A remark on spaces of affine continuous functions on a simplex
aa r X i v : . [ m a t h . F A ] M a r A REMARK ON SPACES OF AFFINE CONTINUOUSFUNCTIONS ON A SIMPLEX
E. CASINI, E. MIGLIERINA, AND Ł. PIASECKI
Abstract.
We present an example of an infinite dimensional separable spaceof affine continuous functions on a Choquet simplex that does not contain asubspace linearly isometric to c . This example disproves a result stated in [8]. Introduction and Preliminaries
In the
Concluding remarks in [8], the author claims that a separable predualof an abstract L space contains a (complemented) copy of c (the Banach spaceof real convergent sequences) if its unit ball has an extreme point. Only a sketchof the proof of this property was indicated. In particular there is no proof thatthe extreme point and the sequence { y n } ( -equivalent to the standard basis of c built in the main theorem of [8]), span a subspace isometric to c . The aim of thispaper is to present a simple example that disproves Zippin’s claim. Moreover, inthe last section of our paper we point out that our example also shows that someknown results, establishing geometrical properties of polyhedral Banach spaces, areincorrect.Let B X ( S X ) denote the closed unit ball (sphere) in a real Banach space X and X ∗ denotes the dual of X . If K is a compact, convex subset of a linear topologicalspace, then by Ext K we denote the set of all extreme points of K . A convexsubset F of B X is called a face of B X if for every x, y ∈ B X and λ ∈ (0 , suchthat (1 − λ ) x + λy ∈ F we have x, y ∈ F . A face F of B X is named a proper face if F = B X . Here c denotes the Banach space of all real convergent sequences and A ( K ) stands for a simplex space , that is, the space of all affine continuous functionson a Choquet simplex K endowed with the supremum norm. It is well known that c ∗ = ℓ and the duality is given by: f ( x ) = f (1) lim x ( i ) + + ∞ X i =1 f ( i + 1) x ( i ) where f = ( f (1) , f (2) , . . . ) ∈ ℓ and x = ( x (1) , x (2) , . . . ) ∈ c . A Banach space X iscalled an L - predual space or a Lindenstrauss space if its dual is isometric to L ( µ ) for some measure µ . It is well known that this class includes all the simplex spaces.Moreover a Lindenstrauss space X is isometric to a simplex space if and only if B X has at least one extreme point (see [7]). Finally, we recall that a Banach space X is polyhedral if the unit balls of all its finite-dimensional subspaces are polytopes(see [4]). 2. A simplex space not containing c We begin by providing a necessary condition for the presence of a copy of c in aseparable Banach space X . Mathematics Subject Classification.
Primary 46B04; Secondary 46B45, 46B25.
Key words and phrases.
Affine functions, Lindenstrauss spaces, Space of convergent sequences,Polyhedral spaces.
Theorem 2.1.
Let X be a separable Banach space. If X contains a subspacelinearly isometric to c , then there exist x ∈ X and a sequence ( e ∗ n ) ⊂ Ext B X ∗ such that ( e ∗ n ) is w ∗ -convergent to e ∗ , e ∗ n ( x ) = e ∗ ( x ) = k e ∗ k = k x k = 1 and k e ∗ n ± e ∗ k = 2 for every n ∈ N .Proof. Assume that X contains an isometric copy of c . Consider x = (1 , , . . . , , . . . ) ∈ c, the sequence ( x n ) n ∈ N ⊂ B c defined by x = ( − , , , . . . , , . . . ) ,x = (1 , − , , , . . . , , . . . ) ,x = (1 , , − , , , . . . , , . . . ) ,. . . and the sequence ( x ∗ n ) n ∈ N ⊂ B c ∗ defined by x ∗ = (0 , , , , . . . , , . . . ) ,x ∗ = (0 , , , , , . . . , , . . . ) ,x ∗ = (0 , , , , , , . . . , , . . . ) ,. . . . Let f x ∗ n denote a norm preserving linear extension of x ∗ n to the whole X . Next, letus define the sets F n , n ∈ N , by F n = { x ∗ ∈ B X ∗ : x ∗ ( x n ) = − and x ∗ ( x m ) = x ∗ ( x ) = 1 for m = n } . It is easy to see that(a) F n = ∅ for every n ∈ N (because f x ∗ n ∈ F n ),(b) F n is a w ∗ -closed proper face of B X ∗ , for every n ∈ N ,(c) F n ∩ F m = ∅ provided m = n .Hence, Ext F n = ∅ by the Krein-Milman Theorem, Ext F n ⊂ Ext B X ∗ for every n ∈ N and Ext F n ∩ Ext F m = ∅ whenever m = n .Let e ∗ n ∈ Ext F n ⊂ Ext B X ∗ . We can assume that ( e ∗ n ) is w ∗ -convergent, let ussay to e ∗ . Then(d) e ∗ n ( x ) = e ∗ ( x ) = k e ∗ k = k x k = 1 for every n ∈ N ,(e) e ∗ ( x i ) = lim n e ∗ n ( x i ) = 1 for every i ∈ N .Consequently, for every n ∈ N , we have ≥ k e ∗ − e ∗ n k ≥ e ∗ ( x n ) − e ∗ n ( x n ) = 1 − ( −
1) = 2 and ≥ k e ∗ + e ∗ n k ≥ e ∗ ( x ) + e ∗ n ( x ) = 1 + 1 = 2 . (cid:3) The previous theorem gives a corollary which allows us to show that the an-nounced example does not contain c . Corollary 2.2.
Let X be a predual of ℓ . If X contains a subspace isometric to c then there exist x ∈ B X and a subsequence ( e ∗ n k ) k ∈ N of the standard basis ( e ∗ n ) n ∈ N in ℓ such that (1) e ∗ n k σ ( ℓ ,X ) −→ e ∗ and supp e ∗ n k ∩ supp e ∗ = ∅ for every k ∈ N , where for x ∗ ∈ ℓ = X ∗ we put supp x ∗ := { i ∈ N : x ∗ ( i ) = 0 } , (2) e ∗ n k ( x ) = e ∗ ( x ) = 1 for every k ∈ N . REMARK ON SPACES OF AFFINE CONTINUOUS FUNCTIONS ON A SIMPLEX 3
Example 2.1.
Let W = ( x = ( x (1) , x (2) , . . . ) ∈ c : lim i x ( i ) = ∞ X i =1 x ( i )2 i ) . The hyperplane W has the following properties: (a) The map φ : ℓ → W ∗ defined by ( φ ( y ))( x ) = + ∞ X j =1 x ( j ) y ( j ) , where y = ( y (1) , y (2) , . . . ) ∈ ℓ and x = ( x (1) , x (2) , . . . ) ∈ W is an ontoisometry. Moreover, if ( e ∗ n ) denotes the standard basis of ℓ , then e ∗ n σ ( ℓ ,W ) −→ e ∗ = (cid:18) , , , . . . (cid:19) (see Theorem 4.3 in [1]). (b) From Corollary 2.2 we conclude that W does not contain a subspace linearlyisometric to c . (c) By Corollary 2 in [6] the set K = ( ( y (1) , y (2) , . . . ) ∈ ℓ : ∞ X i =1 y ( i ) = 1 , y ( i ) ≥ , i = 1 , , . . . ) is an infinite dimensional σ ( ℓ , W ) -closed proper face of B ℓ . (d) It is easy to see that x = (1 , , . . . , , . . . ) ∈ Ext B W . Consequently, as wasobserved in [7], W is isometric to A ( K ) . Nevertheless, in our special casethis property can be shown directly. (e) In order to prove that the space W is polyhedral we need a characterizationof polyhedrality given by Durier and Papini (Theorem 2 in [2]): a Banachspace X is polyhedral if and only if the set C ( x ) = { y ∈ X : ∃ λ > , k x + λ ( y − x ) k ≤ } is a closed set for every x ∈ S X . Moreover, we remark that x ∈ S W if andonly if there exists at least one index i ∈ N such that | x ( i ) | = 1 . Then,an easy computation shows that C ( x ) = { y ∈ W : y ( i ) ≤ for i ∈ I ( x ) and y ( j ) ≥ − for j ∈ J ( x ) } , where I ( x ) = { i : x ( i ) = 1 } and J ( x ) = { j : x ( j ) = − } . Therefore the set C ( x ) is closed for every x ∈ S W .Remark . Example 2.1 shows that property (2) in Corollary 2.2 does not implythat c ⊂ X . Also property (1) in the same corollary does not imply that c ⊂ X .Indeed, to this end is sufficient to consider a different hyperplane of c : V = ( x = ( x (1) , x (2) , . . . ) ∈ c : lim i x ( i ) = ∞ X i =1 ( − i +1 x (2 i − i ) . By using Theorem 4.3 in [1] we have that V ∗ = ℓ and e ∗ n σ ( ℓ ,V ) −→ e ∗ = (cid:18) , , − , , , , − , . . . (cid:19) . It is easy to see that does not exist x ∈ V satisfying the property (2) in Corollary2.2. Therefore V does not contain an isometric copy of c . It would be desirableto understand if the simultaneous validity of conditions (1) and (2) ensures thepresence of a isometric copy of c in a predual of ℓ , but we have not been able todo this. Nevertheless we show that the necessary condition expressed in Theorem REMARK ON SPACES OF AFFINE CONTINUOUS FUNCTIONS ON A SIMPLEX 4 X = ℓ and the sequence ( x ∗ n ) n ∈ N in ℓ ∗ = ℓ ∞ defined by x ∗ = (1 , − , , , . . . , , . . . ) ,x ∗ = (1 , , − , , , . . . , , . . . ) ,x ∗ = (1 , , , − , , , . . . , , . . . ) , . . . . Then(a) x ∗ n σ ( ℓ ∞ ,ℓ ) −→ x ∗ = (1 , , . . . , , . . . ) ,(b) k x ∗ n ± x ∗ k = 2 for every n ∈ N ,(c) x ∗ n , x ∗ ∈ Ext B ℓ ∞ for every n ∈ N ,(d) for e = (1 , , , . . . , , . . . ) ∈ ℓ we have x ∗ ( e ) = x ∗ n ( e ) = 1 but ℓ does not contain c . 3. Final remarks
Different authors refer to Zippin’s statement. We focus on a paper by Lazar thatgives a characterization of polyhedral Lindenstrauss spaces.In [5], the implication (1) ⇒ (3) in Theorem 3 is incorrect. Indeed, W is a poly-hedral space but B W ∗ contains an infinite dimensional w ∗ -closed proper face (seeitems (e) and (c) in Example 2.1). As a consequence of this remark we have thatsome of the implications stated in the theorem mentioned above reveal to be un-proven. For instance, the implication (1) ⇒ (4) has no proof.The result of Lazar has been subsequently used by Gleit and McGuigan in [3] toprovide another characterization of polyhedral Lindenstrauss spaces. We remarkthat, in [3], the implication (3) ⇒ (1) in Theorem 1.2 is incorrect. Indeed, W doesnot contain an isometric copy of c and for x = (1 , , . . . , , . . . ) in W and the w ∗ -limit e ∗ = (cid:0) , , , . . . (cid:1) of the standard basis in ℓ = W ∗ we have (see items (b)and (a) in Example 2.1) e ∗ ( x ) = ∞ X i =1 i = 1 = k x k = k e ∗ k . Also, the implication (3) ⇒ (1) in Corollary 2.7 is incorrect because W is a simplexspace (see item (d) in Example 2.1). References [1] E. Casini, E. Miglierina and Ł. Piasecki. Hyperplanes in the space of convergent sequences andpreduals of ℓ . online first on Canad. Math. Bull., http://dx.doi.org/10.4153/CMB-2015-024-9,4 Mar 2015.[2] R. Durier, P.L. Papini. Polyhedral norms in an infinite-dimensional space. Rocky Mountain J.Math. , (1993), 863-875.[3] A. Gleit, R. McGuigan. A note on polyhedral Banach spaces. Proc. Amer. Math. Soc. ,(1972), 398-404.[4] V. Klee. Polyhedral sections of convex bodies, Acta Math. , (1960), 243-267.[5] A. J. Lazar. Polyhedral Banach spaces and extensions of compact operators.
Israel J. Math. (1969), 357-364.[6] M. A. Japón-Pineda, S. Prus. Fixed point property for general topologies in some Banachspaces. Bull. Austral. Math. Soc. (2004), 229-244.[7] Z. Semadeni. Free compact sets. Bull. Acad. Polon. Sci. Sér Sci. Math. Astronom. Phys. (1964), 141-146.[8] M. Zippin. On some subspaces of Banach spaces whose duals are L spaces. Proc. Amer. Math.Soc. , (1969), 378-385. REMARK ON SPACES OF AFFINE CONTINUOUS FUNCTIONS ON A SIMPLEX 5
Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, via Valleggio11, 22100 Como, Italy
E-mail address : [email protected] Dipartimento di Discipline Matematiche, Finanza Matematica ed Econometria, Uni-versità Cattolica del Sacro Cuore, Via Necchi 9, 20123 Milano, Italy
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