Stanimir Troyanski

Bulgarian Academy of Sciences | BAS · Institute of Mathematics and Informatics

We introduce and study a strict monotonicity property of the norm in solid Banach lattices of real functions that prevents such spaces from having the local diameter two property. Then we show that any strictly convex 1-symmetric norm on $\ell_\infty(\mathbb N)$ possesses this strict monotonicity property. In the opposite direction, we show that an...

We study Orlicz functions that do not satisfy the Δ 2 \Delta _2 -condition at zero. We prove that for every Orlicz function M M such that lim sup t → 0 M ( t ) / t p > 0 \limsup _{t\to 0}M(t)/t^p \!>0 for some p ≥ 1 p\ge 1 , there exists a positive sequence T = ( t k ) k = 1 ∞ T=(t_k)_{k=1}^\infty tending to zero and such that sup k ∈ N M ( c t k )...

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...

We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1 1 -unconditional basis. A norm one element x x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 2 from x x ....

This is Stanimir Troyanski's personal account of the late Professor Rumen Maleev.The author shares his reminescences about his dear friend and colleague.

We prove that a Banach space of continuous functions C ( K ) C(K) has a renorming that is uniformly rotund in every direction (URED) if and only if the compact space K K supports a strictly positive measure.

We prove that the class of positive operators from L∞(μ) to L1(ν) has the Bishop-Phelps-Bollobás property for any positive measures μ and ν. The same result also holds for the pair (c0, ℓ1). We also provide an example showing that not every pair of Banach lattices satisfies the Bishop-Phelps-Bollobás property for positive operators.

We establish a weak version of the three-space property for strictly convex renormings, by using a topological characterization of Orihuela, Smith, and Troyanski of the class of strictly convexifiable Banach spaces.

We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a $1$-unconditional basis. A norm one element $x$ in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance $2$ from $x$. A...

We prove that a Banach space of continuous functions $C(K)$ has a renorming that is uniformly rotund in every direction (URED) if and only if the compact space $K$ supports a strictly positive measure

We show that C(X) admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided X is a Fedorchuk compact of spectral height 3. In other words, X admits a fully closed map f onto a metric compact Y such that f−1(y) is metrizable for all y∈Y. A continuous map of compacts f:X→Y is said to be fully closed if the intersectio...

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on ℓ∞. As a consequence we get that every dual Banach space containing c0 has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of thei...

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in...

We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such that $f^{-1}(y)$ is metrizable for all $y\in Y$ . A continuous map of compacts $f : X \to Y$ is said to be fully...

We construct an LUR-norm on the space C(K) which is lower semicontinuous with respect to the pointwise topology for any Fedorchuk compactum of height 3.

We show that, if $X$ is a closed subspace of a Banach space $E$ and $Z$ is a closed subspace of $E^*$ such that $Z$ is norming for $X$ and $X$ is total over $Z$ (as well as $X$ is norming for $Z$ and $Z$ is total over $X$), then $X$ and the pre-annihilator of $Z$ are complemented in $E$ whenever $Z$ is $w^*$-closed or $X$ is reflexive.

The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The hypotheses of both results are based on decomposing the unit sphere of a Banach space into countably many pieces, su...

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discre...

Given a bounded linear operator T from a separable infinite-dimensional Banach space E into a Banach space Y, an operator range R in E and a closed subspace L ⊂ E such that L ∩ R = {0} and codim (L + R) = ∞, we provide a condition to ensure the existence of an infinite-dimensional closed subspace L1 ⊂ E, containing L as an infinite-codimensional su...

We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Ban...

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties. We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizatio...

A sequence in a separable Banach space X 〈resp. in the dual space X*〉 is said to be overcomplete (OC in short) 〈resp. overtotal (OT in short) on X〉 whenever the linear span of each subsequence is dense in X 〈resp. each subsequence is total on X〉. A sequence in a separable Banach space X 〈resp. in the dual space X*〉 is said to be almost overcomplete...

We present an equivalent midpoint locally uniformly rotund (MLUR) renorming $X$ of $C[0,1]$ on which every weakly compact projection $P$ satisfies the equation $\|I-P\| = 1+\|P\|$ ($I$ is the identity operator on $X$). As a consequence we obtain an MLUR space $X$ with the properties D2P, that every non-empty relatively weakly open subset of its uni...

We show that if $X$ and $Y$ are Banach spaces, where $Y$ is separable and polyhedral, and if $T:X \to Y$ is a bounded linear operator, such that $T^*(Y^*)$ contains a boundary $B$ of $X$. Then $X$ is separable and isomorphic to a polyhedral space. Some corollaries of this result are presented.

The aim of this paper is to present two tools, Theorems 4 and 7, that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The hypotheses of both tools are based on countable decompositions, either of the unit sphere S_X or of certain subsets of the dual ball of a given Banach space X. The su?ci...

Some new classes of compacta $K$ are considered for which $C(K)$ endowed with the pointwise topology has a countable cover by sets of small local norm--diameter.

We prove two theorems giving sufficient conditions for a Banach space to be isomorphically polyhedral.

In terms of fragmentability, we describe a new class of Banach spaces which do not contain weak-G_delta open bounded subsets. In particular, none of these spaces is isomorphic to a separable polyhedral space.

We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_\delta$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterisation of those scattered compact spaces $...

We show that for any probability measure \mu there exists an equivalent norm on the space L^1(\mu) whose restriction to each reflexive subspace is uniformly smooth and uniformly convex, with modulus of convexity of power type 2. This renorming provides also an estimate for the corresponding modulus of smoothness of such subspaces.

We survey the current state of research in renormings of C(K) spaces. En este artículo analizamos el estado actual de la investigaciíon en teoría del renormamiento de espacios C(K). KeywordsAsplund-descriptive compact-Fréchet-Gâteaux-Gruenhage compact-LUR-Namioka-Phelps compact-polyhedral-renorming-Rosenthal compact-rotund-scattered-strictly conve...

We study the relationship between the classical combinatorial inequalities of Simons and the more recent (I)-property of Fonf and Lindenstrauss. We obtain a charac-terization of strong boundaries for Asplund spaces using the new concept of finitely self predictable set. Strong properties for w * -K-analytic boundaries are established as well as a s...

A characterization of the Banach spaces of type C(K) that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when K is a Namioka-Phelps compact or for some particular class of Rosenthal compacta, results which were orig...

We use one-dimensional differential inequalities to estimate the squareness and type of Banach spaces with modulus of convexity of power type two. The estimates obtained are sharp and the constants involved moderate.

We present equivalent conditions for a space $X$ with an unconditional basis to admit an equivalent norm with a strictly convex dual norm.

A completely geometrical approach for the construction of locally uniformly rotund norms and the associated networks on a normed space X is presented. A new proof providing a quantitative estimate for a central theorem by M. Raja, A. Moltó and the authors is given with the only external use of Deville-Godefory-Zizler decomposition method. Presenta...

Given a separable Banach space X with no isomorphic copies of ℓ1 and a separable subspace Y of its bidual, we provide a sufficient condition on Y to ensure that X admits an equivalent norm such that the restriction to Y of the corresponding bidual norm is midpoint locally uniformly rotund. This result applies to the separable subspaces of the bidua...

Banach spaces X with an equivalent sigma(X, F)-lower semicontinuous and locally uniformly rotund norm, for a norming subspace F subset of X*, are those spaces X that admit countably many families of Convex and sigma(X, F)-lower semicontinuous functions {phi(n)(i) : X -> R(+); i is an element of I(n)}(n=1)(infinity) such that there are open subsets...

In this chapter we isolate the topological setting that is suitable for our study. We first present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T) into a met...

All examples of σ-slicely continuous maps are connected somehow with LUR Banach spaces. It is clear that if x is a denting point of a set D and Φ is a norm continuous map at x then Φ is slicely continuous at x. Hence if X is a LUR normed space then every norm continuous map Φ on B X is slicely continuous on S X .

We have extensively considered here the use of Stone's theorem on the paracompactness of metric spaces in order to build up new techniques to construct an equivalent locally uniformly rotund norm on a given normed space X. The discreetness of the basis for the metric topologies gives us the necessary rigidity condition that appears in all the known...

Let us remember that given two Banach spaces X, Y and a bounded linear operator T from X onto Y there exists a continuous (nonlinear in general) mapping B of Y into X such that TBy = y for every y ∈ Y [DGZ93, p. 299]. The operator B is called the Bartle-Graves selector of T −1.

A class of generalized metric spaces is a class of spaces defined by a property shared by all metric αspaces which is close to metrizability in some sense [Gru84]. The s-spaces are defined by replacing the base by network in the Bing-Nagata-Smirnov metrization theorem; i.e. a topological space is a αspace if it has a αdiscrete network. Here we shal...

We prove the existence of equivalent polyhedral norms on a number of classes of non-separable spaces, the majority of which being of the form C(K). In particular, we obtain a complete characterization of those trees T, such that C0(T) admits an equivalent polyhedral norm.

A normed space X is said to be strictly convex if x = y whenever ‖( x + y )/2‖ = ‖ x ‖ = ‖ y , in other words, when the unit sphere of X does not contain non-trivial segments. Our aim in this paper is the study of those normed spaces which admit an equivalent strictly convex norm. We present a characterization in...

We consider two classes of nonlinear maps between normed spaces which are relevant for the study of Banach spaces that admit equivalent locally uniformly rotund and midpoint locally uniformly rotund norms. It turns out to be a useful characterization of such maps in terms of optimization, topology, and probability. Using these characterizations we...

In this paper we study the connections between moduli of asymptotic convexity and smoothness of a Banach space, and the existence of high order differentiable bump functions or equivalent norms on the space. The existence of a high order uniformly differentiable bump function is related to an asymptotically uniformly smooth renorming of power type....

An upper bound q(c) for the best, under equivalent renorming, possible power type of the modulus of smoothness of a Banach space with modulus of convexity satisfying δX(ε)⩾cε2, is found. The estimate is asymptotically sharp and is expressed in terms of linear fractional function q(c).

Approximation and rigidity properties in renorming constructions are characterized with some classes of simple maps. Those maps describe continuity properties up to a countable partition. The construction of such kind of maps can be done with ideas from the First Lebesgue Theorem. We present new results on the relationship between Kadec and locally...

A point x ∈ A ⊂ (X, ∥ · ∥) is quasi-denting if for every ε > 0 there exists a slice of A containing x with Kuratowski index less than ε. The aim of this paper is to generalize the following theorem with a geometric approach, see [19]: A Banach space such that every point of the unit sphere is quasi-denting (for the unit ball) admits an equivalent L...

It is shown that an Orlicz sequence space h M admits an equivalent analytic renorming if and only if it is either isomorphic to ℓ 2n or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent C ∞ -Fréchet norm, but no equivalent analytic norm. In this note, we denote by h M as usual th...

Some new generalizations of locally uniform rotundity in Banach spaces are introduced and studied.

The main goal of this paper is to prove that any Banach space X with the Krein–Milman property such that the weak and the norm topology coincide on its unit sphere admits an equivalent norm that is locally uniformly rotund.

We show that every normed spaceEwith a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm. After obtaining aσ-discrete network of the unit sphereSEfor the weak topology we deduce that the spaceEmust have a countable cover by sets of small local diameter, which in turn implies the renorming conclusion. This solves a...

The dual space of a WUR Banach space is weakly K-analytic. A Banach space is said to be weakly uniformly rotund (WUR -for short) if given sequences (xn) and (yn) in the unit sphere withkxn +ynk! 2 we should have weak-limn(xn yn) = 0. This notion has become more important since H ajek proved that every WUR Banach space must be Asplund (8). To obtain...

It is shown that the order of Gâteaux smoothness of bump functions on a wide class of Banach spaces with unconditional basis is not better than that of Fréchet differentiability. It is proved as well that in the separable case this order for Banach lattices satisfying a lower p-estimate for 1≤p<2 can be only slightly better.

It is proved that C(K) has no equivalent uniformly Gâteaux differentiable norm (UGD) when K-is an uncountable separable scattered com¬pact space. This result is applied to obtain an example of scattered compact K such that K′″ = and C(K) has no UGD renorming.

In this paper a local version is given of a well-known theorem for uniform rotund renorming of superreflexive Banach spaces.

The concept of quasi-denting point has been recently introduced by J. R. Giles and W. B. Moors, and its relevance increased by the fact (proved by the third present author) that a Banach space such that every point in the unit sphere is quasi-denting can be renormed to be LUR. It is known that the denting points of a set are its extreme points of c...

Let X be a Banach space with fundamental biorthogonal system. Then X has an equivalent norm that shares many properties of ℓ 1 (I). For example, the unit ball is the closed convex hull of its strongly exposed points (with a uniformity in the strong exposition), but has no point of locally uniform rotundity similarly to the ball of ℓ 1 (I). If in ad...

Every nonseparable Banach space X with separable norming subspace in X * admits an equivalent norm |·| such that (X,|·|) does not have an Auerbach basis.

Every separable Banach space admits an equivalent norm such that the unit ball with respect to this norm has at most countably many strongly extreme points. Every separable nonreflexive Banach space can be renormed so that its unit ball has at most countably many weakly strongly extreme points.

Equivalent norms with best order of Frechet and uniformly Frechet differentiability in Orlicz spaces are constructed. Classes of Orlicz which admit infinitely many times Frechet differentiable equivalent norm are found.

LetV be a Banach space whose dualV * is Vašák, that is, weakly countably determined. Then an equivalent locally uniformly rotund norm onV is constructed. According to a recent example of Mercourakis, this is a real extension of an earlier result of Godefroy Troyanski, Whitfield and Zizler, whereV * has been a subspace of a weakly compactly generate...

Let $x$ be a PC (point of continuity) for a bounded closed convex set $K$ of a Banach space. Then $x$ is a denting point of $K$ if and only if $x$ is an extreme point (resp. strong extreme point; weak$^\ast$-extreme point) of $K$. A new definition for denting point is also given.

Let a be a PC (point of continuity) for a bounded closed convex set K of a Banach space. Then x is a denting point of K if and only if x is an extreme point (resp. strong extreme point; weak*-extreme point) of K. A new definition for denting point is also given.

If Y is a subspace of a real Banach space X such that X/Y admits an equivalent LUR norm, then X admits an equivalent LUR (strictly convex) norm provided Y also does.

If Y is a subspace of a real Banach space X such that X/Y admits an equivalent LUR norm, then X admits an equivalent LUR (strictly convex) norm provided Y also does.

A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim | x n — x | = 0 for every x n , x ∈ X for which lim2| x | ² + 2 | x n | ² -| xn + x n | ² = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c 0 (Γ) such that T * c * 0 (Γ) is norm dense in X *. This is t...

If X∗ is a weakly compactly generated (WCG) Banach space, then X admits an equivalent C1-smooth norm. If a WCG Banach space X admits a Ck-smooth function with bounded support, then X admits Ck-smooth partitions of unity.