Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and ∆-points

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2024) 118:96 https://doi.org/10.1007/s13398-024-01596-x ORIGINAL PAPER Banach spaces with small weakly open subsets of the unit ball and massive sets of Daugavet and 1-points Christian Cobollo1·Daniel Isert2·Ginés López-Pérez3·Miguel Martín3· Yoël Perreau4·Alicia Quero3,5 ·Andrés Quilis6· Daniel L. Rodríguez-Vidanes7·Abraham Rueda Zoca3 Received: 7 September 2023 / Accepted: 24 March 2024 © The Author(s) 2024 Abstract We prove that there exists an equivalent norm |||·||| on L∞[0,1]with the following properties: (1) The unit ball of (L∞[0,1],|||·|||)contains non-empty relatively weakly open subsets of arbitrarily small diameter; (2) The set of Daugavet points of the unit ball of (L∞[0,1],|||·|||)is weakly dense; (3) The set of ccw -points of the unit ball of (L∞[0,1],|||·|||)is norming. We also show that there are points of the unit ball of (L∞[0,1],|||·|||)which are not -points, meaning that the space (L∞[0,1],|||·|||)fails the diametral local diameter 2 property. Finally, we observe that the space (L∞[0,1],|||·|||)provides both alternative and new examples that illustrate the differences between the various diametral notions for points of the unit ball of Banach spaces. Keywords Daugavet points ·-Points ·Points of continuity ·Renormings ·Space of essentially bounded measurable functions Mathematics Subject Classification 46B04 ·46B20 ·46B22 1 Introduction Recall that a Banach space Xis said to have the Daugavet property if every rank one bounded operator T:X−→ Xsatisfies the Daugavet equation I+T=1+T,(DE) where I:X−→ Xstands for the identity operator. Furthermore, if Xhas the Daugavet property, then every weakly compact operator T:X→Xsatisfies (DE). Since the Daugavet equation is a stress of the operator norm’s triangle inequality, it is natural to expect that it will impose severe restrictions on the underlying operator. As a matter of fact, if Tis an BAbraham Rueda Zoca [email protected] Extended author information available on the last page of the article 0123456789().: V,-vol 123 96 Page 2 of 17 C. Cobollo et al. eigenvalue of T,thenTsatisfies the Daugavet equation, and the converse holds true if the space Xis uniformly convex [7, Lemma 11.3 and Theorem 11.10]. Actually, the Daugavet property puts very strong constraints on the structure of the underlying Banach space. An old result in this line is that a Banach space with the Daugavet property cannot be linearly embedded into any Banach space with an unconditional basis (see e.g. [24, Theorem 3.2]). Further restrictions follow from the celebrated geometric characterisation of the Daugavet property exhibited in [18, Lemma 2.1] stated as follows: a Banach space Xhas the Daugavet property if and only if every point x∈SXsatisfies the following condition: given any slice Sof BXand any ε>0, there exists y∈Ssuch that x−y>2−ε. The latter characterisation, which still holds with respect to non-empty relatively weakly open subsets (resp. convex combinations of slices) [21, Lemma 3], shows that spaces with the Daugavet property live in the universe of Banach spaces far away from Asplundness and Radon–Nikodym property. Indeed, the above characterisation allows to prove that if Xhas the Daugavet property, then Xcontains an isomorphic copy of 1, and every slice, relatively weakly open subset and convex combination of slices of BXhas diameter two. Very recently, local versions of the Daugavet property have been considered in the following sense. Definition 1.1 Let Xbe a Banach space and let x∈SX. We say that xis (1) a Daugavet point if, for every slice Sof BXand every ε>0, there exists y∈Ssuch that y−x>2−ε, (2) a super Daugavet point if, for every non-empty relatively weakly open subset Wof BX and every ε>0, there exists y∈Wsuch that y−x>2−ε, (3) a ccs Daugavet point if, for every convex combination of slices Cof BXand every ε>0, there exists y∈Csuch that y−x>2−ε. A classical result, often known as Bourgain’s lemma, establishes that every non-empty relatively weakly open subset of BXcontains a convex combination of slices of BX(see e.g. [11, Lemma II.1]). As an immediate consequence we infer that every ccs Daugavet point is a “ccw Daugavet point”, meaning that the property of the definition actually holds for every convex combination of non-empty relatively weakly open subsets of BX. In particular, every ccs Daugavet point is a super Daugavet point. Furthermore, it is known that the mere existence of a ccs Daugavet point implies that every convex combination of slices (and of weak open subsets) of the unit ball of the underlying space has diameter 2 [20, Proposition 3.12]. Apart from finite dimensional considerations, this is surprisingly the only known isomorphic obstruction to the existence of diametral points, see below for more details. Variants of the above notions restricting to slices, weakly open subsets, and convex combinations of slices and weakly open subsets containing a given point, were also considered. Definition 1.2 Let Xbe a Banach space and let x∈SX. We say that xis (1) a -point if, for every slice Sof BXwith x∈Sand every ε>0, there exists y∈Ssuch that y−x>2−ε, (2) a super -point if, for every non-empty relatively weakly open subset Wof BXwith x∈Wand every ε>0, there exists y∈Wsuch that y−x>2−ε, (3) a ccs -point if, for every convex combination of slices Cof BXwith x∈Cand every ε>0, there exists y∈Csuch that y−x>2−ε, 123 Banach spaces with small weakly open... Page 3 of 17 96 (4) a ccw -point if, for every convex combination of non-empty relatively weakly open subsets Dof BXwith x∈Dand every ε>0,there exists y∈Dsuch that y−x>2−ε. The notions of Daugavet and -points were introduced in [4, Section 1], whereas the rest of notions go back to [20, Definitions 2.4 and 2.5]. See [1,2,16,19,20,23] for further research on these notions. In particular, note that it is still unknown whether every ccs -point has to be a super -point, and whether the notions of ccs and ccw -points are different. This is due to the subtle failure of a localization of Bourgain’s lemma (see e.g. [20, Remark 2.3]). However, all the other notions are known to be different and can even present extreme differences, see [20] for more details. In view of the fact that the Daugavet property imposes strong restrictions on the geometric structure of the given space, a natural question is how the mere presence of Daugavet or -points affect the geometric structure of the underlying Banach space. Although it was proved in [5] that finite dimensional spaces contain no -points and that the notion strongly negates some isometric properties of Banach spaces (asymptotic uniform smoothness and weak∗asymptotic uniform convexity [2,5], or existence of subsymmetric bases [6]aswell as unconditional bases with small constants of unconditionality which are either shrinking or boundedly complete [2]), surprising examples have recently shown the purely isometric nature of these local notions. To name a few, there exists a space with a 1-unconditional basis and a weakly dense subset of Daugavet point [6], there exists a Lipschitz-free space with the RNP and a Daugavet point which is both isomorphic to 1and isometric to a dual space [2,23], and there exists an equivalent norm on 2for which the unit vector basis e1 is simultaneously a super Daugavet point and a ccw -point [15]. Actually, every infinite dimensional Banach space can be renormed with a -point [2], and every Banach space with a weakly-null unconditional Schauder basis can be renormed with a super Daugavet point [15]. The various -notions can be seen as extreme opposites to the classical notions of denting points, points of continuity and points of strong regularity (also see [9] for precise quantitative formulations of this statement). They are localized versions of the so-called “diametral diameter 2 properties” (DLD2P,DD2P and DSD2P) that have previously appeared in the literature under various names, but were formally introduced in [14]. With this terminology, the DLD2P (resp. DD2P) asks all the elements of the unit sphere of a Banach space to be -points (resp. super -points). The DSD2P was originally defined by asking all the points inside of the unit ball of a Banach space to be ccs -points, but it turned out to be equivalent to the Daugavet property [17]. On the other hand, its restricted version (the restricted DSD2P [20]), as well as the DLD2P and the DD2P, are known to be strictly weaker properties. Although the Daugavet property can be characterized by Daugavet, super Daugavet or ccs Daugavet points, it is currently unknown whether the three remaining diametral properties are equivalent. Furthermore, it is unknown whether the DLD2P forces all the weakly open subsets of the unit ball to have diameter 2 (but note that there exists a space with the DD2P, the restricted DSD2P and convex combinations of slices of arbitrarily small diameter in its unit ball [3]). The example from [6] provides an interesting insight to this question. Indeed, the space that was constructed there with a weakly dense subset of Daugavet points and a 1-unconditional basis admits non-empty relatively weakly open subsets of arbitrarily small diameter in its unit ball. In fact, each of the Daugavet points in the considered weakly dense set is a point of continuity for the identity mapping I:(BX,w)→(BX,·)(in other words, it has relative weak open neighborhoods of arbitrarily small diameter). However, this space cannot contain any point satisfying a stronger diametral condition, as it was proved in [6] (resp. [20]) that 123 96 Page 4 of 17 C. Cobollo et al. spaces with a 1-unconditional basis contain neither super -points nor ccs -points. Thus, at this point, a natural question is how big the set of stronger notions than Daugavet and -points can be in a Banach space where there are non-empty relatively weakly open subsets of arbitrarily small diameter. In view of this fact, during the last week of June 2023, in the framework of the 2023 ICMATIMAG Doc-Course in Functional Analysis, a supervised research program was celebrated at IMAG (Granada), where we considered the following question: How massive can the sets of Daugavet, super , super Daugavet and ccs/ccw -points be in a Banach space having non-empty relatively weakly open subsets of arbitrary small diameter in its unit ball? The main goal of the project was to study the renorming techniques from [12], where it is proved that every Banach space containing c0can be renormed in such a way that all the slices of the new unit ball have diameter 2, whereas it admits weakly open subsets of arbitrarily small diameter, and to try to build a similar renorming in a more suitable context for our study, namely in the space L∞[0,1]. The idea is also inspired by the construction from [20, Section 4.6], where similar techniques were used in order to produce an example of a super Daugavet point that is not a ccs -point. The main aim of the present paper is to present the results obtained in this workshop. We prove that the space L∞[0,1]admits an equivalent renorming such that the new unit ball contains non-empty relatively weakly open subsets of arbitrarily small diameter and such that the sets of Daugavet points and super -points are as big as they can be taking into account that its unit ball contains non-empty weakly open subsets of small diameter. This is a big difference with the above-mentioned example of [6], where the set of super -points is empty. Furthermore, we show that this space also contains points which are simultaneously super Daugavet and ccw , which is the strongest diametral notion we can get in this context. We collect the results in the following theorem. Theorem 1.3 For every ε∈(0,1), there exists an equivalent norm |||·|||εon L∞[0,1]with the following properties: (1) For every f ∈L∞[0,1],f∞≤||| f|||ε≤1 1−εf∞; (2) The unit ball of (L∞[0,1],|||·|||ε)contains non-empty relatively weakly open subsets of arbitrarily small diameter; (3) The set of Daugavet points of the unit ball of (L∞[0,1],|||·|||ε)is weakly dense; (4) The set of ccw -points of the unit ball of (L∞[0,1],|||·|||ε)is norming (in other words, every slice of the unit ball contains a ccw -point); (5) There are points of the unit ball of (L∞[0,1],|||·|||ε)which are: (a) Simultaneously super Daugavet points and ccw -points; (b) Simultaneously Daugavet points and preserved extreme points (hence also ccw - points), but not super Daugavet points; (c) Simultaneously Daugavet points and points of continuity. Furthermore, if εis smaller than 1/7, then there are points of the unit ball of (L∞[0,1],|||·|||ε) which are not -points (in other words, (L∞[0,1],|||·|||ε)fails the DLD2P). In particular, in the above renorming there are Daugavet points which are not super - points and there are ccw -points which are not super Daugavet points. Even though it was already known that these notions are not equivalent (see [20] for references), the various counterexamples from the literature were obtained with different techniques. Theorem 1.3 shows that such counterexamples may live in the same Banach space. Furthermore, it is, to our knowledge, the first example of a Banach space which contains points that are both Daugavet and ccw , but not super Daugavet. 123 Banach spaces with small weakly open... Page 5 of 17 96 2 Notation and preliminary results Given a Banach space X,BX(resp. SX) stands for the closed unit ball (resp. the unit sphere) of X.WedenotebyX∗the topological dual of X. By a slice of BX, we mean any non-empty subset of BXgiven as the intersection of BXwith an open half-space. Every slice Sof BX can be written as S=S(BX,f,δ),where fis a norm one functional on X,δ>0and S(BX,f,δ):= {x∈BX:f(x)>1−δ}. If Ais a subset of a Banach space X,wedenotebycoA(resp. co A) the convex hull (resp. the closure of the convex hull) of A. Recall that a subset Ain the unit ball of Banach space Xis said to be norming if x∗=supx∈A|x∗(x)|for every x∗∈X∗. In particular, if Ais a symmetric subset of BX, then this property is equivalent to Asatisfying BX=co A(in other words, to every slice of BXcontaining an element of A). We deal with real Banach spaces only. Let μbe the Lebesgue measure on the segment [0,1]. Recall that two measurable subsets Aand Bof [0,1]are said to be essentially disjoint if μ(A∩B)=0. The space L∞[0,1] stands for the classical Banach space of all equivalent classes of μ-essentially bounded functions on [0,1]equipped with the norm given by the essential supremum. Recall that the following criteria provides a practical way of testing whether a given sequence in L∞[0,1] is weakly-null (see e.g. [22, Theorem 8.7]). Theorem 2.1 A bounded sequence (un)in L∞[0,1]converges weakly to 0if and only we can find, for every δ>0and (kj)increasing sequence of natural numbers, some J ∈Nsuch that μ⎛ ⎝ J  j=1t∈[0,1]: ukj(t)>δ ⎞ ⎠=0. In particular, every bounded sequence of functions with pairwise essentially disjoint supports in L∞[0,1]is weakly-null. We now recall some classical definitions from Banach space geometry. Given a convex set Ain a vector space X, a point x0∈Ais said to be extreme if the condition x0=y+z 2for y,z∈Aforces y=z=x0. Given a bounded, closed, and convex subset Cof a Banach space X, a point x0∈Cis a preserved extreme point if x0is an extreme point in Cw∗ ,wherethe closure is taken in the w∗-topology of X∗∗. For easy reference, let us point out the following characterisation of preserved extreme points (which proof can be found, for instance, in [10, Proposition 0.1.3]). Proposition 2.2 Let X be a Banach space and let C ⊆X be a bounded, closed, and convex set. Let x0∈C. The following are equivalent: (1) x0is a preserved extreme point of C; (2) The slices of C containing x0form a neighbourhood basis of x0in C for the weak topology; (3) For every pair of nets (ys)and (zs)in C such that ys+zs 2→x0weakly, we have ys→x0 weakly. Given a Banach space Xand a subset A⊆X, a point x0∈Ais said to be a point of continuity if, for every ε>0, there exists a weakly open subset W⊆Awith x0∈Wand diam (W)<ε. Observe that this means that the identity mapping I:(A,w)−→ (A,·) 123 96 Page 6 of 17 C. Cobollo et al. is continuous at x0. In turn, this is equivalent to the fact that if a net (xs)of elements of A satisfies that xs w →x0,thenxs−x0→0. A closed and bounded set B(resp. a closed convex and bounded set C) in a Banach space Xis said to have the point of continuity property (resp. convex point of continuity property (CPCP)) if every closed subset Aof B(resp. every closed and convex subset Aof C) contains a point of continuity. We finally recall the definition of the “Summing Tree Simplex” from [8] that was constructed in order to distinguish between the CPCP and the PCP for subsets of Banach spaces. This set will be the stepping stone for our renorming of L∞[0,1].LetN<ω be the set of all ordered finite sequences of positive integers including the empty sequence denoted by ∅.If α=(α1,...,α n)∈N<ω, the length of αis |α|=nand |∅| = 0. For simplicity, we will sometimes identify N1with Nand denote by ithe sequence (i)with one element i∈N.We use the natural order in N<ω given by: αβif |α|≤|β|and αi=βifor all i∈{1,...,|α|}, and ∅αfor any α∈N<ω.Wedenotebyα4ithe finite sequence resulting from the concatenation of an element α∈N<ω with the sequence i=(i)with only one element i∈N. Let c0(N<ω)be the completion of the space c00(N<ω)of all finitely supported families of real numbers indexed by N<ω with the supremum norm. Then c0(N<ω)is isometric to the usual space c0. From now on, in order to distinguish with the norm from the space L∞[0,1], we will denote by .the supremum norm on c0and c0(N<ω), and by .∞the essential supremum norm on L∞[0,1].Let(eα)α∈N<ω be the unit vector basis of c00(N<ω) and (e∗ α)α∈N<ω be the sequence of biorthogonal functionals. For a given α∈N<ω,let xα:=  βα eβ. We consider the set K:= co{xα}α∈N<ω ⊂S+ c0. Some properties of the set Kare given in [8, Theorem 1.1]. In particular, it is proved there that Khas the CPCP but fails the PCP. We end the present section by providing a few more properties for K. Lemma 2.3 For every x ∈K and for every slice S of K , supy∈Sx−y=1. Proof Observe that for every z∈co{xα}α∈N<ω and for every α∈N<ω,wehave lim z−xα4n=1. Thus, since every slice of Kcontains some xα, and since xα4n→nxα weakly, the conclusion follows from an easy density argument.  From the fact that Khas the CPCP it is immediate to infer that Kcontains non-empty relatively weakly open subsets of arbitrarily small diameter. However, we will describe a particular family of non-empty relatively weakly open subsets of small diameter because they will be useful in order to localise ccw -points which are not super Daugavet points in the final renorming of L∞[0,1](see Remark 3.8). Lemma 2.4 For n ∈Nand ρ∈(0,1/n),let Vn,ρ := n  i=1 {z∈K:e∗ i(z)>1/n−ρ}. 123 Banach spaces with small weakly open... Page 7 of 17 96 Then Vn,ρ is a non-empty relatively weakly open subset of K with diameter smaller than 2/n+2nρ. Proof For i∈{1,...,n},letxi:= x(i).Thenx0:= 1 nn i=1xi∈Vn,ρ. Clearly, it is enough to prove that for every z∈co{xα}α∈N<ω ∩Vn,ρ ,x0−z≤1/n+nρ.Fixsuchaz,and write z=L l=1λlxαlwith λl>0, L l=1λl=1, and αl∈N<ω.Foreveryi∈N,let Ai:= {l:(i)αl}. Since z∈Vn,ρ,wehavee∗ i(z)=l∈Aiλl>1/n−ρfor every i≤n. So observe that for any given j∈{1,...,n},wehave  l∈Aj λl= n  i=1 l∈Ai λl− n  i=1 i= j l∈Ai λl≤1−(n−1)(1/n−ρ) =1/n+(n−1)ρ. Inthesameway,  i>n l∈Ai λl≤ i∈N l∈Ai λl− n  i=1 l∈Ai λl≤1−n(1/n−ρ) =nρ. Now let us define v=x0−z=1 nn i=1xi−L l=1λlxαland let us fix β∈N<ω.We want to evaluate |v(β)|. There are three cases to consider. Case 1. If β=∅,thenv(β) =0, so there is nothing to do. Case 2. If |β|>1, take jβ∈Nsuch that (jβ)β. Then, either there is no l∈{1,...,L} such that (jβ)αlin which case v(β) =0, or there is an l∈{1,...,L}such that (jβ)αl, and |v(β)|=L l=1λlxαl(β)≤l∈Ajβλl. Hence |v(β)|≤max{1/n+(n−1)ρ, nρ}≤ 1/n+nρ. Case 3. If β=(jβ)for some jβ∈N, then either jβ>n,and|v(β)|≤l∈Ajβλl≤nρ, or jβ≤n,and|v(β)|=1/n−l∈Ajβλl≤nρbecause 1/n−ρ<l∈Ajβλl≤ 1/n+(n−1)ρ. It follows that x0−z=supβ∈N<ω |v(β)|≤1/n+nρ,aswewanted.  For easy future reference, we end the section with the following lemma, whose proof follows from [8, P.82]. Lemma 2.5 Every xαis a preserved extreme point of K . 3 Main result The aim of the section is to prove Theorem 1.3.Let{Aα}α∈N<ω be a family of pairwise disjoint non-empty open subsets of [1/2,1]. Then, for every α∈N<ω,let fα:= 1Aαand let Eαbe the norm one functional on L∞[0,1]given by Eα(f):= 1/μ(Aα)·Aα fdμ for every f∈L∞[0,1]. We will use the fαto construct a positive isometric embedding of c0(N<ω)into L∞[0,1].Foreveryz:= (zα)α∈N<ω ∈c00(N<ω),wedefine (z):=  α∈N<ω zαfα. 123 96 Page 8 of 17 C. Cobollo et al. By the disjointness of the support of the functions fα, this map is clearly isometric and sends positive sequences to positive functions. Thus it can be extended by density to a positive isometric embedding :c0(N<ω)→L∞[0,1]. Furthermore, observe that, by construction, satisfies the equation Eα◦=e∗ α,(3.1) that is Eα((z)) =zαfor every z:= (zα)α∈N<ω ∈c0(N<ω). Let K0:= (K)be the image of the subset Kof c0(N<ω)from the preliminary section, and fix ε∈(0,1). We consider the equivalent norm |||·|||εon L∞[0,1]given by the Minkowski functional of the set Bε:= co (2K0−1)∪(−2K0+1)∪(1−ε)BL∞[0,1]+εBker(E0), where 1:= 1[0,1]and E0is the norm one functional on L∞[0,1]given by E0(f):= 4·1/4 0fdμfor every f∈L∞[0,1]. Observe that (1−ε)BL∞[0,1]⊂Bε⊂BL∞[0,1],which means (1−ε) |||·|||ε≤·∞≤|||·|||ε. We will prove that this renorming of L∞[0,1]satisfies all the properties of Theorem 1.3. Let A:= (2K0−1),B:= (1−ε)BL∞[0,1]+εBker(E0)and Aε:= A∪−A∪B.For every α∈N<ω,lethα:= βαfβand uα:= 2hα−1. Observe that A=co{uα}α∈N<ω , and that E0(ψ) =−1foreveryψ∈Aand E0(ϕ) ∈[−1+ε, 1−ε]for every ϕ∈B.We will start by proving that our renorming produces weakly open subsets of arbitrarily small diameter in the new unit ball. Proposition 3.1 The set Bεadmits non-empty relatively weakly open subsets of arbitrarily small diameter. Proof For n∈Nand ρ>0, consider ˜ Vn,ρ := f∈Bε:E0(f)<−1+ρand Ei(f)>21 n−ρ−1 for every i∈{1,...,n}, where istands the sequence (i). Note that for x0:= 1 nn i=1xi,wehavethat f0:= 2(x0)− 1∈˜ Vn,ρ. By density, it is enough to find an upper bound for the distance of fto f0for every f∈˜ Vn,ρ ∩co Aε.Sopicksuchan f, and write f=λ1f1+λ2f2+λ3f3 with λ1,λ 2,λ 3∈[0,1],3 i=1λi=1, f1∈A,f2∈−Aand f3∈B. Then, evaluating against the functional E0,weget −1+ρ>E0(f)≥−λ1+λ2−(1−ε)λ3≥−λ1−(1−ε)λ3, hence 1−ρ<λ 1+(1−ε)λ3. In particular, 1−ρ<λ 1+λ3=1−λ2, and we get λ2<ρ. Furthermore, since λ1=1−λ2−λ3≤1−λ3, we have 1−ρ<1−λ3+(1−ε)λ3, 123 Banach spaces with small weakly open... Page 9 of 17 96 and thus λ3<ρ/ε. Finally λ1=1−λ2−λ3>1−(1+1/ε)ρ. It follows that ||| f−f1|||ε≤(1−λ1)||| f1|||ε+λ2||| f2|||ε+λ3||| f3|||ε<2(1+1/ε)ρ. Write f1=2(z)−1with z∈K.Foreveryi∈{1,...,n},wehave 2(1/n−ρ) −1<Ei(f)=λ1Ei(f1)+λ2Ei(f2)+λ3Ei(f3)<Ei(f1)+2(1+1/ε)ρ, thus zi=Ei((z)) =Ei(f1)+1 2>1 n−(2+1/ε)ρ, which means z∈Vn,˜ρfor ˜ρ=(2+1/ε)ρ. So by Lemma 2.4,weget ||| f1−f0|||ε=2|||(x0)−(z)|||ε≤2 1−ε(x0)−(z)∞ =2 1−εx0−z≤4/n+4n˜ρ 1−ε. The conclusion follows.  Actually, we can say a bit more in that regard: the set Bεadmits points of continuity. Indeed, the latter claim immediately follows from the fact that the set Kitself admits points of continuity (it has the CPCP, see [8, Theorem 1.1(c)]) together with the following result. Proposition 3.2 Let z be a point of continuity of K . Then 2(z)−1is a point of continuity of Bε. Proof Let z∈Kbe a point of continuity. To show that f:= 2(z)−1is a point of continuity of Bε, it is enough by density to prove that for every net (fs)in co Aε,wehavethat fs→f weakly if and only if ||| f−fs|||ε→0. So consider such a net, and for every s, write fs=λ1 sf1 s+λ2 sf2 s+λ3 sf3 s with λ1 s,λ 2 s,λ 3 s∈[0,1],3 i=1λi s=1, f1 s∈A,f2 s∈−Aand f3 s∈B.If fs→f weakly, then E0(fs)→E0(f)=−1. Now since E0(f1 s)=−1, E0(f2 s)=1andE0(f3 s)∈[−1+ε, 1−ε]for every s,it immediately follows that λ1 s→1andλ2 s,λ 3 s→0. From the above we conclude that f1 s→fweakly. For every s,pickzs∈Ksuch that f1 s=2(zs)−1.Then(zs)→(z)weakly, and since is a linear isometry, we get that zs→zweakly. But zis a point of continuity of K, so it follows that z−zs→0. Going back to L∞[0,1], we get that |||(z)−(zs)|||ε→0, and thus ||| f−fs|||ε→0, as we wanted.  We will now prove that the set of Daugavet points of Bεis weakly dense. The following approximation lemma will be useful throughout the rest of the article. 123 96 Page 16 of 17 C. Cobollo et al. Interdisciplinar” (IMI). The research of A. Rueda Zoca was also funded by Fundación Séneca: ACyT Región de Murcia grant 21955/PI/22 and by Generalitat Valenciana project CIGE/2022/97. Funding Funding for open access publishing: Universidad de Granada/CBUA. Data availability Not applicable. 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Authors and Affiliations Christian Cobollo1·Daniel Isert2·Ginés López-Pérez3·Miguel Martín3· Yoël Perreau4·Alicia Quero3,5 ·Andrés Quilis6· Daniel L. Rodríguez-Vidanes7·Abraham Rueda Zoca3 Christian Cobollo [email protected] Daniel Isert [email protected].es Ginés López-Pérez [email protected] Miguel Martín [email protected] Yoël Perreau [email protected] Alicia Quero [email protected] Andrés Quilis [email protected] Daniel L. Rodríguez-Vidanes [email protected] 1Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain 2Departamento de Análisis Matemático, Univeristat de València, Calle Doctor Moliner, 50, Burjassot, 46100 Valencia, Spain 3Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain 4Institute of Mathematics and Statistics, University of Tartu, Narva mnt 18, 51009 Tartu, Estonia 5Present Address: Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00 Prague 6, Czech Republic 6Laboratoire de mathématiques de Besançon, UMR CNRS 6623, Université Franche-Comté, 16, route de Gray, 25000 Besançon, France 7Departamento de Matemática Aplicada a la Ingeniería Industrial, Escuela Técnica Superior de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain 123