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Category: Seminar name

2024.03.20 Taras Banakh

Ivan Franko National University of Lviv and Jan Kochanowski University of Kielce

Geometry of Banakh spaces

Abstract: Following Will Brian, we define a metric space X to be Banakh if all nonempty spheres of positive radius r in X have cardinality 2 and diameter 2r. This notion arose from attempts to find a characterization of the real line in the class of metric spaces. Standard examples of Banakh spaces are subgroups of the real line. However, there exist also some exotic examples of Banakh spaces that are quite different from subsets of the real line. In particular, every Hilbert space of density less or equal to the continuum contains a dense Banakh subgroup.

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Michał Popow

Pomeranian University in Słupsk (Poland) and Vasyl Stefanyk Precarpathian National University, IvanoFrankivsk (Ukraine)

Representation theorems for regular operators

[6.03.2024, 12.15] 


Please find the abstract here

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Antoni Machowski

Jagiellonian University 

Strongly exposing functionals of convex weakly compact sets

[27.02.2024, 12.15. THE MEETING IS HELD HYBRID, STATIONARY PART in 1016]

Abstract: For a closed, convex bounded subset C of a Banach space X we call a point x in C strongly exposed if some functional on X attains its strong maximum in x. Functionals of norm one that satisfy the above condition for some point in C are called strongly exposing functionals of C. In my talk I will present a proof by J. Bourgain of celebrated results by J. Lindenstrauss and S.L. Troyanski stating that a convex and weakly compact subset of a Banach space is the closed convex hull of its strongly exposed points and the set of strongly exposing functionals of C is a dense G-delta subset of the unit sphere of X*.


J. Lindenstrauss, On operators which attain their norm (1963)

S. L. Trojanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces (1970)

J. Bourgain, Strongly exposed points in weakly compact convex sets in Banach spaces (1976)

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Meeting ID: 392 359 686 95

Passcode: HMgHLn

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Seminars in 2023/2034 (Winter term)

24.01.2024 ▴ Jerzy Grzybowski (Adam Mickiewicz University in Poznań):
Maksymalna różnica Demyanova zbiorów wypukłych, a maksymalna ilość punktów ekstremalnych sumy Minkowskiego
10.01.2024 ▴ Tomasz Kania:
Lattice points in Banach spaces
20.12.2023 ▴ Joanna Garbulińska-Węgrzyn (Jan Kochanowski University of Kielce):
On the Darji Matheron universal operator
13.12.2023 ▴ Tomasz Kobos:
Spaces with maximal projection constants revisited
6.12.2023 ▴ Mariusz Niwiński:
Indefinite inner product spaces and Dirac operators
29.11.2023 ▴ Grzegorz Lewicki:
On two-strongly unique best approximation in the complex case
22.11.2023 ▴ Anna Pelczar-Barwacz:
Equivalence of block sequences in Schreier spaces
8.11.2023 ▴ Krystian Kazaniecki (Johannes Kepler University Linz):
Schur property for jump parts of gradient measures
25.10.2023 ▴ Grzegorz Lewicki:
On the maximal hyperplane in ℓ^p_n
18.10.2023 ▴ Hanna Wojewódka-Ściążko (UŚ; PAN):
When asymptotically stable Markov semigroups have the e-property?
11.10.2023 ▴ Jerzy Kąkol:
About Banach-Mazur’s problem from the 1930s
4.10.2023 ▴ Michał Wojciechowski (Instytut Matematyczny Polskiej Akademii Nauk):
Singular pluriharmonic quaternionic measure
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Seminars in 2022/2023

31.05.2023  Tomasz Kochanek (MIMUW):
The Hyers-Ulam property of groups and its connections to group C*-algebras
24.05.2023  Tomasz Kobos:
Lower bounds on the norm of linear projection through smoothness
17.05.2023  Antoni Machowski:
A generalization of a theorem of Yamabe
10.05.2023  Jarosław Swaczyna (Technical University of Łódź):
Slow dynamical systems with large Hausdorff dimension
26.04.2023  Anna Pelczar-Barwacz:
Operator ideals on Schreier spaces of arbitrary order
19.04.2023  Natalia Maślany (UJ):
Differential embeddings into algebras of topological stable rank 1
5.04.2023  Barbara Lewandowska:
On the generalized Grunbaum conjecture (2)
22.03.2023  Barbara Lewandowska:
On the generalized Grunbaum conjecture (1)
8.03.2023  Tomasz Kobos:
An affine dimension of the set of minimal hyperplane projections
25.01.2023  Tomasz Kania:
Isometries of combinatorial Tsirelson spaces
18.01.2023  Grzegorz Lewicki:
Minimal projections onto subspaces generated by sign matrices (2)
11.01.2023  Grzegorz Lewicki:
Minimal projections onto subspaces generated by sign matrices (1)
14.12.2022  Krzysztof Ciosmak (University of Oxford, Fields Institute, University of Toronto):
Multi-dimensional localisation and 1-Lipschitz maps
7.12.2022  Natalia Maślany:
Isometries of combinatorial Tsirelson spaces
30.11.2022  Tomasz Kobos:
On the affine dimension of the set of minimal projections (4)
23.11.2022  Tomasz Kobos:
On the affine dimension of the set of minimal projections (3)
16.11.2022  Bence Horváth (Institute of Mathematics of the Czech Academy of Sciences):
Ring-theoretic (in)finiteness in ultraproducts of Banach algebras
9.11.2022  Anna Pelczar-Barwacz:
A Banach space with an infinite-dimensional reflexive quotient algebra L(X)/SS(X)
26.10.2022  Tomasz Kobos:
On the affine dimension of the set of minimal projections (2)
19.10.2022  Andrzej Kryczka (UMCS):
Regularne metody sumowania dla sum prostych i przestrzeni interpolacyjnych
12.10.2022  Tomasz Kobos:
On the affine dimension of the set of minimal projections (1)
5.10.2022  Spotkanie organizacyjne 
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2022.03.30 Jarosław Swaczyna, Technical University of Łódź

Continuity of coordinate functionals for filter Schauder basis without Large Cardinals

During my talk in November, I presented joint results with Tomasz Kania about using the large cardinals to positively answer the Kadets problem about coontinuity of coordinate functionals related to filter Schauder basis. This time I will present results obtained jointly with Tomasz Kania and Noe de Rancourt, which gives a positive answer to Kadets’ question in ZFC for analytic filters. I will also show that each filter basis with continuous coordinate functionals is also a basis with respect to some analytic filter.

Link to the talk.

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2022.03.23 Kevin Beanland, W&L University

Surjective Isometries of Banach sequence spaces: a survey

Beginning with the work of Banach on \(L_p\) spaces there are numerous papers characterizing the surjective isometries on various classical and non-classical Banach spaces. In this talk, we will give a broad overview of this work including the spaces of James, Schreier, and Tsirelson, and state several interesting open problems.

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2022.03.09 Noé de Rancourt, Charles University, Prague

Guarded Fraïssé Banach spaces

Fraïssé theory, originally developed in the context of model theory, establishes a bijective correspondence between classes of finite structures having good amalgamation properties and so-called Fraïssé structures, i.e. countable structures satisfying a strong homogeneity property. This correspondence has recently been extended to Banach spaces by Ferenczi, Lopez-Abad, Mbombo et Todorcevic, who proved that the spaces \(L_p\), \(1 \leq p \neq 4, 6, 8, … < \infty\), and the Gurarij space, are Fraïssé. They asked whether those are the only examples; this question is related to Mazur’s rotation problem.

In a work in progress with Marek Cúth and Michal Doucha, we introduce a weak version of the Fraïssé correspondence, the so-called guarded Fraïssé correspondence, extending results obtained by Krawczyk and Kubiś in the discrete setting. We prove that a separable Banach space is guarded Fraïssé if and only if its isometry class is \(G_\delta\) for a natural topological coding of separable Banach spaces introduced by Cúth, Doležal, Doucha et Kurka. This links to the above-mentioned question of Ferenczi, Lopez-Abad, Mbombo et Todorcevic with descriptive set theory of Banach

I will present those results and their links with some questions in continuous logic. I will also discuss the existence of new examples of guarded Fraïssé Banach spaces.

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2022.01.26 Colin Petitjean, Université Gustave Eiffel

Some remarks about ‘‘Lipschitz-free operators’’

If M is a metric space, then the so-called Lipschitz-free space over M, usually denoted F(M), is a Banach space which is built around M in such a way that – M is isometric to a subset of F(M); – Lipschitz maps from M into any other Banach space X uniquely extend to bounded linear operators from F(M) into X. An interesting feature of Lipschitz-free spaces is that every Lipschitz map between two metric spaces M and N can be ‘‘linearised’’  in such a way that it becomes a bounded linear operator between the free spaces F(M) and F(N). We refer to these linearisations as ‘‘Lipschitz-free operators’’, or simply ‘‘Lipschitz operators’’. In this talk, we will study how the properties of the Lipschitz maps and their linearisations are related. After a few simple observations, we will mainly focus on some dynamical properties, compactness properties, and injectivity. This talk is based on ongoing works joint with Arafat Abbar (Marne-la-Vallée) and Clément Coine (Caen); Luis García-Lirola (Zaragoza) and Antonín Procházka (Besançon).

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