In [5],  Victor Klee introduced the following definition: a real Banach space E is  polyhedral if the unit ball of each of its finite-dimensional subspaces is a polytope. Subsequently, other different notions of polyhedrality, all implying  Klee's polyhedrality, appeared in the literature; we refer to [4] for the  definitions  and implications  between them. In the paper [6], Joram Lindenstrauss proved that a dual Banach space cannot be polyhedral and, among other questions,  asked whether there exists a polyhedral infinite-dimensional Banach space whose unit ball is the closed convex hull of its extreme points. This problem was solved in the affirmative in the paper [3] by considering a suitable renorming of c0. After that, other examples of polyhedral Banach spaces with the same property but of a different kind were provided in [1]. During the talk, we  present some recent  results contained in [2], which show that every separable  polyhedral Banach space E  admits an e-equivalent (V)-polyhedral norm  such that the corresponding closed unit ball  is the closed convex hull of its extreme points. In the nonseparable case, the same result holds under the additional assumption that E is (IV)-polyhedral.

References

 [1] L. Antunes, K. Beanland and H.V. Chu, On the geometry of higher order Schreier spaces, Illinois J. Math. 65 (2021), 47–69.

 [2] C.A. De Bernardi, Unit balls of polyhedral Banach spaces with many extreme points, Studia Math., to appear.

 [3] C.A. De Bernardi, Extreme points in polyhedral Banach spaces, Israel J. Math., 220 (2017), 547–557.

 [4] V.P. Fonf and L. Veselý, Infinite dimensional polyhedrality, Canad. J. Math. 56 (2004), 472–494.

 [5] V. Klee, Polyhedral sections of convex bodies, Acta Math. 103 (1960), 243–267.

 [6]J. Lindenstrauss, Notes on Klee’s paper: “Polyhedral sections of convex bodies”, Israel J. Math. 4 (1966), 235–242.

After defining the missing notions, we will revisit a famous theorem of the local theory in the asymptotic setting. More precisely, we will see that a Banach space X is asymptotically B-convex iff l1 is not asymptotically finitely representable in X iff X has nontrivial asymptotic infratype iff X has nontrivial asymptotic Rademacher type iff X has nontrivial asymptotic stable type.

In this talk, we discuss the existence of bounded linear operators which never attain their norms. For this, we provide necessary and sufficient conditions for this to happen. We also discuss the possibility of finding very large infinite dimensional closed subspaces inside the (not linear at all) set of all non-norm-attaining operators.

We compute the Borel complexity of some classes of Banach spaces such as different versions of diameter two properties, spaces satisfying the Daugavet equation or spaces with an octahedral norm. In most of the above cases our computation is even optimal, which completes the research done during the last years around this topic for isomorphism classes of Banach spaces.

References

[1] G. López-Pérez, E. Martínez Vañó, and A. Rueda Zoca - Computing Borel complexity of some geometrical properties in Banach spaces. Preprint arXiv:2404.19457 (2024).

We analyze question 18 of J. Lindenstrauss in [4]. We prove that  a Banach space E with a norming subspace F ⊆ E* has an equivalent σ(E,F)-lower semicontinuous LUR norm if,  and only if,  there is a sequence {An: n=1,2,...} of subsets of E such that, given any x∈E and ε>0, there is a σ(E,F)-open half-space H and p ∈ IN such that x∈ H ∩ Ap and the slice H ∩ Ap can be covered with  countable many sets of diameter less than ε. Thus E has an equivalent σ(E,F)-lower semicontinuous LUR norm if, and only if,  it has another one with separable denting faces [8,9].

This result completely solves four problems asked in [6, Question 6.33, p.128] extending Troyanski's fundamental results (see Chapter IV in [1]), and other ones in [2,5]. 

Moreover,  LUR renormings are possible at points of separable faces wich could be glued as a σ-slicely isolated family of faces [6],  of the unit sphere  of E.

Among new examples covered by this results are Banach spaces C(K), where K is  a Rosenthal compact space K ⊆ IR^Γ i.e., a compact space of Baire one functions on a Polish space Γ,  with at most  countably many discontinuity points for every $s ∈ K$,  which solves three problems asked in [6, Question 6.23,  p.125]. Previously,  it was only known for K being separable too,  see [3] where the σ-fragmentability of C(K) was already proved for non separable K, and a conjecture for the pointwise lower semicontinuous and  LUR renorming presented here was posed, details  will appear in [7].

For strictly convex  renormings, we solve a recent question of R. Smith [11] giving a final answer to Lindenstauss question 18 in [4], see [9] and [10]. Indeed, we prove that E admits an equivalent σ(E,F)-lower semicontinuous and strictly convex norm if, and only if, it has another one with separable faces.  A purely topological new characterization follows for dual spaces and dual norms.

References

[1] R. Deville, G. Godefroy, and V. Zizler. Smoothness and renormings in Banach spaces, volume 64 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow, 1993.

[2] F. García, L. Oncina, J. Orihuela, and S. Troyanski. Kuratowski's index of non-compactness and renorming in Banach spaces. J. Convex Anal., 11(2):477-494, 2004.

[3] R. Haydon, A. Molt o, and J. Orihuela. Spaces of functions with countably many discontinuities. Israel J. Math., 158:19-39, 2007.

[4] J. Lindenstrauss. Some open problems in Banach space theory.

In Seminaire Choquet. Initiation  a l' analyse tome 15, 1975-1976, Exp. No. 18, pages 1{9. Secretariat mathématique, Paris, 1975-76.

[5] A. Moltó, J. Orihuela, and S. Troyanski. Locally uniformly rotund renorming and fragmentability. Proc. London Math. Soc.

(3), 75(3):619-640, 1997.

[6] A. Moltó J. Orihuela, S. Troyanski, and M. Valdivia. A non-linear transfer technique for renorming, volume 1951 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009.

[7] A. Moltó, V. Montesinos, J. Orihuela and S. Troyanski. Rosenthal compact spaces and renormings. Work in progress.

[8] V. Montesinos and J. Orihuela. Separable slicing and locally uniformly rotund renormings of Banach spaces. To appear in PAFA, 2024.

[9] V. Montesinos and J. Orihuela. Weak compactness and separability of faces for convex renormings of Banach spaces. Preprint 2024.

[10] J. Orihuela. A separation method through convex sets for strictly convex renormings of Banach spaces. Work in progress.

[11] R.J. Smith. A topological characterization of dual strict convexity in Asplund spaces. J. Math. Anal. Appl. 485 (2020) 123806

In a recent monograph A.J. Guirao, V. Montesinos, and V. Zizler posed several interesting problems on renorming techniques in Banach spaces. The present talk is dedicated to those problems which have been solved recently, with particular attention to the following one: 

Does every infinite-dimensional separable Banach space admit a norm that is rotund and Gâteaux-smooth but not midpoint locally uniformly rotund?

It turns out that the above question admits a positive answer, as we showed in collaboration with C. A. De Bernardi and A. Preti.

Abstract to be announced.

There has been recent interest in free objects in Banach space theory, as evidenced by the survey bearing that name (by García-Sánchez, de Hevia and Tradacete). Operator spaces, being a noncommutative version of Banach spaces, pose another natural setting in which to look for free objects.

Fittingly for this conference, we will talk about how the classical methods based on the Dixmier-Ng-Kaijser theorem can be adapted to the noncommutative world. We emphasize the free holomorphic operator space, since that is the one situation where we already have better results (such as transference of operator space approximation properties), but the same ideas work in several other cases. The key is to state the Dixmier-Ng-Kaijser result in a slightly more precise form that brings convexity into play, so that we can take advantage of noncommutative notions of convexity.

This is joint work with Verónica Dimant (Universidad de San Andrés). 

Let G be a topological group. A G-Banach space is an ordered triple (G, X, u), where X is a Banach space, and u: G → B(X) is a bounded left action of G on X. We say that a G-Banach space (G, X, u) is a GTop-Banach space if the action u is (τG, SOT)-continuous. We will show that, if

0 → (G, X, u) → (G, Z, λ) → (G, Y, v) → 0

is an exact sequence of G-Banach spaces such that (G, X, u) and (G, Y, v) are GTop-Banach spaces, and such that Z is super-reflexive, then (G, Z, λ) is also a GTop-Banach space. Afterwards, we will describe the linear derivations associated to a G-quasi-linear map Ω: Y → X in the case where the quasi-norm induced by Ω on X ⊕Ω Y is super-reflexive (in the sense that it is equivalent to a uniformly convex norm).

(joint work with Gregorz Plebanek and Antonio Avilés)

We start by considering a separable compact line K and its countable discrete extension L (so a compact space of the form K U ω, where ω is a discrete, countable set). We consider the minimal norm of an extension operator E : C(K) → C(L) and ask whether it can be finite, equal to one, infinite, etc. Those questions are inspired by the problem of twisted sums of Banach spaces.

The separable compact line K and its countable discrete extension can be described as an almost chain A = {Ax ⊆ ω: x ∈ X ⊆ [0, 1]}. It turns out that there is a space K of weight ω1 and L ∈ CDE(K) such that the minimal norm of an extension operator is equal to 3. With Antonio Avilés we were able to show that, consistently, if the weight of space K is sufficiently small, then each almost chain as above can be straightened, which means that there is always an extension operator of norm at most 3.

References

[1] M. Korpalski, G. Plebanek, Countable discrete extensions of compact lines, Fund. Math. 184 (2024), available online. 

[2] W. Marciszewski, Modifications of the double arrow space and related Banach spaces C(K), Studia Math. 184 (2008), 249–262.

[3] A.J.\ Ostaszewski, A characterization of compact, separable, ordered spaces, J. Lond. Math. Soc. 7 (1974), 758--760.

While the complementability of c0 in Banach spaces (in particular, in C(K) spaces) has been studied quite extensively, there seems to be very little known about when a Banach space contains complemented copies of other separable C(L) spaces. In the talk, we speak about a joint work in progress in this direction with Damian Sobota. More concretely, let L be a metrizable compact space and E be a Banach space. We present a characterization of the presence of a complemented copy of C(L) in E in terms of existence of a certain tree in E x E*. We also show how this can be applied to deduce that C(L) is complemented in C(K)  for certain nonmetrizable compacta K.

In the talk we focus on the relation of countable tightness of the space P(K) of Radon probabilty measures on a compact Hausdorff space K and of existence of measures in P(K) that have uncountable Maharam type. Recall that a topological space X has countable tightness if any element of the closure of a subset A of X lies in the closure of some countable subset of A. A Maharam type of a Radon probability measure μ is the density of the Banach space L1(μ). 

It was proven by Fremlin that, under Martin's axiom and negation of continuum hypothesis, for a compact Hausdorff space K the existance of a Radon probability of uncountable type is equivalent to the exitence of a continuous surjection from K onto [0,1]^ω1. Hence, under such assumptions, countable tightness of P(K) implies that there is no Radon probability on K which has uncountable type. Later, Plebanek and Sobota showed that, without any additional set-theoretic assumptions, countable tightness of P(KxK) implies that there is no Radon probability on K which has uncountable type as well. It is thus natural to ask whether the implication "P(K) has countable tightness implies every Radon probability on K has countable type" holds in ZFC. 

I will present our joint result with Piotr Koszmider that under diamond principle there is a compact Hausdorff space K such that P(K) has countable tightness but there exists a Radon probability on K of uncountable type.

Preprint with the result: https://arxiv.org/…750

It is well-known that the Banach space C(K) of continuous real-valued functions on a compact space K is not Grothendieck provided that K contains a non-trivial convergent sequence. Using Borel filters and submeasures on the set IN of natural numbers, we will generalize this observation and construct continuum many pairwise non-homeomorphic countable spaces Y with exactly one non-isolated point, no non-trivial convergent sequences, and such that if Y embeds into a compact space K, then C(K) is not a Grothendieck space. We will apply these techniques to obtain new results concerning Grothendieck Banach spaces of the form c0,F.

Let M, N be pointed metric spaces and F(M), F(N) their corresponding Lipschitz-free spaces. It is well known that every Lipschitz map f:M → N such that f(0)=0 admits the free linearization ^f:F(M) → F(N), also called the lipschitz operator associated to f. For example, it can be realized as the pre-adjoint of the composition operator Cf: Lip0(N) → Lip0(M). In this talk, based on a joint work with Gonzalo Flores, Mingu Jung, Gilles Lancien, Colin Petitjean and Andrés Quilis, we will characterize those f for which ^f is completely continuous, i.e. transforms weakly convergent sequences into norm convergent ones.

The content of this talk is based on a joint paper with L. Doktorski and T. Signes, Mathematische Nachrichten, 2023.

We consider Lorentz-Karamata spaces, small and grand Lorentz-Karamata spaces, and the so-called L, R, LL, RL and RR spaces. The quasi-norms for a function f in each of these spaces can be defined via the non-increasing rearrangement f* or via the maximal function f**. We investigate when these quasi-norms are equivalent. Most of the proofs are based on Hardy-type inequalities. As application we demonstrate how our general results can be used to establish interpolation formulae for the grand and small Lorentz-Karamata spaces.