces , that is, contin uous image s of σ 1 (Γ) Λ for any sets Γ and Λ, (this co ncept was intro duced in [8 ] a nd studied ea rlier by Ger lits [7 ]), but he constructed a uniform Ebe rlein compact without pro per ty (Q). Later, B ell [4] provided another example of a uni- form E ber lein co mpact which is not a c ontin uous image of a ny σ 1 (Γ) N but w hich is nevertheless p o lyadic. Our main r esult is the follo wing: Theorem 2. B (Γ) is a c ontinuous image of σ 1 (Γ) N . As a conseq uence, B (Γ) satisfies prop erty (Q) as well as other pr op erties o f the same t yp e in tro duced by Bell in [4] and [3]. How ever, if Γ is uncountable, we show in Theorem 4 that a mo difica tion of one of the exa mples of Bell pr ovides an equiv a - lent norm o n ℓ p (Γ) whose unit ball is not a c ontin uous image of σ 1 (Γ) N , indeed no t satisfying prop erty (Q). In particular, we are showing the existence of equiv alent norms in the nonseparable ℓ p (Γ) whose closed unit balls are not homeomorphic in the weak top ology . This contrasts with the separable case, since the balls of a ll separable reflexive B anach spaces are weakly homeomorphic [1, Theore m 1 .1]. W e refer to [1] for information ab out the problem whether the balls of equiv alent norms in a Banach space are weakly homeomo rphic in the s eparable ca se. Pro of of Theorem 2: F o r a set ∆ we will us e the notation B + (∆) = B (∆) ∩ [0 , 1] ∆ . First, we p oint out that B (Γ) is a c ontin uo us image of B + (Γ). Indeed, if we con- sider Γ o = Γ × { a, b } , w e hav e a contin uous surjection ψ : B + (Γ o ) − → B (Γ) g iven by ψ ( x ) γ = x ( γ , a ) − x ( γ , b ) . In a second step, w e apply the standard proce dure to express the space B + (Γ) as a co n tinuous imag e of a totally disconnected co mpact L 0 . W e fix a sequence ( r n ) ∞ n =0 of positive r eal n umbers such that P ∞ n =0 r n = 1 and such that the con tinuous map φ : { 0 , 1 } N − → [0 , 1] given by φ ( x ) = P ∞ n =0 r n x n is surjective, for example r n = 1 2 n +1 . W e consider the p ow er φ Γ : { 0 , 1 } Γ × N − → [0 , 1] Γ and then we set: L 0 = ( φ Γ ) − 1 ( B + (Γ)) , f = φ Γ | L 0 , so that f : L 0 − → B + (Γ) is a contin uous surjection. It will b e co nv enient to ha ve an explicit description of L 0 . F or x ∈ { 0 , 1 } Γ × N and n ∈ N , w e define N n ( x ) = |{ γ ∈ Γ : x ( γ , n ) = 1 }| . 3 x ∈ L 0 ⇐ ⇒ φ Γ ( x ) ∈ B + (Γ) ⇐ ⇒ X γ ∈ Γ φ Γ ( x ) γ ≤ 1 ⇐ ⇒ X γ ∈ Γ ∞ X n =0 r n x ( γ , n ) ≤ 1 ⇐ ⇒ ∞ X n =0 r n N n ( x ) ≤ 1 . The co mpact space L 0 can be a lternatively descr ibe d as follows. Let Z b e a compact s ubset of N N such that if σ ∈ Z and τ n ≤ σ n for a ll n ∈ N , then τ ∈ Z . Asso ciated to such a set Z we construct the fo llowing s pace: K ( Z, Γ) = { x ∈ { 0 , 1 } Γ × N : ( N n ( x )) n ∈ N ∈ Z } . W e hav e tha t L 0 = K ( Z 0 , Γ) where Z 0 =  s ∈ N N : P i ∈ N r i s i ≤ 1  . Note that Z 0 is indeed compact since it is a closed subset of Q n ∈ N { 0 , . . . , M n } where M n is the integer par t of 1 r n . The pro of will b e complete after the following le mma: Lemma 3. L et Z b e a c omp act subset of N N such that if σ ∈ Z and τ n ≤ σ n for al l n ∈ N , then τ ∈ Z . The n K ( Z, Γ) is a c ontinuous image of σ 1 (Γ) N . PROOF: First we c heck that K ( Z, Γ) is a closed subset o f { 0 , 1 } Γ × N and hence compact. Namely , if x ∈ { 0 , 1 } Γ × N \ K ( Z, Γ), then ( N n ( x )) n ∈ N 6∈ Z and s ince Z is closed in N N , there is a finite set F ⊂ N such tha t σ 6∈ Z w henever σ n = N n ( x ) for all n ∈ F . Indeed, by the definition of Z , if τ ∈ N N and τ n ≥ σ n of all n ∈ F , als o τ 6∈ Z . In this cas e, W = { y ∈ { 0 , 1 } Γ × N : y γ , n = 1 whenev er n ∈ F and x γ , n = 1 } is a neigh b orho o d which separates x from K ( Z, Γ) and this finishes the pro of that K ( Z, Γ) is closed. Since Z is compa ct, for every n ∈ N there exists M n ∈ N such that σ n ≤ M n for a ll σ ∈ Z . W e define the following compact space: L 1 = Z × Y m ∈ N M m Y i =0 σ i (Γ) Note that L 1 is a co n tinuous ima ge of σ 1 (Γ) N . On the o ne hand, since Z is a metrizable compact, it is a contin uo us image o f { 0 , 1 } N and in particular o f σ 1 (Γ) N . On the other hand, for any i ∈ N , the space σ i (Γ) ca n b e view ed as the family of all subsets of Γ of c ardinality at mos t i . I n this wa y , we consider the contin uous surjection p : σ 1 (Γ) i − → σ i (Γ) given by p ( x 1 , . . . , x i ) = x 1 ∪ · · · ∪ x i . F ro m the existence o f such a function follows the fact that any countable pro duct o f space s σ i (Γ) is a co nt inuous image of σ 1 (Γ) N , and in particular, the s econd factor in the expression of L 1 is such an image. It remains to define a co ntin uous surjection g : L 1 − → K ( Z , Γ). W e first fix some notation. An e lement of L 1 will b e wr itten as ( z , x ) wher e z ∈ Z and x ∈ 4 ANTONIO A VIL ´ ES Q m ∈ N Q M m i =0 σ i (Γ). A t the same time, such an x is of the form ( x m ) m ∈ N with x m ∈ Q M m i =0 σ i (Γ) and again ea ch x m is ( x m,i ) M m i =1 where x m,i ∈ σ i (Γ). Finally x m,i = ( x m,i γ ) γ ∈ Γ ∈ σ i (Γ) ⊂ { 0 , 1 } Γ . The function g : L 1 − → K ( Z , Γ) ⊂ { 0 , 1 } Γ × N is defined as follows: g ( z , x ) γ , m = x m,z ( m ) γ Observe that g ( x, z ) maps indeed L 1 onto K ( Z, Γ) b ecause for every m , ( x m,z ( m ) γ ) γ ∈ Γ is an arbitrar y element of σ z ( m ) (Γ).  Theorem 4. L et Γ b e an un c ount able set and 1 < p < ∞ . Ther e exists an e qu iv- alent norm on ℓ p (Γ) whose un it b al l do es not satisfy pr op erty (Q) and henc e it is not p olyadic. PROOF: This is a v a riation of an example of Bell [2], originally a scattered compact, so that to make it a bsolutely conv ex. W e will consider ω 1 as a subset of Γ. Le t φ : ω 1 − → R b e a one-to-one map and G = { ( α, β ) ∈ ω 1 × ω 1 : φ ( α ) < φ ( β ) ⇐ ⇒ α  β } . W e define an equiv a lent nor m on ℓ p (Γ) × ℓ p (Γ) ∼ ℓ p (Γ) b y k ( x, y ) k ′ = sup {k x k p , k y k p , | x α | + | y β | : ( α, β ) ∈ G } . and let K b e its unit ball considered in its weak topo logy . Fix num b ers 1 < ξ 1 < ξ 2 < 2 1 − 1 p . The families of op en sets U α = { ( x, y ) ∈ K : | x α | + | y α | > ξ 2 } , α < ω 1 V α = { ( x, y ) ∈ K : | x α | + | y α | > ξ 1 } , α < ω 1 verify that U α ⊂ V α and that for any α, β < ω 1 , U α ∩ U β = ∅ if and only if ( α, β ) ∈ G if and only if V α ∩ V β = ∅ . Namely , if there is so me ( x, y ) ∈ V α ∩ V β , then | x α | + | y α | + | x β | + | y β | > ξ 1 + ξ 1 > 2 and therefore either | x α | + | x β | > 1 or | y α | + | y β | > 1 a nd t his implies that ( α, β ) 6∈ G since ( x, y ) ∈ K . On the other hand, if ( α, β ) 6∈ G then the elemen t ( x, y ) ∈ ℓ p (Γ) × ℓ p (Γ) which has all c o ordinates zero except x α = x β = y α = y β = 2 − 1 p lies in U α ∩ U β . Since there is no uncountable w ell order ed (or inv ersely w ell or- dered) subset of R ther e is no uncountable subset A of ω 1 such that A × A ⊂ G o r ( A × A ) ∩ G = ∅ . T herefore, the families { U α } a nd { V α } witness the fact that K do es no t hav e prop erty (Q) and hence, it is not p olyadic.  The pres ent work was written during a visit to the Universit y o f W ar saw. The author wishes to thank their hospitality , sp ecia lly to Witold Marciszews ki and Roman Pol, and to Rafa l G´ orak, from the Polish Academy of Sciences. 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M r´ owk a, Mazur the or em and m -adic sp ac es , Bull. Acad. Po lon. Sci. S ´ er. Sci. Math. As- tronom. Phys. 18 (1970), 299–305. Dep art am ento de Mat em ´ aticas, Universidad de Murcia, 30100 Espinardo (Murcia), Sp ain E-mail addr ess : avileslo@um.e s