A Banach space X with closed unit ball B is said to have property 2-beta, repsectively 2-NUC if for every \ep > 0, there exists δ> 0 such that for every \ep-separated sequence (x_n) in the unit ball B, and every x in B, there are distinct indices m and n such that ||x + x_m + x_n|| < 3(1 - δ), respectively, ||x_m + x_n|| < 2(1 - δ). It is shown that a Banach space constructed by Schachermayer has property 2-beta but cannot be renormed to have property 2-NUC.
We prove that if $Y$ is a closed subspace of a Banach space $X$ such that $Y$ and $X/Y$ admit an equivalent asymptotically uniformly smooth norm, then $X$ also admits an equivalent asymptotically uniformly smooth norm. The proof is based on the use of the Szlenk index and yields a few other applications to renorming theory.
A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly midpoint locally uniformly rotund and has the D$2$P. We also prove that for Banach spaces admitting a norm-one finite-co-dimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.
Trees are very agreeable objects to work with, offering a diversity of behaviour within a structure that is sufficiently simple to admit precise analysis. Thus we are able to offer fairly satisfactory necessary and sufficient conditions on a tree $Υ$ for the existence of equivalent LUR or strictly convex norms on $\C_0(Υ)$ and for norms with the Kadec Property. In particular, we show that for a {\sl finitely branching} tree $Υ$ the space $\C_0(Υ)$ admits a Kadec renorming. Since some finitely branching trees fail the condition for strictly convex renormability, we obtain an example of a Banach…
Megrelishvili defines \emph{light groups} of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups $G$ of classical Banach spaces $X$ without the PCP, specially isometry groups, and relate it to the existence of $G$-invariant LUR or strictly convex renormings of $X$.
We study the existence of pointwise Kadec renormings for Banach spaces of the form $C(K)$. We show in particular that such a renorming exists when $K$ is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if $C(K_1)$ has a pointwise Kadec renorming and $K_2$ belongs to the class of spaces obtained by closing the class of compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then $C(K_1\times K_2)$ has a pointwise Kadec renorming. We also prove a version of t…
We introduce a new diametral notion for points of the unit sphere of Banach spaces, that naturally complements the notion of Delta-points, but is weaker than the notion of Daugavet points. We prove that this notion can be used to provide a new geometric characterization of the Daugavet property, as well as to recover -- and even to provide new -- results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Finally, we show that this notion leads to powerful new ideas for renorming questions, and that those ideas can be combined with pr…
We prove that every separable Banach space containing $\ell_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its …
The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ($β$). As applications we prove that up to renorming, property ($β$) is stable under coarse Lipschitz embeddings and coarse quotients.
We study certain twisted sums of Orlicz spaces with non-trivial type which can be viewed as Fenchel-Orlicz spaces on ${\rm {\bf R}}^2$. We then show that a large class of Fenchel-Orlicz spaces on ${\rm {\bf R}}^n$ can be renormed to have property (M). In particular this gives a new construction of the twisted Hilbert space $Z_2$ and shows it has property (M), after an appropriate renorming.
We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and w…
The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character $ω_1$ has a renorming with the Mazur intersection property. Combined with the previous result of Jim\' enez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP normability of Asplund spaces of density $ω_1$ is undecidable in ZFC.
We study four asymptotic smoothness properties of Banach spaces, denoted $\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their description by proving the missing renorming theorem for $\textsf{A}_p$. We prove that asymptotic uniform flattenability (property $\textsf{T}_\infty$) and summable Szlenk index (property $\textsf{A}_\infty$) are three space properties. Combined with the positive results of the first named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three space problem for this family of properties. We also …
We establish a linear variational principle extending the Deville-Godefroy-Zizler's one. We use this variational principle to prove that if $X$ is a Banach space having property $(α)$ of Schachermayer and $Y$ is any banach space, then the set of all norm strongly attaining linear operators from $X$ into $Y$ is a complement of a $σ$-porous set. Moreover, the results of the paper applies also to an abstract class of (linear and nonlinear) operator spaces.
We provide a Laakso construction to prove that the property of having an equivalent norm with the property $(β)$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $(β)$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $(β)$-moduli of the same power-type even under renorming.
In this work, we obtain some necessary and some sufficient conditions for a space to be nicely smooth, and show that they are equivalent for separable or Asplund spaces. We obtain a sufficient condition for the Ball Generated Property (BGP), and conclude that Property $(II)$ implies the BGP, which, in turn, implies the space is nicely smooth. We show that the class of nicely smooth spaces is stable under $c_o$ and $\ell_p$ sums and also under finite $\ell_1$ sums; that being nicely smooth is not a three space property; and that the Bochner $L_p$ spaces are nicely smooth if and only if $X$ is b…
Steinhaus proved that given a~positive integer $n$, one may find a circle surrounding exactly $n$ points of the integer lattice. This statement has been recently extended to Hilbert spaces by Zwoleński, who replaced the integer lattice by any infinite set that intersects every ball in at most finitely many points. We investigate Banach spaces satisfying this property, which we call (S), and characterise them by means of a new geometric property of the unit sphere which allows us to show, e.g., that all strictly convex norms have (S), nonetheless, there are plenty of non-strictly convex norms s…
Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blo…
We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinite-dimensional Banach space contains an infinite subset $A$ with the property that $\|x\pm y\| > 1$ for distinct elements $x,y\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an infinite-dimensional separable dual space or an unconditional basic sequence, the set $A$ may be chosen in a way that $\|x\pm y\| \geqslant 1+\varepsilon$ for some $\varepsilon > 0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as no…
For each countable ordinal $α$ let $\mathcal{S}_α$ be the Schreier set of order $α$ and $X_{\mathcal{S}_α}$ be the corresponding Schreier space of order $α$. In this paper we prove several new properties of these spaces. 1) If $α$ is non-zero then $X_{\mathcal{S}_α}$ possesses the $λ$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $α$ is non-zero and $1<p<\infty$ then the $p$-convexification $X^{p}_{\mathcal{S}_α}$ possesses the uniform $λ$-property of R. Aron and R. Lohman. 3) For each countable ordinal $α$ the space $X^*_{\math…
Texas A&M University（德克萨斯农工大学）；Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Institute of Mathematics, Jagiellonian University（雅盖隆大学数学研究所）
We strengthen the property $Δ$ of a function $f:[ω_2]^2\rightarrow [ω_2]^{\leq ω}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fréchet sm…
The maximal roundness of a metric space is a quantity that arose in the study of embeddings and renormings. In the setting of Banach spaces, it was shown by Enflo that roundness takes on a much simpler form. In this paper we provide simple computations of the roundness of many standard Banach spaces, such as $\ell^{p}$, the Lebesgue-Bochner spaces $\ell^{p}(\ell^{q})$ and the Schatten ideals $S_{p}$. We also introduce a property that is dual to that of roundness, which we call coroundness, and make explicit the relation of these properties to the geometric concepts of smoothness and convexity …
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València（瓦伦西亚理工大学纯数学与应用数学大学研究所）；Departamento de Matemáticas, Universidad de Zaragoza（萨拉戈萨大学数学系）；Departamento de Matemáticas, Universidad de Murcia（穆尔西亚大学数学系）；Universidad de Granada, Facultad de Ciencias. Departamento de Análisis Matemático（格拉纳达大学理学院数学分析系）
We construct a nonseparable Banach space $\mathcal X$ (actually, of density continuum) such that any uncountable subset $\mathcal Y$ of the unit sphere of $\mathcal X$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon$ for some $\varepsilon>0$). This solves in the negative the central problem of the search for a nonseparable version of Kottman's theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets o…
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 $\|Tx\|$ with $x\in C$ is attained are dense, then the $G_δ$ set of those functionals which strongly exposes $C$ is dense in $X^*$. This extends previous results by J.\ Bourgai…
