Daugavet property in projective symmetric tensor products of Banach spaces

Miguel Martín partially supported by Spanish AEI Project PGC2018-093794-B- I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), A-FQM-484-UGR18 (Universidad de Granada and Junta de Analucía/FEDER, UE), FQM-185 (Junta de Andalucía/FEDER, UE), and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. Abraham Rueda Zoca was supported by Juan de la Cierva-Formación fellowship FJC2019-039973, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by Spanish AEI Project PGC2018-093794-B- I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18 and by Junta de Andalucía Grant FQM-0185.We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L1-predual (i.e., X∗ is isometric to an L1-space) or X is a vector-valued L1-space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise “localised” versions of the Daugavet property [i.e., Daugavet points and Δ-points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for L1-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L1-preduals in terms of the absence of Δ-points and also to provide new examples of L1-preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces [in the sense of Werner (J Funct Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L1-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.ACyT Región de Murcia 20797/PI/18Universidad de Granada and Junta de AnalucíaFundación SénecaEuropean CommissionEuropean Regional Development FundJunta de Andalucía FJC2019-039973, FQM-0185, MCIN/AEI/10.13039/501100011033, MTM2017-86182-PUniversity of the East FQM-18

Separating club-guessing principles in the presence of fat forcing axioms

We separate various weak forms of Club Guessing at $\omega_1$ in the presence of $2^{\aleph_0}$ large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with $\omega$-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds $\aleph_1$-many reals but preserves CH

Evaluation of cervical posture improvement of children with cerebral palsy after physical therapy based on head movements and serious games

Background: This paper presents the preliminary results of a novel rehabilitation therapy for cervical and trunk control of children with cerebral palsy (CP) based on serious videogames and physical exercise. Materials: The therapy is based on the use of the ENLAZA Interface, a head mouse based on inertial technology that will be used to control a set of serious videogames with movements of the head. Methods: Ten users with CP participated in the study. Whereas the control group (n=5) followed traditional therapies, the experimental group (n=5) complemented these therapies with a series of ten sessions of gaming with ENLAZA to exercise cervical flexion-extensions, rotations and inclinations in a controlled, engaging environment. Results: The ten work sessions yielded improvements in head and trunk control that were higher in the experimental group for Visual Analogue Scale, Goal Attainment Scaling and Trunk Control Measurement Scale (TCMS). Significant differences (27% vs. 2% of percentage improvement) were found between the experimental and control groups for TCMS (p<0.05). The kinematic assessment shows that there were some improvements in the active and the passive range of motion. However, no significant differences were found pre- and post-intervention. Conclusions:Physical therapy that combines serious games with traditional rehabilitation could allow children with CP to achieve larger function improvements in the trunk and cervical regions. However, given the limited scope of this trial (n=10) additional studies are needed to corroborate this hypothesis

Residuality in the set of norm attaining operators between Banach spaces

This paper was partially written when the first author was visiting the University of Granada and he would like to acknowledge the hospitality that he received there. The authors would like to thank Antonio Avilés, Luis Carlos García-Lirola, Gilles Godefroy, Manuel Maestre, Warren Moors, Vicente Montesinos, and Rafael Payá for kindly answering several inquiries related to the topics of the paper. We also thank the anonymous referee for the careful reading of the manuscript and for providing a number of comments which have improved its final form. M. Jung was supported by NRF (NRF-2019R1A2C1003857), by POSTECH Basic Science Research Institute Grant (NRF-2021R1A6A1A10042944) and by a KIAS Individual Grant (MG086601) at Korea Institute for Advanced Study. M. Martín was supported by Project PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Junta de Andalucía I+D+i grants P20_00255, A-FQM-484-UGR18, and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by MCIN/AEI/10.13039/501100011033. A. Rueda Zoca was supported by Projects MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), PGC2018-093794-B-I00/AEI/10.13039/501100011033 (MCIU/AEI/FEDER, UE), by Fundación Séneca, ACyT Región de Murcia grant 20797/PI/18, by Junta de Andalucía Grant A-FQM-484-UGR18, and by Junta de Andalucía Grant FQM-0185.We study the relationship between the residuality of the set of norm attaining functionals on a Banach space and the residuality and the denseness of the set of norm attaining operators between Banach spaces. Our first main result says that if C is a bounded subset of a Banach space X which admit an LUR renorming satisfying that, for every Banach space Y, the operators T from X to Y for which the supremum of with is attained are dense, then the set of those functionals which strongly exposes C is dense in ⁎. This extends previous results by J. Bourgain and K.-S. Lau. The particular case in which C is the unit ball of X, in which we get that the norm of ⁎ is Fréchet differentiable at a dense subset, improves a result by J. Lindenstrauss and we even present an example showing that Lindenstrauss' result was not optimal. In the reverse direction, we obtain results for the density of the set of absolutely strongly exposing operators from X to Y by requiring that the set of strongly exposing functionals on X is dense and conditions on Y or ⁎ involving RNP and discreteness on the set of strongly exposed points of Y or ⁎. These results include examples in which even the denseness of norm attaining operators was unknown. We also show that the residuality of the set of norm attaining operators implies the denseness of the set of absolutely strongly exposing operators provided the domain space and the dual of the range space are separable, extending a recent result for functionals. Finally, our results find important applications to the classical theory of norm-attaining operators, to the theory of norm-attaining bilinear forms, to the geometry of the preduals of spaces of Lipschitz functions, and to the theory of strongly norm-attaining Lipschitz maps. In particular, we solve a proposed open problem showing that the unique predual of the space of Lipschitz functions from the Euclidean unit circle fails to have Lindenstrauss property A.ACyT Región de Murcia 20797/PI/18Junta de Andalucía I+D+i A-FQM-484-UGR18, FQM-185, MCIN/AEI/10.13039/501100011033, MTM2017-86182-P, P20_00255KIAS MG086601POSTECH Basic Science Research Institute NRF-2021R1A6A1A10042944Institute for Advanced Study PGC2018-093794-B-I00/AEI/10.13039/501100011033Fundación SénecaEuropean CommissionFEDERJunta de Andalucía FQM-018

Diametral notions for elements of the unit ball of a Banach space

The first and third authors were supported by grant PID2021-122126NB-C31 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU, by Junta de Andalucía I+D+i grants P20_00255 and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG (CEX2020-001105-M) funded by MICIU/AEI/10.13039/501100011033. The second named author was supported by the Estonian Research Council grant SJD58.We introduce extensions of Δ-points and Daugavet points in which slices are replaced by relatively weakly open subsets (super Δ-points and super Daugavet points) or by convex combinations of slices (ccs Δ-points and ccs Daugavet points). These notions represent the extreme opposite to denting points, points of continuity, and strongly regular points. We first give a general overview of these new concepts and provide some isometric consequences on the spaces. As examples: (1) If a Banach space contains a super Δ-point, then it does not admit an unconditional FDD (in particular, unconditional basis) with suppression constant smaller than 2. (2) If a real Banach space contains a ccs Δ-point, then it does not admit a one-unconditional basis. (3) If a Banach space contains a ccs Daugavet point, then every convex combination of slices of its unit ball has diameter 2. We next characterize the notions in some classes of Banach spaces, showing, for instance, that all the notions coincide in L1-predual spaces and that all the notions but ccs Daugavet points coincide in L1-spaces. We next comment on some examples which have previously appeared in the literature, and we provide some new intriguing examples: examples of super Δ-points which are as close as desired to strongly exposed points (hence failing to be Daugavet points in an extreme way); an example of a super Δ-point which is strongly regular (hence failing to be a ccs Δ-point in the strongest way); a super Daugavet point which fails to be a ccs Δ-point. The extensions of the diametral notions to points in the open unit ball and consequences on the spaces are also studied. Lastly, we investigate the Kuratowski measure of relatively weakly open subsets and of convex combinations of slices in the presence of super Δ-points or ccs Δ-points, as well as for spaces enjoying diameter-two properties. We conclude the paper with some open problems.MICIU/AEI/10.13039/501100011033 PID2021-122126NB-C31ERDF/EUJunta de Andalucía I+D+i P20_00255, FQM-185MICIU/AEI/10.13039/501100011033 “Maria de Maeztu” (CEX2020-001105-M

Preparation of quantum dots hydrogel nanocomposites with improved cytotoxicity

Nanocomposites are materials with unique properties and a wide range of applications. The combination of different nanostructures with traditional materials gives a variety of possibilities that should be analyzed. Especially, functional fluorescent semiconductor quantum dots (QDs) embedded in polymeric matrices have shown promising fluorescence and biocompatibility properties. These hybrid materials can be used in medical applications such as biodiagnostic and bioimaging. In this study, two hydrogels, one of polyethylene glycol diacrylate (PEGDA) and other of polyacrylamide (PAAm), were prepared with quantum dots of CdTe (4 nm of diameter) and characterized. The aim of this research was to analyze the optical properties of the nanocomposites and their cell viability. QDs nanocomposites were fabricated by a free radical polymerization process. The optical studies showed that the nanocomposites have well defined properties of fluorescence. To study the biocompatibility of the nanocomposites, metastatic B16f10 cell line were used and MTT assay was performed. The nanocomposites had a significant improved cell viability compared with QDs solutions

Slicely countably determined points in Banach spaces

We introduce slicely countably determined points (SCD points) of a bounded and convex subset of a Banach space which extends the notions of denting points, strongly regular points and much more. We completely characterize SCD points in the unit balls of $L_1$-preduals. We study SCD points in direct sums of Banach spaces and obtain that an infinite sum of Banach spaces may have an SCD point despite the fact that none of its components have it. We then prove sufficient conditions to get that an elementary tensor $x\otimes y$ is an SCD point in the unit ball of the projective tensor product $X \widehat{\otimes}_\pi Y$. Regarding Lipschitz-free spaces on compact metric spaces, we show that norm one SCD points of their unit balls are exactly the ones that can be approximated by convex combinations of strongly exposed points of the unit ball. Finally, as applications, we prove a new inheritance result for the Daugavet property to its subspaces, we show that separable Banach spaces for which every convex series of slices intersects the unit sphere must contain an isomorphic copy of $\ell_1$, and we get pointwise conditions on an operator on a Banach space with the Daugavet property to satisfy the Daugavet equation.Comment: version 2 - Improvement on Theorem 5.