On the local moduli of squareness

[EN] We introdu e the notions of pointwise modulus of squareness and lo al modulus of squareness of a normed spa e X . This answers a question of C. Benítez, K. Przesªawski and D. Yost about the de nition of a sensible lo alization of the modulus of squareness. Geometri al properties of the norm of X (Fré het smoothness, Gâteaux smoothness, lo al uniform onvexity or stri t onvexity) are hara terized in terms of the behaviour of these moduli.The author wishes to express his gratitude to Professor R. Deville for helpful omments and suggestions whi h improved the ontent of this paperGuirao Sánchez, AJ. (2008). On the local moduli of squareness. Studia Mathematica. 184(2):175-189. doi:10.4064/sm184-2-6S175189184

On the moduly of convexity

[EN] It is known that, given a Banach space (X, parallel to center dot parallel to), the modulus of convexity associated to this space delta X is a non-negative function, nondecreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation delta x(epsilon)/epsilon(2) 0 is a constant. We show that, given a function f satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to f, in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.The first author was supported by grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain), and AP2003-4453 of MECD (Spain).Guirao Sánchez, AJ.; Hajek, P. (2007). On the moduly of convexity. Proceedings of the American Mathematical Society. 135(10):3233-3240. https://doi.org/10.1090/S0002-9939-07-09030-2S323332401351

The Bishop-Phelps-Bollobás property for numerical radius in l(1)(C)

We show that the set of bounded linear operators from X to X admits a Bishop Phelps Bollobas type theorem for numerical radius whenever X is l(1)(C) or c(0)(C). As an essential tool we provide two constructive versions of the classical Bishop-Phelps-Bollobas theorem for l(1)(C).The research of the first named author was supported in part by MICINN and FEDER (project MTM2011-25377), by Fundacion Seneca (project 08848/PI/08), by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project PAID-06-09-2829). The research of the second named author is supported by Kent State UniversityGuirao Sánchez, AJ.; Kozhushkina, O. (2013). The Bishop-Phelps-Bollobás property for numerical radius in l(1)(C). Studia Mathematica. 218(1):41-54. https://doi.org/10.4064/sm218-1-3S4154218

Schauder bases under uniform renormings

[EN] Let X be a separable superreflexive Banach space with a Schauder basis. We prove the existence of an equivalent uniformly smooth (resp. uniformly rotund) renorming under which the given basis is monotone.First author supported by the grants MTM2005-08379 of MECD (Spain), 00690/PI/04 of Fundación Séneca (CARM, Spain) and AP2003-4453 of MECD (Spain), Second author supported by AV0Z10190503 and A100190502.Guirao Sánchez, AJ.; Hajek, P. (2007). Schauder bases under uniform renormings. Positivity. 11(4):627-638. https://doi.org/10.1007/s11117-007-2067-9S627638114R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys 64, Longman Ed (1993).M. Fabian, P. Habala, P. Hájek, V. Montesinos, J. Pelant, V. Zizler, Functional analysis and infinite dimensional geometry. Canadian Math. Soc. Books, Springer Verlag, (2001).M. Fabian, V. Montesinos, V. Zizler, Smoothness in Banach spaces. Selected problems. Rev. R. Acad. Cien. Serie A Mat. 100, (2006), 101–125.T. Figiel, On the moduli of convexity and smoothness. Studia Math. 56, (1976), 121–155.M. Zippin, A remark on bases and reflexivity in Banach spaces. Isr. J. Math. 6, (1968), 74–79.P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm. Isr. J. Math 13, (1972), 281–288

On the preserved extremal structure of Lipschitz-free spaces

[EN] We characterize preserved extreme points of the unit ball of Lipschitz-free spaces F(X) in terms of simple geometric conditions on the underlying metric space (X,d). Namely, the preserved extreme points are the elementary molecules corresponding to pairs of points p,q in X such that the triangle inequality d(p,q)<=d(p,r)+d(q,r) is uniformly strict for r away from p,q. For compact X, this condition reduces to the triangle inequality being strict. As a consequence, we give an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points, and we show that all extreme points are preserved for several classes of compact metric spaces X, including Hölder and countable compacta.The research of the second author was partially supported by MINECO grant MTM2014-57838-C2-1-P and Fundacion Seneca, Region de Murcia grant 19368/PI/14.Aliaga, RJ.; Guirao Sánchez, AJ. (2019). On the preserved extremal structure of Lipschitz-free spaces. Studia Mathematica. 245(1):1-14. https://doi.org/10.4064/sm170529-30-11S114245

Remarks on the set of norm-attaining functionals and differentiability

[EN] We use the smooth variational principle and a standard renorming to give a short direct proof of the classical Bishop-Phelps-Bollobas theorem on the density of norm-attaining functionals for weakly compactly generated Banach spaces. Then we show that a slight adjustment of a known Preiss-Zajfcek differentiability argument provides a simple, useful characterization of individual norms on separable Banach spaces admitting residual sets of norm-attaining functionals in terms of Frechet differentiability of their dual norms.This research was partly supported by MICINN and FEDER Projects MTM2014-57838-C2-1-P, MTM2014-57838-C2-2-P. The first named author was also partly supported by Fundacion Seneca, Region de Murcia grant 19368/PI/14.Guirao Sánchez, AJ.; Montesinos Santalucia, V.; Zizler, V. (2018). Remarks on the set of norm-attaining functionals and differentiability. Studia Mathematica. 241(1):71-86. https://doi.org/10.4064/sm8768-6-2017S7186241

A note on extreme points of C∞C^\infty-smooth balls in polyhedral spaces

[EN] Morris (1983) proved that every separable Banach space $X$ that contains an isomorphic copy of $c_0$ has an equivalent strictly convex norm such that all points of its unit sphere $S_X$ are unpreserved extreme, i.e., they are no longer extreme points of $B_{X^{**}}$. We use a result of Hájek (1995) to prove that any separable infinite-dimensional polyhedral Banach space has an equivalent $C^{\infty}$-smooth and strictly convex norm with the same property as in Morris' result. We additionally show that no point on the sphere of a $C^2$-smooth equivalent norm on a polyhedral infinite-dimensional space can be strongly extreme, i.e., there is no point $x$ on the sphere for which a sequence $(h_n)$ in $X$ with $\Vert h_n\Vert\not \to 0$ exists such that $\Vert x\pm h_n\Vert\to 1$.The first author’s research was supported by Ministerio de Econom´ıa y Competitividad and FEDER under project MTM2011-25377 and the Universitat Polit`ecnica de Val`encia. The second author’s research was supported by Ministerio de Econom´ıa y Competitividad and FEDER under project MTM2011-22417 and the Universitat Polit`ecnica de Val`enciaGuirao Sánchez, AJ.; Montesinos Santalucia, V.; Zizler, V. (2015). A note on extreme points of $C^\infty$-smooth balls in polyhedral spaces. Proceedings of the American Mathematical Society. 143(8):3413-3420. https://doi.org/10.1090/S0002-9939-2015-12617-2S34133420143

A note on Mackey topologies on Banach spaces

[EN] There is a maybe unexpected connection between three apparently unrelated notions concerning a given w*-dense subspace Y of the dual X* of a Banach space X: (i) The norming character of Y, (ii) the fact that (Y, w*) has the Mazur property, and (iii) the completeness of the Mackey topology mu(X, Y), i.e., the topology on X of the uniform convergence on the family of all absolutely convex w*-compact subsets of Y. To clarify these connections is the purpose of this note. The starting point was a question raised by M. Kunze and W. Arendt and the answer provided by J. Bonet and B. Cascales. We fully characterize mu(X, Y)-completeness or its failure in the case of Banach spaces X with a w*-angelic dual unit ball in particular, separable Banach spaces or, more generally, wealdy compactly generated ones-by using the norming or, alternatively, the Mazur character of Y. We characterize the class of spaces where the original Kunze-Arendt question has always a positive answer. Some other applications are also provided. (C) 2016 Elsevier Inc. All rights reserved.Supported in part by MICINN (Project MTM2014-57838-C2-2-P).Guirao Sánchez, AJ.; Montesinos Santalucia, V.; Zizler, V. (2017). A note on Mackey topologies on Banach spaces. Journal of Mathematical Analysis and Applications. 445(1):944-952. https://doi.org/10.1016/j.jmaa.2016.08.030S944952445

On the Bishop-Phelps-Bollobás property for numeri9cal radius in C(K) spaces

[EN] We study the Bishop-Phelps-Bollobás property for numerical radius within the framework of C(K) spaces. We present several sufficient conditions on a compact space K ensuring that C(K) has the Bishop-Phelps-Bollobás property for numerical radius. In particular, we show that C(K) has such property whenever K is metrizable.Research supported by Ministerio de Economía y Competitividad and FEDER under project MTM2011-25377. A. Avilés was supported by Ramón y Cajal contract (RYC-2008-02051). A.J. Guirao was supported by Generalitat Valenciana (GV/2010/036)Avilés López, A.; Guirao Sánchez, AJ.; Rodríguez, J. (2014). On the Bishop-Phelps-Bollobás property for numeri9cal radius in C(K) spaces. Journal of Mathematical Analysis and Applications. 419(1):395-421. doi:10.1016/j.jmaa.2014.04.039S395421419