We prove that the diametral strong diameter 2 property of a Banach space
(meaning that, in convex combinations of relatively weakly open subsets of its
unit ball, every point has an "almost diametral" point) is stable under 1-sums,
i.e., the direct sum of two spaces with the diametral strong diameter 2
property equipped with the 1-norm has again this property.Comment: 4 page

Haller, Rainis

Pirk, Katriin

Põldvere, Märt

Acta et Commentationes Universitatis Tartuensis de Mathematica

English

arXiv

We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property.</jats:p

Rainis Haller

Katriin Pirk

Märt Põldvere

Crossref

Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm

We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property

Journals from University of Tartu

ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICAVolume 20, Number 1, June 2016Available online at http://acutm.math.ut.eeDiametral strong diameter two propertyof Banach spaces is stable underdirect sums with 1-normRainis Haller, Katriin Pirk, and Märt PõldvereAbstract. We prove that the diametral strong diameter 2 property of aBanach space (meaning that, in convex combinations of relatively weaklyopen subsets of its unit ball, every point has an “almost diametral”point) is stable under 1-sums, i.e., the direct sum of two spaces withthe diametral strong diameter 2 property equipped with the 1-norm hasagain this property.All Banach spaces considered in this note are over the real field. The closedunit ball and the unit sphere of a Banach space X will be denoted by BX andSX , respectively. Whenever referring to a relative weak topology, we meansuch a topology on the closed unit ball of the space under consideration.Diameter 2 properties for a Banach space mean that certain subsets ofits unit ball (e.g., slices, nonempty relatively weakly open subsets, or convexcombinations of weakly open subsets) have diameter equal to 2. In recentyears, these properties have been intensively studied (see, e.g., [1–11] forsome typical results and further references).To clarify the gap between the well-studied Daugavet property [12] andknown diameter 2 properties, the diametral diameter 2 properties were in-troduced and studied in the recent preprint [7]. In particular, the stabilityunder p-sums of diametral diameter 2 properties was analyzed. The ques-tion whether the 1-sum of two Banach spaces enjoying the diametral strongdiameter 2 property also has this property, was posed as an open problemin [7]. Below, we shall answer this question in the affirmative.Received March 31, 2016.2010 Mathematics Subject Classification. 46B20, 46B22.Key words and phrases. Banach space, diameter 2 property, Daugavet property, 1-sum,relatively weakly open set.http://dx.doi.org/10.12697/ACUTM.2016.20.08101102 RAINIS HALLER, KATRIIN PIRK, AND MÄRT PÕLDVEREDefinition (see [7]). A Banach space X is said to have the diametralstrong diameter 2 property (DSD2P) if, given n ∈ N, relatively weaklyopen subsets U1, . . . , Un of BX , λ1, . . . , λn ∈ [0, 1] with∑ni=1 λi = 1, x ∈∑ni=1 λiUi, and ε > 0, there is a u ∈∑ni=1 λiUi satisfying‖x− u‖ ≥ ‖x‖+ 1− ε.Theorem. Suppose that Banach spaces X and Y have the DSD2P. Thenalso the 1-sum X ⊕1 Y has the DSD2P.Our proof of Theorem makes use of the following observation:(•) in Definition, one may assume that the element x is of the formx =∑ni=1 λixi where xi ∈ SX ∩ Ui.For (•), first notice that the space X may be assumed to be infinite dimen-sional (because clearly no finite dimensional space can have the DSD2P)and the sets U1, . . . , Un to be convex (because, since x =∑ni=1 λiui whereui ∈ Ui, for every i ∈ {1, . . . , n}, it suffices to consider in the role of Ui aconvex relatively weakly open neighbourhood Vi of ui satisfying Vi ⊂ Ui).Now, for (•), it suffices to observe that(◦) every a ∈ Ui can be written in the form a = (1− µi)yi + µizi whereµi ∈ [0, 1] and yi, zi ∈ SX ∩ Ui,because, if (◦) holds, then the element x can be written asx =n∑i=1λi(1− µi)yi +n∑i=1λiµiziand (by the convexity of U1, . . . , Un)n∑i=1λi(1− µi)Ui +n∑i=1λiµiUi ⊂n∑i=1λiUi.It remains to prove (◦). Let i ∈ {1, . . . , n} and let a ∈ Ui, ‖a‖ < 1. Letm ∈ N, x∗1, . . . , x∗m ∈ X∗, and δ > 0 be such thatUi ⊃{b ∈ BX : |x∗j (b)− x∗j (a)| < δ, j = 1, . . . ,m}.Choose a non-zero c ∈⋂mj=1 kerx∗j (such a c exists when the space X isinfinite dimensional), and consider the function f(t) = ‖a + tc‖, t ∈ R.Since f(0) = ‖a‖ < 1 and f(t) −−−−→t→±∞∞, there are s, t ∈ (0,∞) such thatf(−s) = f(t) = 1, but now yi := a− sc, zi := a+ tc, and µi :=ss+ tdo thejob.Proof of Theorem. Put Z := X ⊕1 Y , and let n ∈ N, let W1, . . .Wn be re-latively weakly open subsets of BZ , let λ1, . . . , λn ∈ [0, 1] satisfy∑ni=1λi=1,and let z =∑ni=1 λizi where zi = (xi, yi) ∈ SZ ∩ Wi. We must find aDIAMETRAL STRONG DIAMETER TWO PROPERTY OF BANACH SPACES 103w = (u, v) ∈∑ni=1 λiWi so that ‖z − w‖ ≥ ‖z‖ + 1 − ε, i.e., putting x :=∑ni=1 λixi and y :=∑ni=1 λiyi (now one has z = (x, y)),‖x− u‖+ ‖y − v‖ ≥ ‖x‖+ ‖y‖+ 1− ε.For every i ∈ {1, . . . , n}, puttingx̂i =xi‖xi‖, if xi 6= 0,0, if xi = 0,and ŷi =yi‖yi‖, if yi 6= 0,0, if yi = 0,there are relatively weakly open neighbourhoods Ui ⊂ BX and Vi ⊂ BY ofx̂i and ŷi, respectively, such that(‖xi‖Ui)×(‖yi‖Vi)⊂Wi. Indeed, lettingm ∈ N, z∗j = (x∗j , y∗j ) ∈ SZ∗ , j = 1, . . . ,m, and δ > 0 be such thatWi ⊃{w ∈ BZ : |z∗j (w)− z∗j (zi)| < δ, j = 1, . . . ,m},and definingUi :={u ∈ BX : |x∗j (u)− x∗j (x̂i)| < δ, j = 1, . . . ,m},Vi :={v ∈ BY : |y∗j (v)− y∗j (ŷi)| < δ, j = 1, . . . ,m},one has, whenever u ∈ Ui and v ∈ Vi, for every j ∈ {1, . . . ,m},∣∣∣z∗j (‖xi‖u,‖yi‖v)− z∗j (zi)∣∣∣ = ∣∣∣z∗j (‖xi‖u, ‖yi‖v)− z∗j (xi, yi)∣∣∣=∣∣∣x∗j(‖xi‖u)+ y∗j (‖yi‖v)− x∗j (xi)− y∗j (yi)∣∣∣=∣∣∣x∗j(‖xi‖u)+ y∗j (‖yi‖v)− x∗j(‖xi‖x̂i)− y∗j (‖yi‖ŷi)∣∣∣=∣∣‖xi‖x∗j (u− x̂i) + ‖yi‖y∗j (v − ŷi)∣∣≤ ‖xi‖∣∣x∗j (u− x̂i)∣∣+ ‖yi‖ ∣∣y∗j (v − ŷi)∣∣<(‖xi‖+ ‖yi‖)δ = ‖zi‖δ= δ.Putα :=n∑i=1λi‖xi‖ and β :=n∑i=1λi‖yi‖.Notice thatα+ β =n∑i=1λi(‖xi‖+ ‖yi‖)=n∑i=1λi‖zi‖ =n∑i=1λi = 1.We only consider the case when both α 6= 0 and β 6= 0. (The case whenα = 0 or β = 0 can be handled similarly and is, in fact, simpler.)For every i ∈ {1, . . . , n}, lettingαi :=λi‖xi‖αand βi :=λi‖yi‖β,104 RAINIS HALLER, KATRIIN PIRK, AND MÄRT PÕLDVEREone has αi, βi ∈ [0, 1], and∑ni=1 αi =∑ni=1 βi = 1. Since X and Y have theDSD2P, observing thatxα=n∑i=1λi‖xi‖αx̂i ∈n∑i=1αiUi andyβ=n∑i=1λi‖yi‖βŷi ∈n∑i=1βiVi,there are u0 ∈∑ni=1 αiUi and v0 ∈∑ni=1 βiVi such that∥∥∥xα− u0∥∥∥ ≥ 1α‖x‖+ 1− ε and∥∥∥ yβ− v0∥∥∥ ≥ 1β‖y‖+ 1− ε.Finally, puttingu := αu0 ∈n∑i=1ααiUi =n∑i=1λi‖xi‖Ui,v := βv0 ∈n∑i=1ββiVi =n∑i=1λi‖yi‖Vi,one has(u, v) ∈n∑i=1λi((‖xi‖Ui)×(‖yi‖Vi))⊂n∑i=1λiWiand‖x− u‖+ ‖y − v‖ ≥ ‖x‖+ ‖y‖+ (α+ β)(1− ε) = ‖x‖+ ‖y‖+ 1− ε,as desired. Thus the stability of the diametral strong diameter 2 property under 1-and ∞-sums is similar to that of the Daugavet property. In fact, amongall 1-unconditional sums of two Daugavet spaces only the 1- and ∞-sumhave the Daugavet property. Whether the diametral strong diameter twoproperty and the Daugavet property coincide remains an open question.AcknowledgementsThis research was supported by institutional research funding IUT20-57of the Estonian Ministry of Education and Research.The authors thank the referee for a careful reading of the paper, andvaluable comments and suggestions which improved the presentation.References[1] T. A. Abrahamsen, P. Hájek, O. Nygaard, J. Talponen, and S. Troyanski, Diameter2 properties and convexity, arXiv:1506.05237v1 [math.FA] (2015).[2] T. A. Abrahamsen, V. Lima, and O. Nygaard, Remarks on diameter 2 properties, J.Conv. Anal. 20 (2013), 439–452.[3] M. D. Acosta, J. Becerra Guerrero, and G. López Pérez, Stability results of diametertwo properties, J. Convex Anal. 22 (2015), 1–17.DIAMETRAL STRONG DIAMETER TWO PROPERTY OF BANACH SPACES 105[4] J. Becerra Guerrero, G. López-Pérez, and A. Rueda Zoca, Big slices versus big rela-tively weakly open subsets in Banach spaces, J. Math. Anal. Appl. 428 (2015), 855–865.[5] J. Becerra Guerrero, G. López-Pérez, and A. Rueda Zoca, Extreme differences betweenweakly open subsets and convex combinations of slices in Banach spaces, Adv. Math.269 (2015), 56–70.[6] J. Becerra Guerrero, G. López-Pérez, and A. Rueda Zoca, Octahedral norms in spacesof operators, J. Math. Anal. Appl. 427 (2015), 171–184.[7] J. Becerra Guerrero, G. López-Pérez, and A. Rueda Zoca, Diametral and Lipschitzdiameter two properties, arXiv:1509. 02061v3 [math.FA] (2016).[8] R. Haller and J. Langemets, Two remarks on diameter 2 properties, Proc. EstonianAcad. Sci. 63 (2014), 2–7.[9] R. Haller, J. Langemets, and M. Põldvere, On duality of diameter 2 properties, J.Convex Anal. 22 (2015), 465–483.[10] G. López-Pérez, The big slice phenomena in M-embedded and L-embedded spaces,Proc. Amer. Math. Soc. 134 (2005), 273–282.[11] O. Nygaard and D. Werner, Slices in the unit ball of a uniform algebra, Arch. Math.76 (2001), 441–444.[12] D. Werner, Recent progress on the Daugavet property, Irish. Math. Soc. Bull. 46(2001), 77–97.Institute of Mathematics and Statistics, University of Tartu, J. Liivi 2,50409 Tartu, EstoniaE-mail address: rainis.haller@ut.eeE-mail address: katriinp@ut.eeE-mail address: mart.poldvere@ut.ee

Diametral strong diameter two property of Banach spaces is stable under direct sums with 1-norm

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