STABILITY of DIAMETRAL DIAMETER TWO PROPERTIES 11 Aforementioned Question Is Positive

arXiv:2012.09492v1 [math.FA] 17 Dec 2020 l ftesie fteui alhv imtrto hni is it then two, diameter have ball unit the of slices the the of all h 2 n h 2 mle h DP oee,bt fteconv the of both However, LD2P. the implies D2P the and D2P the al(.. lcs aedaee w se[6) aahspa Banach A [26]). the (see have two to diameter weasaid have is relatively it slices) nonempty Moreover, (e.g., all two. ball that diameter satisfy has ball property unit Daugavet the of slice every that rpryhsatatdalto teto n ayeape h examples many and attention of lot a C attracted has property rudteDuae rpry(e .. 2,[] 1] [15] [10], inspi [5], has 1.1 [2], it Lemma e.g., Moreover, (see property property. Daugavet Daugavet the the around with spaces of whenever h rtt bev httespace the that observe to first the pcsfraols measure atomless for spaces nyi o every for if only rjcietno product. tensor Projective where T ( : bev htLma11imdaeygvsta nasaewith space a in that gives immediately 1.1 Lemma that Observe h olwn emti hrceiainhsbe eyus very a been has characterisation geometric following The e od n phrases. and words Key 2010 hswr a upre yteEtna eerhCuclgra Council Research Estonian the by supported was work This aahspace Banach A K X oa imtrtoproperty two diameter local k ) TBLT FDAERLDAEE W PROPERTIES TWO DIAMETER DIAMETRAL OF STABILITY x → pcsfracmatHudr space Hausdorff compact a for spaces I ahmtc ujc Classification. Subject Mathematics F Abstract. h imta imtrtopoete r tbeudrtef the under (e.g., stable spaces are Köthe–Bochner properties ing assumptions two geometric diameter strong diametral under the superspace the to ideal h rjcietno rdc ftoBnc pcshssm d some has spaces Banach two of property. product tensor projective the − : X iel (e.g., -ideals X y M ≥ k aifistenr equality norm the satisfies → salnt pc l aeteDuae property. Daugavet the have all – space length a is se[9 em 2.1]) Lemma [19, (see 2 X imtrtoproperty two diameter − x eoe h dniyoeao.I 93 .K agvt(e [ (see Daugavet K. I. 1963, In operator. identity the denotes epoeta h imta imtrtopoete r inhe are properties two diameter diametral the that prove We ∈ ε. X M S iel) nteohrhn,teepoete r itdfro lifted are properties these hand, other the On -ideals). ssi ohv the have to said is X OANLNEESADKTINPIRK KATRIIN AND LANGEMETS JOHANN imtrtopoet,Duae property, Daugavet property, two Diameter vr slice every , µ n isht-respaces Lipschitz-free and , L2 nsot.Cery h agvtpoet implies property Daugavet the Clearly, short). in (LD2P k 1. . I C aahspace Banach A L + S Introduction [0 p 60,4B0 62,46B42. 46B22, 46B20, 46B04, Bcnrsae) ial,w netgt when investigate we Finally, spaces). -Bochner T , of DPi hr) faBnc pc aifisthat satisfies space Banach a If short). in (D2P 1] k agvtproperty Daugavet B + 1 = a h agvtpoet.Snete this then Since property. Daugavet the has 1 K X n every and , ihu sltdpoints, isolated without k T k X , a h agvtpoet fand if property Daugavet the has F t PG8)ad(PRG877). and (PSG487) nts 1] [23]). [16], , > ε eas hwta l of all that show also We . ( M l pnsbeso h unit the of subsets open kly raino correspond- of ormation nw htsae ihthe with spaces that known flto oelreteclass the enlarge to tool eful M feeyrn-n operator rank-one every if aerldaee two diameter iametral ewt uhapoet is property a such with ce ) 0 e aynwproperties new many red iel öh–ohe space, Köthe–Bochner -ideal, , n hi ul Lip duals their and adta h pc has space the that said hr exists there h agvtproperty Daugavet the v mre.Indeed, emerged. ave re r o rein true not are erses L 1 an m ( ie by rited µ ) y and M ∈ 3)was 13]) - S L 0 ∞ such ( M ( µ ) ) 2 JOHANNLANGEMETSANDKATRIINPIRK general. The space c0 is an example of a space with the D2P and failing the Daugavet property. The existence of a space with the LD2P and without the D2P was first proven in [9], another example can be found in [4]. In [18] another weakening of the Daugavet property was considered: a Banach space X is said to be a space with bad projections (SBP in short) if for every rank-one projection P : X → X one has that kI − P k≥ 2. Y. Ivakhno and V. Kadets obtained the following geometrical characterization of a space with bad projections.

Theorem 1.2 (see [18, Theorem 1.4]). A Banach space X is SBP if and only if for every slice S of BX , every unit sphere element x ∈ S, and every ε> 0, there exists y ∈ S such that kx − yk≥ 2 − ε.

It is clear that an SBP space has the LD2P in a strong sense since for any slice S of the unit ball and any unit sphere element x in S there exists a point y in S almost diametral to x. Inspired by this the authors of [10] called the equivalent formulation of an SBP space appearing in Theorem 1.2 the diametral local diameter two property (DLD2P in short). Note that the DLD2P has also appeared under the name of LD2P+ in [1]. From now on we will use the name DLD2P. In [10] also the diametral analogue of the D2P was introduced and studied: a Banach space X is said to have the diametral diameter two property (DD2P in short) if for every nonempty relatively weakly open subset U of BX , every unit sphere element x ∈ U, and every ε > 0 there exists y ∈ U such that kx − yk ≥ 2 − ε. From [26, Lemma 3] one has that the Daugavet property implies even the DD2P. An example of a Banach space with the DD2P and failing the Daugavet property was given in [10, Example 2.2]. The questions whether there is a space with the DLD2P and without the DD2P is still open (see [10, Question 4.1]). We need also a very recent deﬁnition of a ∆-point emerging from [3] before we can start to describe the contents of this paper. Observe that a Banach space X has the DLD2P if and only if every point in the unit sphere of X is a ∆-point (see Lemma 2.1). A Banach space X has the convex diametral local diameter two property (convex DLD2P in short) if its unit ball is the closed convex hull of its ∆-points. From the discussion above it is clear that the DLD2P implies the convex DLD2P, however ℓ∞ is an example of a space with the convex DLD2P and failing the DLD2P (see [3, Corollary 5.4]). That the convex DLD2P implies the LD2P was shown in [3, Proposition 5.2]. Finally, note that c0 has the LD2P, but fails the convex DLD2P [3, Remark 5.5]. We summarize all of the aforementioned properties and their relations in the following diagram.

Daugavet convex DD2P DLD2P DLD2P property ?

D2P LD2P STABILITY OFDIAMETRALDIAMETER TWO PROPERTIES 3

The main aim of this paper is to study stability properties of diametral diameter two properties in the context of F -ideals, Köthe–Bochner spaces, and projective tensor products of Banach spaces. It is known that the Daugavet property passes down from the superspace to its M- ideals (see [19]). The Daugavet property lifts from an M-ideal to the superspace if the quotient space with respect to the given M-ideal also has the Daugavet property (see [19]). On the other hand, it is known that the LD2P and the D2P always lift from an M-ideal to the superspace (see [14]). In Section 3, we prove that the DD2P and the DLD2P pass down to a wide class of F -ideals, thus also to M-ideals (see Theorem 3.2) and that these properties lift from an M-ideal to the superspace whenever the quotient space also has the corresponding diametral diameter two property (see Proposition 3.4). However, the convex DLD2P behaves similarly to the regular diameter two property (see Proposition 3.5). Recall that if X has the D2P, then so does Lp(µ, X) for every 1 ≤ p ≤ ∞ (see [6] and [8]). For the Daugavet property it is known that L1(µ, X) and L∞(µ, X) have the Daugavet property whenever X has it or µ is atomless. However, if 1

2. Notation and preliminary results All Banach spaces considered in this paper are nontrivial and over the real field. The closed unit ball of a Banach space X is denoted by BX and its unit sphere by SX . The ∗ ∗∗ dual space of X is denoted by X and the bidual by X . By a slice of BX we mean a set of the form ∗ ∗ S(BX ,x , α) := {x ∈ BX : x (x) > 1 − α}, ∗ where x ∈ SX∗ and α> 0. Notice that nonempty finite intersections of slices of the unit ball form a basis for the inherited weak topology of BX . Therefore, in the definitions of the D2P and the DD2P, it suffices to consider nonempty finite intersections of slices instead of nonempty relatively weakly open subsets. For x ∈ SX and ε> 0 we denote X ∆ε (x) := {y ∈ BX : kx − yk≥ 2 − ε} 4 JOHANNLANGEMETSANDKATRIINPIRK and X ∆X = {x ∈ SX : x ∈ conv∆ε (x) for all ε> 0}. X Recall from [3] that that an element x ∈ SX is said to be a ∆-point if BX = conv∆ε (x) for every ε> 0. Note that ∆-points can be also characterized in terms of slices.

Lemma 2.1 (see [3, Lemma 2.1]). Let X be a Banach space and x ∈ SX . The following assertions are equivalent: (i) x is a ∆-point; (ii) for every slice S of BX with x ∈ S and every ε> 0 there exists y ∈ S such that kx − yk≥ 2 − ε. Thus, by Lemma 2.1, it is clear that a Banach space X has the DLD2P if and only if SX =∆X . On the other hand, since the convex DLD2P of X is deﬁned by the fact that BX = conv∆X , then by a Hahn–Banach separation argument one has the following. Lemma 2.2. Let X be a Banach space. Then X has the convex DLD2P if and only if S ∩ ∆X 6= ∅ for every slice S of BX . We recall that a norm F on R2 is called absolute (see [12]) if F (a, b)= F (|a|, |b|) for all (a, b) ∈ R2 and normalized if F (1, 0) = F (0, 1) = 1.

For example, the ℓp-norm k · kp is absolute and normalized for every p ∈ [1, ∞]. If F is an absolute normalized norm on R2 (see [12, Lemmata 21.1 and 21.2]), then 2 • k(a, b)k∞ ≤ F (a, b) ≤ k(a, b)k1 for all (a, b) ∈ R ; • if (a, b), (c, d) ∈ R2 with |a| ≤ |c| and |b| ≤ |d|, then F (a, b) ≤ F (c, d); • the dual norm F ∗ on R2 deﬁned by F ∗(c, d)= max (|ac| + |bd|) for all (c, d) ∈ R2 F (a,b)≤1 is also absolute and normalized. Note that (F ∗)∗ = F . If X and Y are Banach spaces and F is an absolute normalized norm on R2, then we denote by X ⊕F Y the product space X × Y with respect to the norm

k(x,y)kF = F (kxk, kyk) for all x ∈ X and y ∈ Y . ∗ In the special case where F is the ℓp-norm, we write X ⊕p Y . Note that (X ⊕F Y ) = ∗ ∗ X ⊕F ∗ Y . The following lemma is easily veriﬁed from the deﬁnitions. Lemma 2.3. Let F be an absolute normalized norm on R2 such that (1, 0) is an extreme point of the unit ball B(R2,F ). Then (1, 0) is a strongly exposed point of B(R2,F ), which is strongly exposed by the functional (1, 0) ∈ B(R2,F ∗). In particular, for every ε > 0 there is a γ > 0 such that, whenever (a, b) ∈ B(R2,F ) and a> 1 − γ, then |b| < ε. STABILITY OFDIAMETRALDIAMETER TWO PROPERTIES 5

Finally, given two Banach spaces X and Y , we will denote by X⊗πY the projective tensor product of X and Y . Recall that the space L(Y, X∗) of bounded linear operators ∗ b from Y to X is linearly isometric to the topological dual of X⊗πY . We refer to [25] for a detailed treatment and applications of tensor products. b 3. F -ideals We denote the annihilator of a subspace Y of a Banach space X by Y ⊥ = {x∗ ∈ X∗ : x∗(y) = 0 for all y ∈ Y }. According to the terminology in [17], a closed subspace Y ⊂ X is called an M-ideal if there exists a norm-1 projection P on X∗ with ker P = Y ⊥ and kx∗k = kPx∗k + kx∗ − Px∗k for all x∗ ∈ X∗. ∗ ∗ ∗ ⊥ Thus, if Y is an M-ideal in X, then X decomposes as X = Y ⊕1 Y . In [19], the heritance of the Daugavet property with respect to M-ideals was studied and the authors obtained the following result. Theorem 3.1 (see [19, Propositions 2.10 and 2.11]). Let X be a Banach space and Y an M-ideal in X. (a) If X has the Daugavet property, then Y has the Daugavet property; (b) If Y and X/Y share the Daugavet property, then X has the Daugavet property. Therefore, the Daugavet property passes down from the superspace to its M-ideals. On the other hand, it is known that the regular diameter two properties always lift from an M-ideal to the superspace (see [14]). Our aim in this section is to show that the diametral diameter two properties behave similarly to the Daugavet property, while the convex DLD2P resembles the regular diameter two properties with respect to M-ideals. We will now consider more general types of ideals (see [17, p.45–46]). Let F be an absolute and normalized norm on R2. A closed subspace Y ⊂ X is called an F-ideal if there exists a norm-1 projection P on X∗ with ker P = Y ⊥ and kx∗k = F ∗(kPx∗k , kx∗ − Px∗k) for all x∗ ∈ X∗.

Therefore, M-ideals are just the k · k∞-ideals. We are now ready to prove our main result in this section. Theorem 3.2. Let X be a Banach space, F be an absolute normalized norm on R2 such that (1, 0) is an extreme point of B(R2,F ∗), and Y an F -ideal in X. (a) If X has the DD2P, then Y has the DD2P; (b) If X has the DLD2P, then Y has the DLD2P. ∗ ∗ ⊥ ∗ ∗ ⊥ Proof. (a) By our assumptions X = Y ⊕F ∗ Y . Let P : X → X with ker P = Y ∗ ∗ n ∗ be the F -ideal projection. Let ε> 0, yi ∈ SY , and y ∈ i=1 S(BY ,yi , ε) with kyk = 1. n ∗ Our aim is to show that there exists u ∈ i=1 S(BY ,yi ,T ε) such that ky − uk≥ 2 − ε. ∗ Pick δ > 0 such that yi (y) > 1 − ε + Tδ for every i ∈ {1,...,n}. Consider the slices ∗ ∗ ∗ S(BX ,Pxi , ε − δ), where xi are norm preserving extensions of yi . n ∗ Since X has the DD2P and y ∈ i=1 S(BX ,Pxi , ε − δ) we can ﬁnd an element w ∈ n ∗ ∗ ∗ ∗ i=1 S(BX ,Pxi , ε − δ) such that kyT− wk≥ 2 − ε/3. Find z ∈ X with kz k = 1 such T 6 JOHANNLANGEMETSANDKATRIINPIRK that z∗(y − w) ≥ 2 − ε/3, which gives that z∗(y) = P z∗(y) ≥ 1 − ε/3. Therefore, by Lemma 2.3, ε ε kP z∗k≥ 1 − and kz∗ − P z∗k≤ . 3 3 ∗ Now, similarly to [17, Remark I.1.13], one can show that BY is σ(X,Y )-dense in BX . Thus, we can ﬁnd a u ∈ BY such that for every i ∈ {1,...,n} ε |Px∗(w − u)| < δ and |P z∗(w − u)| < . i 3 n ∗ Then u ∈ i=1 S(BY ,yi , ε), because for every i ∈ {1,...,n} we have T ∗ ∗ yi (u) = (Pxi )(u) ∗ ∗ = (Pxi )(w) − (Pxi )(w − u) > 1 − (ε − δ) − δ = 1 − ε. Notice that ky − uk≥ z∗(y − u) = z∗(y − w)+ P z∗(w − u) + (z∗ − P z∗)(w) ε ε ε ≥ 2 − − − 3 3 3 = 2 − ε. Therefore, Y has the DD2P. (b) Take n = 1 in the proof of (a). As an application of Theorem 3.2, we obtain the known stability results for the diametral diameter two properties in M-ideals (see [22, Propositions 2.3.4 and 2.3.5]). Corollary 3.3. Let X be a Banach space and Y an M-ideal in X. (a) If X has the DD2P, then Y has the DD2P; (b) If X has the DLD2P, then Y has the DLD2P. In [22, Problem 8] it was asked whether a similar statement as Theorem 3.1, (b), also holds for the diametral diameter two properties. In the following Proposition we will give a positive answer to this question. Proposition 3.4. Let X be a Banach space and Y an M-ideal in X. (a) If Y and X/Y share the DD2P, then X has the DD2P; (b) If Y and X/Y share the DLD2P, then X has the DLD2P. We omit the proof of Proposition 3.4, because it is very similar to the proof of [19, Proposition 2.11]. We end this section by studying how the convex DLD2P behaves in the context of M- ideals. Recall from [3] that the convex DLD2P is not inherited by M-ideals in general, since c0 is an M-ideal in ℓ∞, but even though the latter has the convex DLD2P, c0 does not. On the other hand, we will now show that the convex DLD2P carries over to the whole space from its proper M-ideal similarly to the the regular diameter two properties [14]. STABILITY OFDIAMETRALDIAMETER TWO PROPERTIES 7

Proposition 3.5. Let X be a Banach space and Y an M-ideal in X. If Y has the convex DLD2P, then X has the convex DLD2P. ∗ ∗ ∗ ⊥ Proof. Let S(BX ,x , α) be a slice and let ε> 0. Let P : X → X with ker P = Y be the M-ideal projection. Deﬁne Px∗ ε(1 − kPx∗k)+ ε2 y∗ := and β := > 0. kPx∗k kPx∗k ∗ Since Y has the convex DLD2P there exist y ∈ S(BY ,y , β) ∩ ∆Y . Thus we can ﬁnd n ∈ N and y1,...,yn ∈ BY such that n ky − 1/n yik≤ ε and kyi − yk≥ 2 − ε for all i ∈ {1,...,n}. Xi=1 The choice of β means that Px∗(y) > (kPx∗k− ε)(1 + ε).

Find x ∈ BX such that (x∗ − Px∗)(x) > (kx∗ − Px∗k− ε)(1 + ε). ∗ By Proposition 2.3 in [27] there is a net zγ in Y such that zγ → x in the σ(X,Y )- topology and lim sup ky + (x − zγ)k≤ 1 for all y ∈ BY . Hence we may choose z ∈ Y such that

kyi + x − zk < 1+ ε ky + x − zk < 1+ ε |Px∗(x − z)| < ε. Deﬁne y + x − z y + x − z u := i and u := . i 1+ ε 1+ ε Then x∗(y + x − z) x∗(u)= 1+ ε Px∗(y) + (x∗ − Px∗)(x)+ Px∗(x − z) = 1+ ε (kPx∗k− ε)(1 + ε) + (kx∗ − Px∗k− ε)(1 + ε) − ε > 1+ ε > kx∗k− 3ε = 1 − 3ε. n Also ku − 1/n i=1 uik ≤ ε(1 + ε) and kui − uk(1 + ε) = kyi − yk ≥ 2 − ε for all i ∈ {1,...,n}. P Note that we can assure kuk = 1 by scaling argument. Moreover, since ε > 0 is ∗ arbitrary we can choose it as small as we like so that u ∈ S(BX ,x , α) ∩ ∆X , that is, X has the convex DLD2P.

Remark 3.6. Note that one cannot generalize Proposition 3.5 to F -ideals, because Y = ℓ∞ has the convex DLD2P and is an F -ideal in X = ℓ∞ ⊕2 R, however X even fails the LD2P. 8 JOHANNLANGEMETSANDKATRIINPIRK

4. Köthe-Bochner spaces Let (S, A,µ) be a complete, σ-ﬁnite measure space. A Köthe function space over (S, A,µ) is a Banach space (E, k · kE) of real-valued measurable functions on S modulo equality µ-a.e. such that

(1) χA ∈ E for every A ∈A with µ(A) < ∞; (2) for every f ∈ E and every A ∈A with µ(A) < ∞ one has that f is µ-integrable over A; (3) if g is measurable and f ∈ E such that |g(t)| ≤ |f(t)| µ-a.e., then g ∈ E and kgkE ≤ kfkE. If X is a Banach space, then a function f : S → X is called simple if there are ﬁnitely many disjoint sets A1,...,An ∈ A such that µ(Ai) < ∞ for every i ∈ {1,...,n}, f n is constant on each Ai, and f(t) = 0 if t ∈ S \ ( i=1 Ai). The function f is said to be Bochner-measurable if there exists a sequence S(fn) of simple functions such that limn→∞ kfn(t) − f(t)k = 0 µ-a.e. For a Köthe function space E and a Banach space X we denote by E(X) the space of all Bochner-measurable functions f : S → X such that kf(·)k∈ E. If we equip E(X) with the norm kfkE(X) = kkf(·)kkE , then E(X) is a Banach space and it is called the Köthe–Bochner space. The classical examples of Köthe–Bochner function spaces are the Lebesgue–Bochner spaces Lp(µ, X) for 1 ≤ p ≤∞. We refer the reader to [20] for more information on Köthe function spaces. We are now ready to prove that all of the diametral diameter two properties are stable under the formation of Köthe–Bochner spaces. Theorem 4.1. Let (S, A,µ) be a complete, σ-ﬁnite measure space, E a Köthe function space over (S, A,µ) and X a Banach space such that the simple functions are dense in E(X). (a) If X has the DD2P, then E(X) also has the DD2P; (b) If X has the DLD2P, then E(X) also has the DLD2P; (c) If X has the convex DLD2P, then E(X) also has the convex DLD2P.

n ∗ ∗ ∗ Proof. (a) Let f ∈ i=1 S(BE(X),fi , αi) with kfkE(X) = 1, where fi ∈ (E(X)) are with n ∗ norm 1, and ε> 0.T We need to ﬁnd a g ∈ i=1 S(BE(X),fi , αi) such that kf −gk≥ 2−ε. m Since the simple functions are dense in ET(X), we may assume that f = k=1 xkχAk , where Ak are pairwise disjoint measurable sets of A and xk ∈ X. P ∗ ∗ ∗ ∗ ∗ Deﬁne xik by xik(x)= fi (x·χAk ). Note that xik ∈ X for every i ∈ {1,...,n}. Clearly, ∗ xik is linear and it is even continuous. Indeed, if x ∈ BX , then for every i ∈ {1,...,n} we have ∗ ∗ ∗ |xik(x)| = |fi (x · χAk )| ≤ kfi k · kx · χAk k ∗ ∗ = kfi k · kxk · kχAk k ≤ kfi k · kχAk k. ∗ ∗ Thus kxikk ≤ kfi kkχAk k. m Now we want to construct g = k=1 ykχAk . If xk = 0, we take yk = 0. If xk 6= 0, then consider the slices P ∗ ∗ Sik = {y ∈ kxkkBX : xik(y) >xik(xk) − αik}, STABILITY OFDIAMETRALDIAMETER TWO PROPERTIES 9

m ∗ where αik > 0 are such that k=1 αik ≤ fi (f) − (1 − αi). Since X has the DD2P, there n are yk ∈ i=1 Sik such that kPxk − ykk > (2 − ε)kxkk for every k ∈ {1,...,m} for which xk 6= 0. T From kykk ≤ kxkk, we conclude that kg(·)k ≤ kf(·)k. This implies that g ∈ E(X) and kgk≤ 1. n ∗ Observe that g ∈ i=1 S(BE(X),fi , αi). Indeed, we have for every i ∈ {1,...,n} that T m m m m ∗ ∗ ∗ ∗ fi (g)= fi ykχAk = xik(yk) > xik(xk) − αik Xk=1 Xk=1 Xk=1 Xk=1 m ∗ = fi (f) − αik ≥ 1 − αi. Xk=1

Finally, if s ∈ Ak, then kf(s) − g(s)k = kxk − ykk > (2 − ε)kxkk. Therefore, kf − gk > (2 − ε)kfk = 2 − ε, and we are done. (b) Take n = 1 in the proof of (a). ∗ ∗ (c) Let f ∈ SE(X) and α,ε > 0. Our aim is to show that there exists f ∈ ∗ S(BE(X),f , ε) ∩ SE(X) such that f is a ∆-point in X, i.e. f ∈ conv ∆ε(f) for every ε> 0. ∗ Fix an arbitrary f0 ∈ S(BE(X),f , α). Since the simple functions are dense in E(X), m we may assume that f0 = k=1 xkχAk , where Ak are pairwise disjoint measurable sets of A and xk ∈ X. P ∗ ∗ ∗ Deﬁne xk by xk(x)= f (x · χAk ). Note that, analogically to part (a) it can be shown ∗ ∗ that xk ∈ X . m Now we want to construct f = k=1 ykχAk . If xk = 0, we take yk = 0. If xk 6= 0, then consider the slices P ∗ ∗ Sk = {y ∈ kxkkBX : xk(y) >xk(xk) − αk}, m ∗ where αk > 0 are such that k=1 αk ≤ f (f) − (1 − α). Let δ > 0. Since X has the convex DLD2P,P there are yk ∈ Sk such that yk/kxkk is a ∆-point for every k ∈ {1,...,m}, hence, there exist yki ∈ kxkkBX such that n 1 yk − yki < δkxkk n Xi=1 and kyk − ykik≥ (2 − ε)kxkk for every ε> 0 and every k. From kykk ≤ kxkk, we conclude that kf(·)k ≤ kf0(·)k. This implies that f ∈ E(X) and kfk≤ 1. ∗ Observe that f ∈ S(BE(X),f , α). Indeed, m m m m ∗ ∗ ∗ ∗ f (f)= f ykχAk = xk(yk) > xk(xk) − αk Xk=1 Xk=1 Xk=1 Xk=1 m ∗ = f (f) − αk ≥ 1 − α. Xk=1 10 JOHANNLANGEMETSANDKATRIINPIRK

Note that now 1 − α ≤ kfk≤ 1. In the following we may assume that f is of norm 1, if necessary, we use f/kfk instead. Now we will show, that f is indeed a ∆-point, i.e. n there exist fi ∈ BE(X) such that kf − 1/n i=1 fik < δ and for every i ∈ {1,...,n} we have kf − fik≥ 2 − ε. P m Deﬁne fi = k=1 ykiχAk . From kykik ≤ kxkk, we conclude that kfi(·)k ≤ kf0(·)k. This implies thatPfi ∈ E(X) and kfik≤ 1 for every i ∈ {1,...,n}. Secondly, it is not hard to notice that if s ∈ Ak then n n 1 1 f(s) − fi(s) = yk − yki < δkxkk, n n Xi=1 Xi=1 n and therefore kf − 1/n i=1 fik < δkf0k≤ δ. Finally, if s ∈ Ak, thenP

kf(s) − fi(s)k = kyk − ykik≥ (2 − ε)kxkk, and therefore, kf − fik > (2 − ε)kf0k = 2 − ε, which completes the proof.

In [6, Proposition 2.6] it was shown that the diameter two property is preserved under the formation of Lebesgue–Bochner spaces. From Theorem 4.1 we immediately have the following. Corollary 4.2. Let X be a Banach space, (Ω, A,µ) be a ﬁnite measure space, and 1 ≤ p ≤∞. If X has the DD2P (resp. DLD2P, convex DLD2P), then Lp(µ, X) also has the DD2P (resp. DLD2P, convex DLD2P) whenever Lp(µ) 6= {0}.

Remark 4.3. If µ does not have any atoms, then both L1(µ, X) and L∞(µ, X) even have the Daugavet property for any Banach space X (see [28, p. 81]).

5. Projective tensor product We start this section by recalling a very recent definition of the weak operator Dau- gavet property, which is (formally) a strengthening of the Daugavet property (see [21, Remark 5.3]). Definition 5.1 ([21, Definition 5.2]). Let X be a Banach space. It is said that X has the weak operator Daugavet property (WODP in short) if, given x1,...,xn ∈ SX , ε> 0, a slice S of BX , and z ∈ BX , there is a u ∈ S and T : X → X with kT k ≤ 1+ ε, kT u − zk≤ ε, and kTxi − xik≤ ε for every i ∈ {1,...,n}.

Examples of spaces satisfying WODP include L1-preduals with the Daugavet property and L1(µ, X) whenever µ is atomless and X is an arbitrary Banach space (see [24]). It is an open question whether X⊗πY has the Daugavet property whenever X and Y both have the Daugavet property (see [28, Section 6, Question (3)]). In [21, Theorem 5.4], M. Martín and A. Rueda Zoca provedb that the WODP is stable by forming the projective tensor product, hence providing a large class of Banach spaces where the answer to the STABILITY OF DIAMETRAL DIAMETER TWO PROPERTIES 11 aforementioned question is positive. However, for the DLD2P of X⊗πY it suﬃces to assume that X has the WODP (cf. [24, Proposition 5.7, Remark 5.9]). b Proposition 5.2. Let X and Y be Banach spaces. If X has the WODP, then X⊗πY has the DLD2P. b Proof. Let B be a norm one bilinear form on X × Y , α> 0, and z ∈ S(B b ,B,α) ∩ X⊗π Y S b . Fix ε ∈ (0, 1). Our aim is to show that there is an element w ∈ S(B b ,B,α) X⊗πY X⊗πY such that kz − wk ≥ 2 − ε. Let δ ∈ (0, min{α, ε}). Find xˆ ∈ SX and yˆ ∈ SY such that B(ˆx, yˆ) > 1 − δ. As the convex hull of S ⊗ S is dense in B b we can ﬁnd x ∈ S X Y X⊗π Y i X an yi ∈ SY such that n z − λixi ⊗ yi ≤ δ.

Xi=1 B(·,yˆ) Since X has the WODP, we can find u ∈ S(BX , kB(·,yˆ)k , δ) and Φ: X → X such that kΦk≤ 1+ δ, kΦ(u) +x ˆk≤ δ, and kΦ(xi) − xik≤ δ for every i ∈ {1,...,n}. Define T : X⊗πY → R by T (u ⊗ v)= B(Φ(u), v) for all u ∈ X and v ∈ Y . Note that kT k≤ 1+ δ, indeed if u ∈ BX and v ∈ BY , then b |B(Φ(u), v)| ≤ kBk · kΦ(u)k · kvk≤ 1+ δ. Thus, n T (z) ≥ λiB(Φ(xi),yi) − δkT k Xi=1 n n ≥ λiB(xi,yi) − λikBk · kΦ(xi) − xik · kyik− δ(1 + δ) Xi=1 Xi=1 > 1 − δ(3 + δ). Also, T (−u ⊗ yˆ)= B(−Φ(u), yˆ) ± B(ˆx, yˆ) > 1 − δ − δ = 1 − 2δ. Finally, it suffices to take w := u ⊗ yˆ, because B(u, yˆ) > kB(·, yˆ)k · (1 − δ) > (1 − δ)2 and T (z − u ⊗ yˆ) kz − wk≥ 1+ δ 1 − δ(3 + δ) + 1 − 2δ 2 − 6δ ≥ ≥ . 1+ δ 1+ δ Therefore, by choosing δ small enough, we can assure that w is the element we were looking for.

It is known that for the LD2P of X⊗πY it suﬃces to assume that X has the LD2P (see [5, Theorem 2.7]). Yet we do not know whether in Proposition 5.2 it suﬃces to assume that X has the DLD2P. Fortunately,b it turns out that the convex DLD2P behaves similarly to the LD2P. 12 JOHANNLANGEMETSANDKATRIINPIRK

Proposition 5.3. Let X and Y be Banach spaces. If X has the convex DLD2P, then X⊗πY has the convex DLD2P too.

Proof. Consider a slice S(B b , A, α), where A is a norm one operator from Y to X∗ b X⊗π Y and α> 0. Let ε> 0. We need to show that there are z, z ,...,z ∈ B b such that 1 n X⊗π Y n z ∈ S(B b , A, α) ∩ ∆ b , kz − 1/n z k ≤ ε, and kz − z k ≥ 2 − ε for every X⊗πY X⊗πY i=1 i i i ∈ {1,...,n}. P ∗ Find y ∈ SY such that (1 − α/2)kAyk≥ 1 − α. Take x = Ay/kAyk and consider the ∗ slice S(BX ,x , α/2). Since X has the convex DLD2P, then there are x,x1,...,xn ∈ BX ∗ n such that x ∈ S(BX ,x , α/2) ∩ ∆X , kx − 1/n i=1 xik ≤ ε, and kx − xik ≥ 2 − ε for every i ∈ {1,...,n}. P Denote by z = x ⊗ y and zi = xi ⊗ y for for every i ∈ {1,...,n}. Then, clearly z, z ∈ B b , and z ∈ S(B b , A, α), because i X⊗πY X⊗πY A(z) = (Ay)x = kAykx∗(x) ≥ kAyk(1 − α/2) ≥ 1 − α.

Moreover, kz − zik = kx − xik≥ 2 − ε and n n n z − 1/n zi = (x − 1/n xi) ⊗ y = x − 1/n xi ≤ ε.

Xi=1 Xi=1 Xi=1

From the arbitrariness of ε we conclude that X⊗πY has the convex DLD2P. b Remark 5.4. Notice that it follows easily from the proof of Proposition 5.3 that if x ∈ SX is a ∆-point in X then x ⊗ y is a ∆-point in X⊗πY for every y ∈ SY . In order to apply Proposition 5.3 in vector-valuedb Lipschitz-free spaces, we need to introduce some notation. Given a metric space M with a designated origin 0 and a Banach space X, we will denote by Lip0(M, X) the Banach space of all X-valued Lipschitz functions on M which vanish at 0 under the standard Lipschitz norm kf(x) − f(y)k kfk := sup : x,y ∈ M,x 6= y . d(x,y) ∗ Note that, Lip0(M, X ) is itself a dual Banach space. In fact, the map ∗ R δm,x : Lip0(M, X ) −→ f 7−→ f(m)(x) deﬁnes a linear and bounded map for each m ∈ M and x ∈ X. In other words, δm,x ∈ ∗ ∗ Lip0(M, X ) . If we deﬁne

F(M, X) := span({δm,x : m ∈ M,x ∈ X}), ∗ ∗ R then we have that F(M, X) = Lip0(M, X ). We write F(M) := F(M, ). It is known that F(M, X)= F(M)⊗πX (see [11, Proposition 2.1]). Corollary 5.5. Let Mbbe a complete metric space. Then the following assertions are equivalent: (i) F(M) does not have any strongly exposed point; (ii) F(M) has the convex DLD2P; STABILITY OF DIAMETRAL DIAMETER TWO PROPERTIES 13

(iii) F(M, X) has the convex DLD2P for every nontrivial Banach space X. Proof. (i) ⇔ (ii) follows from [7, Theorem 1.5]. (ii) ⇒ (iii) Let X be a Banach space. Since F(M, X) = F(M)⊗πX and F(M) has the convex DLD2P, then F(M, X) has the convex DLD2P by Proposition 5.3. (iii) ⇒ (ii) is clear. b Remark 5.6. Given a metric space M, the existence of a Banach space X such that F(M, X) has the convex DLD2P does not imply that F(M) has the convex DLD2P. Indeed, given X = L1([0, 1]), it follows that

F(M, X)= F(M)⊗πX = F(M)⊗πL1([0, 1]) = L1([0, 1], F(M)) has the Daugavet property (seeb Remark 4.3),b hence also the convex DLD2P, for every metric space M.

References

[1] T. A. Abrahamsen, P. Hájek, O. Nygaard, J. Talponen, and S. Troyanski, Diameter 2 properties and convexity, Studia Math., 232 (2016), pp. 227–242. [2] T. A. Abrahamsen, J. Langemets, and V. Lima, Almost square Banach spaces, J. Math. Anal. Appl. 434 (2016), pp. 1549–1565. [3] T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk, Delta- and Daugavet points in Banach spaces, Proc. Edinb. Math. Soc., II. Ser. 63 (2020), pp. 475–496. [4] T. A. Abrahamsen, V. Lima, A. Martiny, and S. Troyanski, Daugavet- and delta-points in Banach spaces with unconditional bases, preprint (2020), https://arxiv.org/abs/2007.04946. [5] T. A. Abrahamsen, V. Lima, and O. Nygaard, Remarks on diameter 2 properties, J. Convex Anal. 20 (2013), pp. 439–452. [6] M. D. Acosta, J. Becerra-Guerrero, and G. López-Pérez, Stability results of diameter two properties, J. Convex Anal. 22 (2015), pp. 1–17. [7] A. Avilés and G. Martínez-Cervantes, Complete metric spaces with property (Z) are length spaces, J. Math. Anal. Appl. 473 (2019), pp. 334–344. [8] J. Becerra Guerrero and G. López-Pérez, Relatively weakly open subsets of the unit ball in functions spaces, J. Math. Anal. Appl. 315 (2006), No. 2, pp. 544–554. [9] J. Becerra Guerrero, G. López-Pérez, and A. Rueda Zoca, Big slices versus big relatively weakly open subsets in Banach spaces, J. Math. Anal. Appl. 428 (2015), pp. 855–865. [10] , Diametral diameter two properties in Banach spaces, J. Convex Anal. 25 (2018), pp. 817–840. [11] , Octahedrality in Lipschitz-free Banach spaces, Proc. R. Soc. Edinb., Sect. A, Math. 148 (2018), pp. 447–460. [12] F. F. Bonsall and J. Duncan, Numerical Ranges II, vol. 10, Cambridge University Press, Cam- bridge. London Mathematical Society, London, 1973. [13] I. K. Daugavet, A property of complete continuous operators in the space C, Usp. Mat. Nauk 18 (1963), pp. 157–158. [14] R. Haller and J. Langemets, Two remarks on diameter 2 properties, Proc. Est. Acad. Sci. 63 (2014), pp. 2–7. [15] R. Haller, J. Langemets, V. Lima, and R. Nadel, Symmetric strong diameter two property, Mediterr. J. Math. 16 (2019). [16] R. Haller, J. Langemets, V. Lima, R. Nadel, and A. Rueda Zoca, On Daugavet indices of thickness, J. Func. Anal (2020) in press. [17] P. Harmand, D. Werner, and W. Werner, M-ideals in Banach Spaces and Banach Algebras, vol. 1547, Berlin: Springer-Verlag, 1993. [18] Y. Ivakhno and V. Kadets, Unconditional sums of spaces with bad projections, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh., 645 (2004), pp. 30–35. 14 JOHANNLANGEMETSANDKATRIINPIRK

[19] V. M. Kadets, R. V. Shvidkoy, G. G. Sirotkin, and D. Werner, Banach spaces with the Daugavet property, Trans. Am. Math. Soc. 352 (2000), pp. 855–873. [20] P. K. Lin, Köthe–Bochner Function Spaces, Birkhäuser, Boston, USA, 2004. [21] M. Martin and A. R. Zoca, Daugavet property in projective symmetric tensor products of Banach spaces, preprint (2020), https://arxiv.org/abs/2010.15936. [22] K. Pirk, Diametral diameter two properties, Daugavet- and ∆-points in Ba- nach spaces, Dissertationes Mathematicae Universitatis Tartuensis 133 (2020), https://dspace.ut.ee/handle/10062/68458. [23] A. Ostrak, On the duality of the symmetric strong diameter 2 property in Lipschitz spaces, preprint (2020), https://arxiv.org/abs/2008.03163. [24] A. Rueda Zoca, P. Tradacete, and I. Villanueva, Daugavet property in tensor product spaces, J. Inst. Math. Jussieu, (2019), p. 1–20. [25] R. A. Ryan, Introduction to Tensor Products of Banach Spaces, London: Springer, 2002. [26] R. V. Shvydkoy, Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2000), pp. 198– 212. [27] D. Werner, M-ideals and the “basic inequality”, J. Approx. Theory 76 (1994), pp. 21–30. [28] , Recent progress on the Daugavet property, Ir. Math. Soc. Bull. 46 (2001), pp. (2001), 77–97.

Institute of Mathematics and Statistics, University of Tartu, Narva mnt 18, 51009 Tartu, Estonia Email address: [email protected], [email protected] URL: https://johannlangemets.wordpress.com/