Diametral diameter two properties in Banach spaces
Julio Becerra Guerrero, Ginés López Pérez, Abraham Rueda Zoca
aa r X i v : . [ m a t h . F A ] A p r DIAMETRAL DIAMETER TWO PROPERTIES INBANACH SPACES
JULIO BECERRA GUERRERO, GIN´ES L ´OPEZ-P´EREZ AND ABRAHAM RUEDAZOCA
Abstract.
The aim of this note is to provide several variants of thediameter two properties for Banach spaces. We study such propertieslooking for the abundance of diametral points, which holds in the settingof Banach spaces with the Daugavet property, for example, and we intro-duce the diametral diameter two properties in Banach spaces, showingfor these new properties stability results, inheritance to subspaces andcharacterizations in terms of finite rank projections. Introduction
We recall that a Banach space X satisfies the strong diameter two pro-perty (SD2P), respectively diameter two property (D2P), slice-diameter twoproperty (LD2P), if every convex combination of slices, respectively everynonempty relatively weakly open subset, every slice, in the unit ball of X hasdiameter 2. The weak-star slice diameter two property ( w ∗ -LD2P), weak-star diameter two property ( w ∗ -D2P) and weak-star strong diameter twoproperty ( w ∗ -SD2P) for a dual Banach space are defined as usual, changingslices by w ∗ -slices and weak open subsets by w ∗ - open subsets in the unitball. It is known that the above six properties are extremely different as itis proved in [5].Even though diameter two property theory is a very recent topic in ge-ometry of Banach spaces, a lot of nice results have appeared in the last fewyears (e.g. [1, 5, 6, 4, 8]). Moreover, it turns out that there are quite lot ofexamples of Banach spaces with such properties as infinite-dimensional C ∗ -algebras [4], non-reflexive M -embedded spaces [9] or Daugavet spaces [11].Last example is quite important because Banach spaces with the Daugavetproperty actually satisfy the diameter two properties in a stronger way. In-deed, as it was pointed out in [10], Banach spaces with Daugavet property Mathematics Subject Classification. verify that each slice of the unit ball S has diameter two and each norm-oneelement of S is diametral, i.e. given x ∈ S ∩ S X it follows that(1.1) diam ( S ) = sup y ∈ S k y − x k . We will say that a Banach space X has the diametral local diameter twoproperty (DLD2P) whenever X verifies the above condition. It is knownthat this property is stable by taking ℓ p sums [10] and that is inheritedto almost isometric ideals [2]. Moreover, this property is different to theDaugavet property (see again [10]).The aim of this note is to provide extensions of the diameter two propertiesin the way exposed above and make an intensive study of such properties.Indeed, in sections 2 and 3, we shall analyze extensions of the D2P andSD2P, respectively, by the existence of diametral points. Whilst we shalldefine the diametral diameter two property by the obvious generalizationin view of the diametral slice diameter two property, we provide a naturalextension of the SD2P in terms of diametrality in some different way. Givena Banach space X we will say that X has the diametral strong diameter twoproperty (DSD2P) whenever given C a convex combination of non-emptyrelatively weakly open subsets of B X , x ∈ C and ε ∈ R + we can find y ∈ C such that k y − x k > k x k − ε . This alternative definition is given becausea convex combination of non-empty relatively weakly open subsets of theunit ball of a Banach space does not have to intersect to the unit sphereand it is quite clear that (1.1) implies k x k = 1. We will get some resultsof stability of diametral diameter two properties in terms of ℓ p sums andinheritance to subspaces. Moreover, we will exhibit some characterizationsof such properties in terms of finite-rank projections or weakly convergentnets. Finally, section 4. is devoted to exhibit some open problems andremarks.We shall introduce some notation. We consider real Banach spaces, B X (resp. S X ) denotes the closed unit ball (resp. sphere) of the Banach space X . If Y is a subspace of a Banach space X , X ∗ stands for the dual space of X . A slice of a bounded subset C of X is a set of the form S ( C, f, α ) := { x ∈ C : f ( x ) > M − α } , where f ∈ X ∗ , f = 0, M = sup x ∈ C f ( x ) and α >
0. If X = Y ∗ is a dualspace for some Banach space Y and C is a bounded subset of X , a w ∗ -sliceof C is a set of the form S ( C, y, α ) := { f ∈ C : f ( y ) > M − α } , where y ∈ Y , y = 0, M = sup f ∈ C f ( y ) and α > w (resp. w ∗ ) denotes theweak (resp. weak-star) topology of a Banach space.It is proved in [6, Corollary 2.2] that a Banach space X has the SD2P if,and only if, X ∗ has an octahedral norm. iametral diameter two properties in Banach spaces. 3 Moreover, it is proved in [8, Theorems 3.2 and 3.4] that a Banach space X has the LD2P (respectively the D2P) if, and only if, X ∗ has a locallyoctahedral (respectively weakly octahedral) norm.Let X be a Banach space and Y ⊆ X a closed subspace. According to[3], we will say that Y is an almost isometric ideal in X if for each ε > E ⊆ X a finite-dimensional subspace there exists a linear and boundedoperator T : E −→ Y satisfying the following conditions:(1) T ( e ) = e for each e ∈ E ∩ Y .(2) For each e ∈ E one has11 + ε k e k ≤ k T ( e ) k ≤ (1 + ε ) k e k . In spite of the fact that almost isometric ideals in Banach spaces do nothave to be closed, by a perturbation argument it follows that a non-closedsubspace is an almost isometric ideal if, and only if, its closure is also analmost isometric ideal. Hence, we will consider only closed almost isometricideals.In [3] is proved that each diameter two property as well as Daugavetproperty are inherited to almost isometric ideals from the whole space. Thisis a consequence of the following
Theorem 1.1. [3, Theorem 1.4]
Let X be a Banach space and Y ⊆ X analmost ideal in X . Then there exists ϕ : Y ∗ −→ X ∗ a Hahn-Banach operatorsuch that, for each ε > , for each E ⊆ X finite-dimensional subspace andeach F ⊆ Y ∗ finite dimensional subspace, there exists T : E −→ Y verifyingthe following: (1) T ( e ) = e for each e ∈ E ∩ Y . (2) For each e ∈ E one has
11 + ε k e k ≤ k T ( e ) k ≤ (1 + ε ) k e k . (3) For each e ∈ E and f ∈ F it follows ϕ ( f )( e ) = f ( T ( e )) . We shall also exhibit the following known result which will be used severaltimes in the following. A proof can be found in [10, Lemma 2.1].
Lemma 1.2.
Let X be a Banach space. Consider x ∗ ∈ S X ∗ , ε ∈ R + and x ∈ S ( B X , x ∗ , ε ) ∩ S X . Then, given < δ < ε , there exists y ∗ ∈ S X ∗ suchthat x ∈ S ( B X , y ∗ , δ ) ⊆ S ( B X , x ∗ , ε ) . Similarly, a dual version of Lemma above can be statedas follows.
Lemma 1.3.
Let X be a Banach space. Consider x ∈ S X , ε ∈ R + and x ∗ ∈ S ( B X ∗ , x, ε ) ∩ S X ∗ . Then, given < δ < ε , there exists y ∈ S X suchthat x ∗ ∈ S ( B X ∗ , y, δ ) ⊆ S ( B X ∗ , x, ε ) . J. Becerra, G. L´opez and A. Rueda Diametral diameter two property and stability results
We shall start by giving the following
Definition 2.1.
Let X be a Banach space.We will say that X has the diametral diameter two property (DD2P) ifgiven W a non-empty relatively weakly open subset of B X , x ∈ W ∩ S X and ε ∈ R + there exists y ∈ W such that(2.1) k x − y k > − ε. If X is a dual Banach space we will say that X has the weak-star diametraldiameter two property ( w ∗ -DD2P) if given W a non-empty relatively weakly-star open subset of B X , x ∈ W ∩ S X and ε ∈ R + there exists y ∈ W satisfying(2.1).From [11, Lemma 2.3] we get that each Banach space enjoying to haveDaugavet property satisfies DD2P. However, there are Banach spaces withthe DD2P which do not enjoy to have the Daugavet property. Example 2.2.
Let X be the renorming of C ([0 , X is MLUR, has the DLD2P and X fails SD2P. Then X fails theDaugavet property. However, X has the DD2P because X has the DLD2P,applying the well known Choquet lemma [ ? , Lemma 3.40].It is known that a Banach space X has the D2P if, and only if, X ∗∗ hasthe w ∗ -D2P. However, this fact is far from being true for the DD2P. Indeed,applying the weak-star lower semicontinuity of a bidual norm is easy to getthe following Proposition 2.3.
Let X be a Banach space. If X ∗∗ has the w ∗ -DD2P, then X has the DD2P.Remark . The converse of Proposition 2.3 is not true. Indeed consider X := C ( K ), for an infinite compact Hausdorff and perfect topological space K . Now X has the DD2P as being a Daugavet space. However, B X ∗ hasdenting points, so X ∗ fails the DLD2P and, consequently, X ∗∗ fails the w ∗ -DLD2P [2, Theorem 3.6].Now we shall provide several characterizations of the DD2P. First of all,we shall show a useful characterization of the DD2P in terms of weaklyconvergent nets which will be used in order to prove the stability of theDD2P by ℓ p sums. Proposition 2.5.
Let X be a Banach space. The following assertions areequivalent: (1) X has DD2P. (2) For each x ∈ S X there exists a net { x s } ⊂ B X which convergesweakly to x and such that {k x − x s k} → . iametral diameter two properties in Banach spaces. 5 Proof. (1) ⇒ (2). Pick U a neighborhood system of x in the weak topologyrelative to B X . Now, for each U ∈ U and every ε ∈ R + , choose x ( U,ε ) ∈ U such that k x − x ( U,ε ) k ≥ − ε. Such x ( U,ε ) exists because X has the DD2P. Now, considering in U × R + the partial order given by the reverse inclusion in U and the inverse naturalorder in R we conclude that { x ( U,ε ) } ( U,ε ) ∈U× R + → x in the weak topology of B X . It is also clear that {k x − x ( U,ε ) k} ( U,ε ) ∈U× R + → ⇒ (1). Pick W a non-empty relatively weakly open subset of B X , x ∈ W ∩ S X and ε ∈ R + and let us prove that there exists y ∈ W such that k x − y k > − ε . By assumption there exists { x s } a net in B X such that k x − x s k → , and { x s } w → x. From both convergences then there exists s such that x s ∈ W and k x − x s k > − ε . Now (1) follows choosing y := x s .Now, for dual Banach spaces we have the following characterization ofthe w ∗ -DD2P, as the above one. Corollary 2.6.
Let X be a dual Banach space. The following assertionsare equivalent: (1) X has w ∗ -DD2P. (2) For each x ∈ S X there exists a net { x s } in B X which converges to x in the weak-star topology such that {k x − x s k} → . Remark . In view of Proposition 2.5, Daugavet property can also be easilycharacterized in terms of weakly convergent nets. Indeed it is straighforwardto prove from [11, Lemma 2.3] that a Banach space X has the Daugavetproperty if, and only if, given x, y ∈ S X there exists { y s } a net in B X weakly convergent to y such that {k x − y s k} → . In [10] it is proved a characterization of DLD2P in terms of the behaviorof rank-one projections in a Banach space. It turns out to be also truethat DD2P can be characterized regarding the behaviour of the rank-oneprojections. In fact, we have the following characterization of the DD2P.
Proposition 2.8.
Let X be a Banach space. The following assertions areequivalent: (1) X has the DD2P. J. Becerra, G. L´opez and A. Rueda (2)
For each x ∗ , . . . , x ∗ n ∈ S X ∗ and x ∈ X such that x ∗ i ( x ) = 0 , if wedefine p i := x ∗ i ⊗ xx ∗ i ( x ) ∀ i ∈ { , . . . , n } one has that, for each ε ∈ R + , there exists y ∈ B X such that k y − p i ( y ) k > − ε ∀ i ∈ { , . . . , n } and x ∗ i ( y ) x ∗ i ( x ) ≥ ∀ i ∈ { , . . . , n } (3) Given S := S ( B X , x, δ ) a weak-star slice of B X ∗ and x ∗ , . . . , x ∗ n ∈ S ∩ S X ∗ there exist y ∗ ∈ S and y ∈ S X such that ( x ∗ i − y ∗ )( y ) > − δ ∀ i ∈ { , . . . , n } . Proof. (1) ⇒ (2).Consider x ∗ , . . . , x ∗ n ∈ S X , x ∈ X and p i := x ∗ i ⊗ xx ∗ i ( x ) for each i ∈{ , . . . , n } .Consider ε > ε <
2. Note that x k x k ∈ W := (cid:26) y ∈ B X : (cid:12)(cid:12)(cid:12)(cid:12) x ∗ i ( y ) x ∗ i ( x ) − k x k (cid:12)(cid:12)(cid:12)(cid:12) k x k < ε (cid:27) , where W is a relatively weakly open subset of B X . Moreover x k x k ∈ S X . As X has the DD2P we can assure the existence of an element y ∈ W such that (cid:13)(cid:13)(cid:13) y − x k x k (cid:13)(cid:13)(cid:13) > − ε .Now, on the one hand, as y ∈ W , given i ∈ { , . . . , n } , one has (cid:12)(cid:12)(cid:12)(cid:12) x ∗ i ( y ) x ∗ i ( x ) − k x k (cid:12)(cid:12)(cid:12)(cid:12) k x k < ε ⇒ x ∗ i ( y ) x ∗ i ( x ) > k x k − ε k x k = 1 − ε k x k ≥ . On the other hand, given i ∈ { , . . . , n } , it follows k y − p i ( y ) k ≥ (cid:13)(cid:13)(cid:13)(cid:13) y − x k x k (cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13) x k x k − p i ( y ) (cid:13)(cid:13)(cid:13)(cid:13) > − ε − (cid:13)(cid:13)(cid:13)(cid:13) x k x k − x ∗ i ( y ) x ∗ i ( x ) x (cid:13)(cid:13)(cid:13)(cid:13) > − ε since y ∈ W . So (2) follows.(2) ⇒ (1).Consider W := n T i =1 S ( B X , y ∗ i , ε i ) a non-empty relatively weakly open sub-set of B X and pick x ∈ W ∩ S X . In order to prove that X has the DD2Pchoose 0 < ε < min ≤ i ≤ n ε i . By Lemma 1.2 we can find, for each i ∈ { , . . . , n } ,a functional x ∗ i ∈ S X ∗ such that x ∈ S ( B X , x ∗ i , ε ) ⊆ S ( B X , y ∗ i , ε i ) ∀ i ∈ { , . . . , n } , iametral diameter two properties in Banach spaces. 7 and so x ∈ n T i =1 S ( B X , x ∗ i , ε ) ⊆ W . Consider η ∈ R + small enough to satisfy x ∗ i ( x )(1 − η ) > − ε for each i ∈ { , . . . , n } . For each i ∈ { , . . . , n } define p i := x ∗ i ⊗ xx ∗ i ( x ) . From the hypothesis we can find y ∈ B X such that k y − p i ( y ) k > − η ∀ i ∈ { , . . . , n } and x ∗ i ( y ) x ∗ i ( x ) ≥ . Now, on the one hand, one has1 − η < k p i ( y ) k = (cid:13)(cid:13)(cid:13)(cid:13) x ∗ i ( y ) x ∗ i ( x ) x (cid:13)(cid:13)(cid:13)(cid:13) = x ∗ i ( y ) x ∗ i ( x ) ∀ i ∈ { , . . . , n } . So x ∗ i ( y ) > (1 − η ) x ∗ i ( x ) > − ε for each i ∈ { , . . . , n } and, consequently, y ∈ n T i =1 S ( B X , x ∗ i , ε ) ⊆ W . Moreover, chosen i ∈ { , . . . , n } , it follows k y − x k ≥ k y − p i ( y ) k−k p i ( y ) − x k > − η − (cid:12)(cid:12)(cid:12)(cid:12) x ∗ i ( y ) x ∗ i ( x ) − (cid:12)(cid:12)(cid:12)(cid:12) = 2 − η −| x ∗ i ( y ) − x ∗ i ( x ) | x ∗ i ( x ) > − η − ε. As 0 < ε < min ≤ i ≤ n ε i was arbitrary we conclude the desired result.(1) ⇒ (3). Let S and x ∗ , . . . , x ∗ n be as in the hypothesis and pick 0 < η < δ .Now given i ∈ { , . . . , n } one has x ∗ i ∈ S ⇔ x ∗ i ( x ) > − δ ⇔ x ∈ S ( B X , x ∗ i , δ ) . So x ∈ n T i =1 S ( B X , x ∗ i , δ ) ∩ S X . As X has the DD2P then there exists y ∈ n T i =1 S ( B X , x ∗ i , δ ) ∩ S X such that k x − y k > − η ⇒ ∃ y ∗ ∈ S X ∗ / y ∗ ( x ) − y ∗ ( y ) > − η ⇒ (cid:26) y ∗ ( x ) > − ηy ∗ ( y ) < − η . So y ∗ ( x ) > − δ and thus y ∗ ∈ S . In addition, given i ∈ { , . . . , n } , itfollows ( x ∗ i − y ∗ )( y ) = x ∗ i ( y ) − y ∗ ( y ) > − δ + 1 − η = 2 − δ − η. From the arbitrariness of 0 < η < δ we get the desired result by a pertur-bation argument, if necessary.(3) ⇒ (1). Let W := n T i =1 S ( B X , y ∗ i , ε i ) be a non-empty relatively weaklyopen subset of B X and consider x ∈ W ∩ S X . Pick 0 < δ < min ≤ i ≤ n ε i . From J. Becerra, G. L´opez and A. Rueda
Lemma 1.2 we can find, for each i ∈ { , . . . , n } , an element x ∗ i ∈ S X suchthat x ∈ S ( B X , x ∗ i , δ ) ⊆ S ( B X , y ∗ i , ε i )holds for each i ∈ { , . . . , n } . Now x ∗ , . . . , x ∗ n ∈ S ( B X ∗ , x, δ ). From assump-tions we can find y ∗ ∈ S ( B X ∗ , x, δ ) and y ∈ S X such that( x ∗ i − y ∗ )( y ) > − δ holds for each i ∈ { , . . . , n } . Now, on the one hand x ∗ i ( y ) > − δ ⇒ y ∈ n \ i =1 S ( B X , x ∗ i , δ ) ⊆ W. Moreover, as y ∗ ∈ S ( B X ∗ , x, δ ), it follows k x − y k ≥ y ∗ ( x ) − y ∗ ( y ) > − δ + 1 − δ = 2(1 − δ ) . From the arbitrariness of 0 < δ < min ≤ i ≤ n ε i we have that X has the DD2P,as desired. Remark . Note that given p , . . . , p n rank one projections as in aboveProposition one has k I − p i k ≥ X enjoys to have the DLD2P. However, if X also satisfies theDD2P these projections can be “normed” by a common point of the space.A dual version of above Proposition is the following Proposition 2.10.
Let X be a Banach space. The following assertions areequivalent: (1) X ∗ has the w ∗ -DD2P. (2) For each x , . . . , x n ∈ S X and x ∗ ∈ X ∗ such that x ∗ ( x i ) = 0 , if wedefine p i := x ∗ x ∗ ( x i ) ⊗ x i ∀ i ∈ { , . . . , n } one has that, for each ε ∈ R + , there exists y ∗ ∈ B X ∗ such that k y ∗ − p i ( y ∗ ) k > − ε ∀ i ∈ { , . . . , n } and y ∗ ( x i ) x ∗ ( x i ) ≥ ∀ i ∈ { , . . . , n } (3) Given S := S ( B X , x ∗ , δ ) a slice of B X and x , . . . , x n ∈ S ∩ S X thereexist y ∈ S and y ∗ ∈ S X ∗ such that y ∗ ( x i − y ) > − δ ∀ i ∈ { , . . . , n } . iametral diameter two properties in Banach spaces. 9 It is known that DLD2P is stable under taking ℓ p -sums. Indeed, giventwo Banach spaces X and Y and 1 ≤ p ≤ ∞ , the Banach space X ⊕ p Y hasthe DLD2P if, and only if, X and Y enjoy to have the DLD2P [10, Theorem3.2].Our aim is to establish the same result for the DD2P. We shall begin withthe stability result Theorem 2.11.
Let
X, Y be Banach spaces which satisfy the DD2P and let ≤ p ≤ ∞ . Then X ⊕ p Y enjoys to have the DD2P.Proof. Define Z := X ⊕ p Y , pick ( x , y ) ∈ S Z and let us apply Proposition2.5.On the one hand, if p = ∞ , then either k x k = 1 or k y k = 1. Assume,with no loss of generality, that k x k = 1. As X has the DD2P then thereexists { x s } a net in B X such that { x s } → x in the weak topology of X and k x − x s k → . Then we have that { ( x s , y ) } → ( x , y )in the weak topology of B Z (note that, from the definition of the norm on Z we have that each term of the above net belongs to B Z ). In addition, given s one has2 ≥ k ( x , y ) − ( x s , y ) k ∞ = max {k x − x s k , k y k} ≥ k x − x s k . As {k x − x s k} → {k ( x , y ) − ( x s , y s ) k} → p < ∞ . As ( x , y ) ∈ S Z we have that( k x k p + k y k p ) p = 1 . Now x is an element of k x k S X . As X has the DD2P then by Proposition2.5 there exists { x s } s ∈ S a net in k x k B X such that { x s } → x in the weak topology of X and {k x − x s k} → k x k . In addition, as Y also has the DD2P, then there exists a net { y t } t ∈ T in k y k B Y such that { y t } t ∈ T → y in the weak topology of Y and such that {k y − y t k} t ∈ T → k y k . Now we have { ( x s , y t ) } ( s,t ) ∈ S × T → ( x , y ) in the weak topology of Z . More-over, given s ∈ S, t ∈ T one has k ( x s , y t ) k p = ( k x s k p + k y t k p ) p ≤ ( k x k p + k y k p ) p = 1 , so ( x s , y t ) ∈ B Z for each s ∈ S, t ∈ T . Finally, given s ∈ S, t ∈ T it follows k ( x , y ) − ( x s , y t ) k p = ( k x − x s k p + k y − y t k p ) p → ((2 k x k ) p + (2 k y k ) p ) p = 2 . Now let us prove the converse of the above result.
Proposition 2.12.
Let
X, Y be Banach space and define Z := X ⊕ p Y for ≤ p ≤ ∞ . If X fails to have DD2P so does Z .Proof. As X fails the DD2P then there exists U a non-empty relativelyweakly open subset of B X , x ∈ U ∩ S X and ε ∈ R + such that k x − y k ≤ − ε ∀ y ∈ U. Now we shall argue by cases:(1) If p = ∞ define the weak open subset of B Z given by W := { ( x, y ) ∈ B Z : x ∈ U ∩ B X } , and pick ( x , ∈ W . Then for each ( x, y ) ∈ W one has k ( x , − ( x, y ) k = max {k x − x k , k y k} ≤ max { − ε , } < , as x ∈ U ∩ B X .(2) If p < ∞ , given ε ∈ R + , there exists δ > − δ < | r | ≤ , | s | ≤ , ( | r | p + | s | p ) p ≤ ) ⇒ | s | p < ε. Define W := { ( x, y ) ∈ B Z : x ∈ U ∩ B X and k x k > − δ } , which is a weakly open subset of B Z from the lower weakly semi-continuity of the norm on X . Consider ( x , ∈ W . Now, given( x, y ) ∈ W we have from (2.2) that k y k p ≤ ε . In addition, as x ∈ U ∩ B X we conclude k x − x k ≤ − ε . Hence k ( x , − ( x, y ) k = ( k x − x k p + k y k p ) p ≤ ((2 − ε ) p + ε ) p . So, taking ε small enough, we conclude that sup ( x,y ) ∈ W k ( x , − ( x, y ) k <
2, so we are done.Even though Example 2.2 shows that Daugavet property and DD2P aredifferent, above results provide us more examples of such Banach spaces.Indeed, given 1 < p < ∞ , Z := X ⊕ p Y has the DD2P and fails to haveDaugavet property (actually, Z fails to have the strong diameter two prop-erty [1, Theorem 3.2]) whenever X, Y are Banach spaces with the DD2P (inparticular, Daugavet spaces). iametral diameter two properties in Banach spaces. 11
Finally we shall study the following problem: when a subspace of a Banachspace having the DD2P inherits DD2P? In order to give a partial answer, ithas been recently proved in [7] that D2P is hereditary to finite-codimensionalsubspaces. Bearing in mind the ideas of the proof of that result, we can provethe following
Theorem 2.13.
Let X be a Banach space which satisfies the DD2P. If Y is a closed subspace of X such that X/Y is finite-dimensional then Y hasthe DD2P.Proof. Consider W := { y ∈ Y : | y ∗ i ( y − y ) | < ε i ∀ i ∈ { , . . . , n }} , for n ∈ N , ε i ∈ R + , y ∗ i ∈ Y ∗ for each i ∈ { , . . . , n } and y ∈ Y such that W ∩ B Y = ∅ . Pick y ∈ W ∩ S Y and let us find, for each δ ∈ R + , a point z ∈ W ∩ B Y suchthat k y − z k > − δ . To this aim pick an arbitrary δ ∈ R + . Assume that y ∗ i ∈ X ∗ for each i ∈ { , . . . , n } . Observe that there is no loss of generalityfrom the Hahn-Banach theorem.Define U := { x ∈ X : | y ∗ i ( x − y ) | < ε i ∀ i ∈ { , . . . , n }} , which is a weakly open set in X such that U ∩ B X = ∅ .Let p : X −→ X/Y be the quotient map, which is a w − w open map.Then p ( U ) is a weakly open set in X/Y . In addition ∅ 6 = p ( U ∩ B X ) ⊆ p ( U ) ∩ p ( B X ) ⊆ p ( U ) ∩ B X/Y . Defining A := p ( U ) ∩ B X/Y , then A is a non-empty relatively weakly openand convex subset of B X/Y which contains to zero. Hence, as
X/Y is finite-dimensional, we can find a weakly open set V of X/Y , in fact a ball centeredat 0, such that V ⊂ A and that(2.3) diam ( V ∩ p ( U ) ∩ B X/Y ) = diam ( V ) < δ . As V ⊂ A then B := p − ( V ) ∩ U ∩ B X = ∅ . Hence B is a non-empty relativelyweakly open subset of B X . Moreover y ∈ p − ( V ) because p ( y ) = 0 ∈ V , so y ∈ B ∩ S X . Using that X satisfies the DD2P we can assure the existenceof v ∈ B such that(2.4) k v − y k > − δ . Note that v ∈ B implies p ( v ) ∈ V = V ∩ P ( U ) ∩ B X/Y . In view of (2.3) itfollows k p ( v ) k ≤ diam ( V ∩ p ( U ) ∩ B X/Y ) < δ . Hence there exists u ∈ Y such that k u − v k < δ and so k u k < δ .Letting z = u k u k , we have that k v − z k ≤ k u − v k + (cid:13)(cid:13)(cid:13)(cid:13) u − u k u k (cid:13)(cid:13)(cid:13)(cid:13) < δ
16 + k u k ( k u k − < δ
16 + (cid:18) δ (cid:19) δ
16 == δ (cid:18) δ (cid:19) . So(2.5) k v − z k < δ . Note that given i ∈ { , . . . , n } and bearing in mind (2.5) one has | y ∗ i ( z − y ) | ≤ | y ∗ i ( z − v ) | + | y ∗ i ( v − y ) | ≤ k y ∗ i k δ ε i , using that v ∈ U . Thus, if we define W δ := (cid:26) y ∈ Y : | y ∗ i ( y − y ) | < ε i + k y ∗ i k δ ∀ i ∈ { , . . . , n } (cid:27) it follows that v ∈ W δ ∩ B Y . On the other hand, in view of (2.4) and (2.5)we can estimate k y − z k ≥ k y − v k − k v − z k > − δ − δ > − δ. From here we can conclude the desired result. Indeed, for each i ∈ { , . . . , n } we can find b ε i ∈ R + and δ ∈ R + such that b ε i + δ k y ∗ i k < ε i ∀ i ∈ { , . . . , n } , and that y ∈ c W := { z ∈ Y : | y ∗ i ( z − y ) | < b ε i ∀ i ∈ { , . . . , n }} . For 0 < δ < δ one has c W δ := (cid:26) y ∈ Y : | y ∗ i ( y − y ) | < b ε i + k y ∗ i k δ ∀ i ∈ { , . . . , n } (cid:27) ⊆ W. The arbitrariness of δ in the above argument allow us to conclude the desiredresult.As it is done in [7] for the w ∗ -D2P, we can conclude a stability result forthe w ∗ -DD2P. Corollary 2.14.
Let X be a Banach space and let Y ⊆ X a closed subspace.If X ∗ has the w ∗ -DD2P and Y is finite-dimensional, then ( X/Y ) ∗ has the w ∗ -DD2P. iametral diameter two properties in Banach spaces. 13 Proof.
Consider W a weakly-star open subset of Y ◦ = ( X/Y ) ∗ such that W ∩ B Y ◦ = ∅ , and pick z ∗ ∈ W ∩ S Y ◦ . Now we can extend W to a weak-star open subsetof X ∗ , say U , as it is done in Theorem 2.13 satisfying z ∗ ∈ U ∩ S X ∗ .Let p : X ∗ −→ X ∗ /Y ◦ be the quotient map, which is a w ∗ − w ∗ open map.Then p ( U ) is a weakly-star open set of X ∗ /Y ◦ which meets with B X ∗ /Y ◦ .If we define A := p ( U ) ∩ B X ∗ /Y ◦ , then we have that A is a relativelyweak-star open and convex subset of B X ∗ /Y ◦ which contains to zero.As X ∗ /Y ◦ = Y ∗ is finite-dimensional, we can find V a weak-star openset of X ∗ /Y ◦ , in fact a ball centered at zero, such that V ⊂ A and whosediameter is as closed to zero as desired.From here, it is straightforward to check that computations of Theorem2.13 work and allow us to conclude thatsup x ∗ ∈ W ∩ B Y ◦ k z ∗ − x ∗ k = 2 , so Y ◦ = ( X/Y ) ∗ has the w ∗ -DD2P as desired.As we have pointed out in the Introduction, the D2P is inherited to almostisometric ideals from the whole space [3, Proposition 3.2]. Now, followingsimilar ideas, we get the following Proposition 2.15.
Let X be a Banach space and let Y ⊆ X a closed almostisometric ideal. If X has the DD2P, so does Y .Proof. Take n = 1 in the proof of Proposition 3.123. Diametral strong diameter two property and stabilityresults
Now we shall introduce the natural extension of the SD2P in the sameway the DD2P is defined.
Definition 3.1.
Let X be a Banach space. We will say that X has thediametral strong diameter two property (DSD2P) if given C a convex com-bination of non-empty relatively weakly open subsets of B X , x ∈ C and ε ∈ R + then there exists y ∈ C such that(3.1) k x − y k > k x k − ε. If X is a dual space, we will say that X has the weak-star diametral strongdiameter two property ( w ∗ -DSD2P) if given C a convex combination of non-empty relatively weakly-star open subsets of B X , x ∈ C and ε ∈ R + thenthere exists y ∈ C satisfying (3.1). Remark . On the one hand, note that the above definition extends thestrong diameter two property from the Bourgain lemma [ ? ]. On the other hand, the condition (1.1) is replaced with (3.1) to get theimplication DSD2P ⇒ SD2P. Indeed, consider X the Banach space of Ex-ample 2.2 and C := P ni =1 λ i W i a convex combination of non-empty rel-atively weakly open subsets of B X . If C ∩ S X = ∅ then there exists x := P ni =1 λ i x i ∈ C ∩ S X . As X is a strictly convex space we concludethat x = x = . . . = x n . Consequently x ∈ n T i =1 W i ⊆ C and, as X has theDD2P, we can find, for each ε >
0, an element y ∈ n T i =1 W i ⊆ X such that k y − x k > − ε . However, X fails to have the SD2P.As in the DD2P, the first example of Banach space with the DSD2P comesfrom Daugavet spaces. Example 3.3.
Daugavet Banach spaces enjoy to have DSD2P.
Proof.
Consider X to be a Banach space enjoying to have the Daugavetproperty. From the proof of [11, Lemma 2.3] it follows that given C aconvex combination of non-empty relatively weakly open subsets of B X , x ∈ S X and ε ∈ R + we can find y ∈ C such that k x + y k > − ε. From here let us prove that X enjoys to have the DSD2P. To this aim pick C := P ni =1 λ i W i a convex combination of non-empty relatively weakly opensubsets of B X . Let x ∈ C such that x = 0. From Daugavet property we canfind y ∈ P ni =1 λ i ( − W i ) such that (cid:13)(cid:13)(cid:13)(cid:13) x k x k + y (cid:13)(cid:13)(cid:13)(cid:13) > − ε. Now − y ∈ C . Moreover k x − ( − y ) k ≥ (cid:13)(cid:13)(cid:13)(cid:13) x k x k + y (cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13) x k x k − x (cid:13)(cid:13)(cid:13)(cid:13) > − ε − k x k (cid:12)(cid:12)(cid:12)(cid:12) k x k − (cid:12)(cid:12)(cid:12)(cid:12) =2 − ε − | − k x k| = 2 − ε − k x k = 1 + k x k − ε. In order to conclude the proof assume that 0 ∈ C . As diam ( C ) = 2 (see theproof of [6, Lemma 2.3]) we can find x, y ∈ C such that k x − y k > − ε k x k ≤ (cid:27) ⇒ k y − k = k y k > − ε = 1 + k k − ε. From the arbitrariness of C we conclude that X has the DSD2P.Given a Banach space X , it is true that X has the DSD2P whenever X ∗∗ has the w ∗ -DSD2P by a similar argument to the one given in Proposition 2.3.Again, the converse is not true, because the example exhibited in Remark2.4 also works for the DSD2P.Moreover, DSD2P admits a characterization in terms of weakly conver-gent nets as DD2P does. Indeed, we have the following iametral diameter two properties in Banach spaces. 15 Proposition 3.4.
Let X be a Banach space. The following assertions areequivalent: (1) X has the DSD2P. (2) For each x , . . . , x n ∈ B X and each λ , . . . , λ n ∈ R + such that P ni =1 λ i = 1 it follows that, for each i ∈ { , . . . , n } , there exists { x is } s ∈ S a net in B X weakly convergent to x i such that ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x is ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Proof. (1) ⇒ (2). Pick U a system of neighborhoods of 0. Now, for each U ∈ U and ε ∈ R + , pick x iU,ε for each i ∈ { , . . . , n } such that x iU,ε ∈ ( x i + U ) ∩ B X and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x iU,ε ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε, which can be done because X has the DSD2P.Now it is quite clear that, given i ∈ { , . . . , n } , then { x U,ε } ( U,ε ) ∈U× R + → x i in the weak topology of X . Moreover, it is clear that ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x is ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) from triangle inequality.(2) ⇒ (1). Is similar to Proposition 2.5.Now we can establish a dual version for the result above. Proposition 3.5.
Let X be a dual Banach space. The following assertionsare equivalent: (1) X has the w ∗ -DSD2P. (2) For each x , . . . , x n ∈ B X and each λ , . . . , λ n ∈ R + such that P ni =1 λ i = 1 it follows that, for each i ∈ { , . . . , n } , there exists { x is } s ∈ S a net in B X convergent to x i in the weak-star topology of X such that ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x is ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . As we have checked, DLD2P and DD2P have strong links with the rankone projections. This fact turns out to be also true for the DSD2P when weconsider finite-rank projections.
Proposition 3.6.
Let X be a Banach space. Assume that X has theDSD2P. Then for each p := P ni =1 x ∗ i ⊗ x i projection we have k I − p k ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Proof.
Pick p := P ni =1 x ∗ i ⊗ x i a finite rank projection and let ε ∈ R + .Then n X i =1 n x i k x i k ∈ n X i =1 n ( y ∈ B X : (cid:12)(cid:12)(cid:12) x ∗ i ( y ) − k x i k (cid:12)(cid:12)(cid:12) k x i k < ε | x j ( y ) |k x j k < ε ∀ j = i. ) As X has the DSD2P then, for each i ∈ { , . . . , n } , there exists y i ∈ ( y ∈ B X : (cid:12)(cid:12)(cid:12) x ∗ i ( y ) − k x i k (cid:12)(cid:12)(cid:12) k x i k < ε | x j ( y ) |k x j k < ε ∀ j = i. ) such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n (cid:18) y i − x i k x i k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n x i k x i k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε . Then k I − p k ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n y i − p n X i =1 n y i !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n (cid:18) y i − x i k x i k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n x i k x i k − p n X i =1 n y i !!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Now (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n x i k x i k − p n X i =1 n y i !!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n X i =1 n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x i k x i k − n X j =1 x ∗ j ( y i ) x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n X i =1 n (cid:12)(cid:12)(cid:12)(cid:12) k x i k − x ∗ i ( x i ) (cid:12)(cid:12)(cid:12)(cid:12) k x i k + X j = i | x ∗ j ( y i ) |k x j k < ε . Thus k I − p k ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 n x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Now, as we have done in Theorem 2.11 for the DD2P, we will focus onanalysing the DSD2P in the ℓ p sum of two Banach spaces. As every Banachspace enjoying to have the DSD2P has the strong diameter two property, weconclude that the ℓ p sum of two Banach spaces does not have the DSD2Pwhenever 1 < p < ∞ [1, Theorem 3.2]. Nevertheless, we will prove that, aswell as happens with Daugavet spaces, DSD2P has a nice behavior in thecase p = ∞ . We shall begin proving the following iametral diameter two properties in Banach spaces. 17 Proposition 3.7.
Let
X, Y be Banach spaces and assume that X ⊕ p Y hasthe DSD2P for p ∈ { , ∞} . Then X and Y enjoy to have the DSD2P.Proof. In order to prove the Proposition, assume that X does not satisfythe DSD2P. Then there exists C := P ni =1 λ i n i T j =1 S ( B X , x ∗ ij , η ij ) a convexcombination of non-empty relatively weakly open subsets of B X , an element P ni =1 λ i x i ∈ C and ε ∈ R + satisfying that(3.2) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε ∀ n X i =1 λ i y i ∈ C. Obviously we will assume the non-trivial case (i.e. P ni =1 λ i x i = 0), so wecan assume, taking δ < ε if necessary, that k P ni =1 λ i x i k − ε ≥ η ij are equal (say η ) and that η < ε ∀ i ∈ { , . . . , n } . Define C := n X i =1 λ i n i \ j =1 S ( B X ⊕ p Y , ( x ∗ ij , , η ) . If p = 1 we have from [1, Theorem 3.1, equation (3.1)] that(3.3) C ⊆ C × ηB Y . So consider P ni =1 λ i ( x i , ∈ C and pick P ni =1 λ i ( x ′ i , y ′ i ) ∈ C . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i (( x i , − ( x ′ i , y ′ i )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x ′ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y ′ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Now on the one hand, from (3.2), we have the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x ′ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε. On the other hand we have from (3.3) the following (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y ′ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < η. So combining both previous inequalities and keeping in mind that η < ε weconclude (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i (( x i , − ( x ′ i , y ′ i )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε . From the arbitrariness of P ni =1 λ i ( x ′ i , y ′ i ) ∈ C we conclude that X ⊕ Y failsthe DSD2P, so we are done in the case p = 1.The case p = ∞ is quite easier than the above one. Indeed, pick P ni =1 λ i ( x ′ i , y ′ i ) ∈C . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i (( x i , − ( x ′ i , y ′ i )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x ′ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) ≤ max ( (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε, ) = 1 + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε, where the last inequality holds from the assumption k P ni =1 λ i x i k − ε ≥ X ⊕ ∞ Y does not have the DSD2P, so we are done.Now we shall establish the converse of the result above for p = ∞ . Theorem 3.8.
Let
X, Y be a Banach spaces. If X and Y have the DSD2Pso does Z := X ⊕ ∞ Y .Proof. Pick n ∈ N , ( x , y ) , . . . , ( x n , y n ) ∈ B Z and λ , . . . , λ n ∈ R + suchthat P ni =1 λ i = 1. In order to prove that Z has the DSD2P we shall useProposition 3.4. As (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i , y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ = max ((cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)) , then either k P ni =1 λ i ( x i , y i ) k ∞ = k P ni =1 λ i x i k or k P ni =1 λ i ( x i , y i ) k ∞ = k P ni =1 λ i y i k .We shall assume, with no loss of generality, that k P ni =1 λ i ( x i , y i ) k ∞ = k P ni =1 λ i x i k . Now, as X has the DSD2P, we have from Proposition 3.4that, for each i ∈ { , . . . , n } , there exists { x is } a net weakly convergent to x i such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x is ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Now we have that { ( x is , y i ) } → ( x i , y i ) in the weak topology of Z for each i ∈ { , . . . , n } . Moreover, from the definition of the norm on Z , we havethat ( x is , y i ) ∈ B Z for each i ∈ { , . . . , n } and for each s . Finally, given s one has 1 + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i , y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i (( x i , y i ) − ( x is , y i )) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − x is ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 1 + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i , y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ . So (cid:13)(cid:13)P ni =1 λ i (( x i , y i ) − ( x is , y i )) (cid:13)(cid:13) ∞ → k P ni =1 λ i ( x i , y i ) k ∞ . Consequently, Z has the DSD2P applying Proposition 3.4, so we are done.Finally we will analyze the inheritance of DSD2P to subspaces. Again in[7] it is proved that given X a Banach space with the SD2P and Y ⊆ X aclosed subspace such that X/Y is strongly regular, then Y has the SD2P.Following similar ideas we have the following Theorem 3.9.
Let X be a Banach space and Y ⊆ X be a closed subspace. If X has the DSD2P and X/Y is strongly regular then Y also has the DSD2P. iametral diameter two properties in Banach spaces. 19 Proof.
Let C := n X i =1 λ i W i = n X i =1 λ i (cid:8) y ∈ B Y : | y ∗ ij ( y − y i ) | < η ij ≤ j ≤ n i (cid:9) be a convex combination of non-empty relatively weakly open subsets of B Y ,where y ∗ ij ∈ B Y ∗ for each i ∈ { , . . . , n } , j ∈ { , . . . , n i } and y i ∈ Y for each i ∈ { , . . . , n } . Pick P ni =1 λ i x i ∈ C , ε ∈ R + and let us prove that thereexists P ni =1 λ i y i ∈ C such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε. To this aim pick 0 < δ such that(3.4) | y ∗ ij ( x i − y i ) | + δ
32 + (cid:18) δ (cid:19) −
11 + δ ! < η ij ∀ i ∈ { , . . . , n } and(3.5) δ
16 + 2 (cid:18) δ (cid:19) −
11 + δ ! + δ < ε. Let π : X −→ X/Y the quotient map. We have no loss of gener-ality, by Hahn-Banach theorem, if we assume that y ∗ ij ∈ B X ∗ for each i ∈ { , . . . , n } , j ∈ { , . . . , n i } . Consider W i the non-empty relatively weaklyopen subset of B X defined by W i for each i ∈ { , . . . , n } .For each i ∈ { , . . . , n } consider A i := π ( W i ), which is a convex subset of B X/Y containing to zero. By [ ? , Proposition III.6] then A i is equal to theclosure of the set of its strongly regular points. As a consequence, for each i ∈ { , . . . , n } , there exists a i a strongly regular point of A i such that(3.6) k a i k < δ . For every i ∈ { , . . . , n } we can find m i ∈ N , µ i , . . . , µ im i ∈ ]0 ,
1] such that P m i j =1 µ ij = 1 and ( a i ) ∗ , . . . , ( a im i ) ∗ ∈ S ( X/Y ) ∗ , α ij ∈ R + satisfying that a i ∈ m i X j =1 µ ij ( S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i )and(3.7) diam m i X j =1 µ ij ( S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i ) < δ . It is clear that, for i ∈ { , . . . , n } and j ∈ { , . . . , m i } , one has S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i = ∅ ⇒ S ( B X , π ∗ (( a ij ) ∗ ) , α ij ) ∩ W i = ∅ . Check that we can not still apply the hypothesis because we do not knowwhether P ni =1 λ i x i ∈ C := P ni =1 λ i P m i j =1 µ ij ( W i ∩ S ( B X , π ∗ (( a ij ) ∗ ) , α ij )). Now, in order to finish the proof, we need to find points in C close enoughto P ni =1 λ i x i . This will be done in the following Claim 3.10.
We can find, for each i ∈ { , . . . , n } , an element z i ∈ B X suchthat P ni =1 λ i z i ∈ C and that (3.8) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − z i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < δ
32 + (cid:18) δ (cid:19) −
11 + δ ! . Proof.
Pick i ∈ { , . . . , n } . As k π ( x i ) − a i k = k a i k < δ we can find z i ∈ X such that π ( z i ) = a i and such that(3.9) k x i − z i k < δ . Now a i ∈ P m i j =1 µ ij ( S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i ) so, for each j ∈ { , . . . , m i } ,we can find b ij ∈ S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i such that a i = P m i j =1 µ ij b ij . Foreach j ∈ { , . . . , m i } we can find, considering a perturbation argument ifnecessary, an element z ij ∈ B X such that π ( z ij ) = b ij . Finally, as π ( z i ) − P m i j =1 µ ij π ( z ij ) = 0 we can find, by definition of the norm on X/Y , an element y i ∈ Y such that(3.10) z i = m i X j =1 µ ij z ij + y i , and(3.11) k y i k < δ . We shall prove that P ni =1 λ i z i δ works. First of all we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i z i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + n X i =1 λ i k x i − z i k (3.9) < δ . So z i δ ∈ B X for each i ∈ { , . . . , n } .Moreover, given i ∈ { , . . . , n } , j ∈ { , . . . , n i } , one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ∗ ij z i δ − y i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | y ∗ ij ( x i − y i ) | + k x i − z i k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) z i − z i δ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (3.9) < | y ∗ ij ( x i − y i ) | + δ
32 + (cid:18) δ (cid:19) −
11 + δ ! (3.4) < η ij . Finally, pick i ∈ { , . . . , n } and j ∈ { , . . . , m i } . Then by (3.10) one has z i = m i X j =1 µ ij ( z ij + y i ) . In addition π ∗ ( a ∗ ij )( z ij + y i ) = a ∗ ij ( b ij ) + a ∗ ij ( π ( y i )) . iametral diameter two properties in Banach spaces. 21 On the one hand, as y i ∈ Y then π ( y i ) = 0. On the other hand a ∗ ij ( b ij ) > − α ij . Now, up to consider a smaller positive number in (3.9) (check thatthe choice of b ij does not depend on the one of z , . . . , z n ), we can assumethat a ∗ ij ( b ij ) > (1 − α ij ) (cid:0) δ (cid:1) , so P ni =1 λ i z i δ ∈ C . Now the claim followsjust considering z i δ instead of z i .Now, as X has the DSD2P, we can find P ni =1 λ i z ′ i ∈ C such that(3.12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( z i − z ′ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i z i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − δ . Given i ∈ { , . . . , n } we have that π ( z ′ i ) ∈ m i X j =1 µ ij ( S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i ) ⇒ k π ( z ′ i ) k ≤ k a i k + diam m i X j =1 µ ij ( S ( B X/Y , ( a ij ) ∗ , α ij ) ∩ A i ) (3.6)(3.7) < δ . Now, as it is done in Proposition 2.13, we can find y i ∈ B Y such that(3.13) k y i − z ′ i k < δ . Now, on the one hand, given i ∈ { , . . . , n } and j ∈ { , . . . , n i } , one has | y ∗ ij ( y i − y ) | ≤ | y ∗ ij ( z ′ i − y ) | + | y ∗ ij ( y i − z i ) | < η ij + δ . On the other hand, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − y i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( z i − z ′ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( x i − z i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( y i − z ′ i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (3.12) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i z i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − δ − n X i =1 λ i k y i − z ′ i k − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( z i − x i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (3.8)(3.13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i z i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − δ − δ − δ − (cid:18) δ (cid:19) −
11 + δ ! (3.8) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − δ − δ − (cid:18) δ (cid:19) −
11 + δ ! (3.5) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε. From the arbitrariness of ε we conclude that Y has the DSD2P by a pertur-bation argument similar to the one done in Proposition 2.13. Now we have a weak-star version of Theorem 3.9.
Corollary 3.11.
Let X be a Banach space and let Y ⊆ X be a closedsubspace. If X ∗ has the w ∗ -DSD2P and Y is reflexive then ( X/Y ) ∗ has the w ∗ -DSD2P.Proof. Consider C := P ni =1 λ i W i a convex combination of non-empty rela-tively weakly-star open subsets of B Y ◦ and pick P ni =1 λ i z ∗ i ∈ C .Define W i to be the weak-star open subset of B X ∗ define by W i for each i ∈ { , . . . , n } .Let π : X ∗ −→ X ∗ /Y ◦ the quotient map and define A i := π ( W i ).As X ∗ /Y ◦ = Y ∗ is reflexive, then X ∗ /Y ◦ is strongly regular, so we canfind, for each i ∈ { , . . . , n } , a i a point of strong regularity point of A i whosenorm is as close to zero as desired. Given i ∈ { , . . . , n } , as a i is a point ofstrong regularity, we can find convex combination of slices containing a i andwhose diameter is as small as wanted. In addition, because of reflexivityof X ∗ /Y ◦ , convex combination of slices are indeed convex combination ofweak-star slices, so we can actually find convex combination of weak-starslices containig to a i and whose diameter is a closed to zero as desired foreach i ∈ { , . . . , n } .Using the previous ideas, the result can be concluded following word byword the proof of Theorem 3.9.Now we shall prove the inheritance of the DSD2P to almost isometricideals. Proposition 3.12.
Let X be a Banach space and let Y ⊆ X a closed almostisometric ideal. If X has the DSD2P, so does Y .Proof. Pick C := P ni =1 λ i n i T j S ( B Y , y ∗ ij , α ij ) a convex combination of non-empty relatively weakly open subsets of B Y , choose P ni =1 λ i y i ∈ C and pick ε >
0. Our aim is to find P ni =1 λ i z i ∈ C such that k P ni =1 λ i ( y i − z i ) k > k P ni =1 λ i y i k− ε . Assume, with no loss of generality, that max ≤ i ≤ n max ≤ j ≤ n i y ∗ ij ( y i ) <
1. Choose µ > < µ < µ ⇒ y ∗ ij ( y i )1 + µ > − α ij ∀ i ∈ { , . . . , n } , j ∈ { , . . . , n i } . and(3.15) 0 < µ < µ ⇒ µ (1 + k P ni =1 λ i y i k − µ ) − µ µ > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε iametral diameter two properties in Banach spaces. 23 Now consider 0 < µ < µ and ϕ : Y ∗ −→ X ∗ a Hahn-Banach operatorsatisfying the properties described in Theorem 1.1. Define b C := n X i =1 λ i n i \ j =1 S ( B X , ϕ ( y ∗ ij ) , − y ∗ ij ( y i )) . As X has the DSD2P and clearly P ni =1 λ i y i ∈ b C we can conclude the exis-tence of an element P ni =1 λ i x i ∈ b C such that(3.16) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i ( y i − x i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − µ. Now for µ , E := span { x , . . . , x n , y , . . . , y n } ⊆ E and F := span { y ∗ ij / i ∈{ , . . . , n } , j ∈ { , . . . , n i }} ⊆ Y ∗ consider T the operator satisfying theproperties described in Theorem 1.1. Given i ∈ { , . . . , n } one has k T ( x i ) k ≤ (1 + µ ) k x i k ≤ µ. So, if we define z := P ni =1 λ i T ( x i )1+ µ , it is clear that z ∈ B Y . We will provethat indeed z ∈ C . To this aim, pick i ∈ { , . . . , n } and j ∈ { , . . . , n i } .Hence y ∗ ij (cid:18) T ( x i )1 + µ (cid:19) = y ∗ ij ( T ( x i ))1 + µ = ϕ ( y ∗ ij )( x i )1 + µ > − (1 − y ∗ ij ( y i ))1 + µ = y ∗ ij ( y i )1 + µ (3.14) > − α ij . Thus z ∈ C . Finally, we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y i − z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i (cid:18) y i − T ( x i )1 + µ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k P ni =1 λ i ( y i − T ( x i ) + µy i ) k µ ≥ k T ( P ni =1 λ i ( y i − x i )) k − µ P ni =1 λ i k y i k µ > µ k P ni =1 λ i ( y i − x i ) k − µ µ (3.16) > µ (1 + k P ni =1 λ i y i k − µ ) − µ µ (3.15) > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 λ i y i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ε. As ε was arbitrary we conclude that Y has the DSD2P, so we are done.4. Some remarks and open questions.
Let us consider the following diagram DP (1) = ⇒ DSD P (2) = ⇒ DD P (3) = ⇒ DLD P ⇓ (4) ⇓ (5) ⇓ (6) w ∗ − DSD P (7) = ⇒ w ∗ − DD P (8) = ⇒ w ∗ − DLD P where the last row only make sense in dual Banach spaces. By Example2.2 or Theorem 2.11, neither the converse implication of (2) nor the one of(7) holds. In addition, there are Banach spaces with the Daugavet propertywhose dual unit ball have denting points (e.g. C ([0 , Question 1.
Does the converse of (1),(3) or (8) hold?
It is known that a Banach space X has the DLD2P if, and only if, X ∗ has the w ∗ -DLD2P. This fact arise two questions. Question 2.
Let X be a Banach space. Is it true that X has the DD2P if,and only if, X ∗ has the w ∗ -DD2P? A similar question remains open for the DSD2P.
Question 3.
Let X be a Banach space. Is it true that X has the DSD2Pif, and only if, X ∗ has the w ∗ -DSD2P? Check that a positive answer to the above question would provide, byProposition 3.5 and a similar argument to the one done in Theorem 3.8 tothe dual space, a positive answer to the following
Question 4.
Let
X, Y
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Universidad de Granada, Facultad de Ciencias. Departamento de An´alisisMatem´atico, 18071-Granada (Spain)
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