# Seminar on Geometry of Banach Spaces Posts

## 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).

## Interpolation of isomorphisms and Fredholm operators

We will discuss the interpolation of isomorphisms and Fredholm operators acting between Banach spaces generated by abstract interpolation methods. We will present general results regarding the stability of interpolated isomorphisms and the uniqueness of inverse operators between scales of interpolation spaces. We will show applications to PDE’s. The talk is based on the joint works with Irina Asekritova and Natan Kruglyak.

## A complemented subspace of a C(K)-space which is not a C(K)-space

We present a construction of two separable compacta K and L such that C(L) is a direct sum of C(K) and some Banach space X which is not isomorphic to a space of continuous functions. Joint work with Alberto Salguero Alarcon (Badajoz).

Let $$E$$ be a Hausdorff topological vector space. A subset $$Z$$ of $$E$$ is said to be admissible if for every compact subset $$K$$ of $$Z$$ and for every neighbourhood $$V$$ of zero in $$E$$ there exists a continuous mapping $$H\colon K → Z$$ such that $$\dim({\rm span} [H(K)])< \infty$$ and $$x − Hx \in V$$ for every $$x \in K$$ . If $$Z = E$$ we say that the space $$E$$ is admissible. This notion plays an important role in the fixed point theory. The aim of this talk is to show the admissibility of some classes of Frechet spaces. As an application, it will be proved the admissibility of a large class modular spaces equipped with F-norms and norms which are certain generalizations of the classical Luxemburg F-norm and the classical Luxemburg and Orlicz norms in the convex case Also a linear version of admissibility (so-called metric approximation property) for order continuous symmetric spaces will be demonstrated. The talk is based on a joint work with Maciej Ciesielski, Instytut Matematyki Politechniki Poznańskiej.

## Star-finite coverings of Banach spaces

In the first part of the talk, we provide a brief overview of tilings and coverings of Banach spaces. Then, some recent results in this context will be discussed. In particular, we will focus on star-finite coverings of Banach spaces by bodies. A family of subsets of a real normed space is called star-finite if each member of the family intersects at most finitely many other members. Part of the contents of the talk is included in joint work with C.A. De Bernardi, and L. Vesely.

## Chalmers-Metcalf operator and minimal projections

The Chalmers-Metcalf operator is a powerful tool in the theory of minimal projections. It was introduced by two American mathematicians Bruce Chalmers and Frederic Metcalf and applied in finding a formula for minimal projection from $$C_R[a,b]$$ onto the subspace polynomials of degree less than 2. During my talk, I would like to present various applications of this technique in the theory of minimal projections.

## Getting continuity of coordinate functionals related to F-basis with logic tools

Given a filter of subsets of natural numbers $$\mathcal{F}$$ we say that a sequence $$(x_n)$$ is $$\mathcal{F}$$ -convergent to $$x$$ if for every $$\varepsilon>0$$ condition $$\{n \in \mathbb N\colon d(x_n , x) < \varepsilon\} \in \mathcal{F}$$ holds. We may use this notion to generalize the idea of Schauder basis, namely we say that a sequence $$(e_n)$$ is an $$\mathcal{F}$$-basis if for every $$x \in X$$ there exists a unique sequence of scalars $$(\alpha_n)$$ s.t. $$\sum_{n,\mathcal{F}} \alpha_n e_n =x$$, which means that the sequence of partial sums is $$\mathcal{F}$$-convergent to $$x$$ Once such a notion is introduced it is natural to ask whenever corresponding coordinate functionals are continuous. Such a question was posed by V. Kadets during the 4th conference Integration, Vector Measures, and Related Topics held in 2011 in Murcia. Surprisingly, there is an obstacle related to the lack of uniform boundedness of functionals related to $$\mathcal{F}$$ basis, due to which we can not find proof of continuity analogous to the classical case. During my talk, I will discuss the problem and provide proof of continuity of considered functionals under some large cardinal assumptions. This is joint work with Tomasz Kania.

## Vector-valued invariant means and projections from the bidual space

Invariant means on amenable groups are an important tool in many parts of mathematics, especially in harmonic analysis. Invariant means and their generalizations for vector-valued functions play also an important role in the stability of functional equations and selections of set-valued functions.

Some generalization of the invariant mean for vector-valued functions was investigated in [2] and the existence of such means is connected with reflexive spaces.

Some generalized definition of an invariant mean has been used by many mathematicians as folklore (e.g. by A. Pełczyński [5]. The explicit form of this definition we can find e.g. in the work of R. Ger [3].

Definition:

Let $$(S,+)$$ be a left [right] amenable semigroup, $$X$$ be a real Banach space. A linear map $$M$$ from the space of all bounded functions from a set $$S$$ into a Banach space $$X$$ into $$X$$ is called left [right] $$X$$–valued invariant mean if
1) norm of $$M$$ is 1;
2) $$M$$ on a constant function is equal this constant,
3) $$M$$ is invariant under a left [right] shifts of the argument of the function.

If $$M$$ is a left and right invariant mean, then $$M$$ is called $$X$$-valued invariant mean.

If in the above definition norm of map $$M$$ is equal at most $$\lambda \geq 1$$, then $$M$$ is called $$X$$-valued invariant $$\lambda$$-mean.

The existence of such invariant means for a fixed Banach space and for all amenable semigroups has been studied by H.B. Domecq [1] (his paper has a gap in the proof which was corrected by T. Kania [4]).

In this talk, we will show a connection between the existence of $X$ -valued invariant $$\lambda$$-means on a Banach $$X$$ and projections from subspaces of bidual space $$X^{**}$$ (with large enough density) onto $$X$$ .

References:
[1] H. Bustos Domecq, Vector-valued invariant means revisited, J. Math. Anal. Appl. 275(2) (2002) 512–520.
[2] Z. Gajda, Invariant means and representations of semigroups in the theory of functional equations, Prace Naukowe Uniwersytetu Śląskiego, Katowice (1992).
[3] R. Ger, The singular case in the stability behavior of linear mappings, Grazer Math. Ber. {316} (1992), 59–70.
[4] T. Kania, Vector-valued invariant means revisited once again, J. Math. Anal. Appl. 445 (2017), 797–802.
[5] A. Pełczyński, Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Rozprawy Mat. 58 (1968), 92 pp.

## O charakteryzacji (i własnościach) $$\Delta$$-przestrzeni $$X$$ (w sensie Reeda) w terminach przestrzeni funkcji ciągłych $$C(X)$$

The seminar will take place in room 1016.

W 1975 roku Reed (zob. [7], [4]) wyróżnił (pod nazwą $$\Delta$$-zbiorów) te nieprzeliczalne podzbiory $$D$$ przestrzeni liczb rzeczywistych $$\mathbb{R}$$ (z topologią naturalną), które mają następujęcą a własność:

Dla każdego malejącego ciągu $$(H_{n})_{n}$$ zbiorów w $$D$$ takich, że $$\bigcap_n H_{_n}=\emptyset$$ istnieje ciąg $$G_{\delta}$$ -zbiorów w $$D$$ taki, że $$H_{n}\subset V_{n},\, n\in\mathbb{N}$$ oraz $$\bigcap_{n}V_{n}=\emptyset.$$

Przymusiński pokazał [6], że istnienie $$\Delta$$ -zbioru w $$\mathbb{R}$$ jest równoważne istnieniu przeliczalnie parazwartej ośrodkowej przestrzeni Moora nie będącej normalną.

Badania wokół $$\Delta$$-zbiorów ściśle związane z badaniami $$\mathbb{Q}$$-zbiorów (w sensie Hausdorffa), te ostatnie nadal stanowią fundamentalne wyzwania w teorii mnogości.

W pracy [2] pojęcie $$\Delta$$ -zbioru zostało rozszerzone do dowolnej przestrzeni topologicznej $$X$$ :

Mówimy, że przestrzeń topologiczna $$X$$ jest $$\Delta$$-przestrzenią a jeżeli dla każdego malejącego ciągu $$(H_{n})_{n}$$ zbiorów w $$X$$ takich, że $$\bigcap_n H_{_n}=\emptyset$$ istnieje ciąg $$(V_{n})_{n}$$ otwartych zbiorów w $$X$$ takich, że $$H_{n}\subset V_{n},\, n\in\mathbb{N}$$, oraz $$\bigcap_{n}V_{n}=\emptyset$$.

Pojęcie to pozwoliło uzyskać charakteryzację tych przestrzeni $$C_{p}(X)$$ (tj. przestrzeni funkcji ciągłych z topologią zbieżności punktowej określonych na przestrzeni Tichonowa $$X$$ ), które są wyróżnione (distingushed spaces). Ta ostatnia własność była intensywnie badana w klasie przestrzeni Frécheta, w szczególności przestrzeni Köthego $$\lambda_{p}(A)$$ .

W pracy [2] pokazuje się, że przestrzeń $$X$$ jest $$\Delta$$ -przestrzenią wtedy i tylko wtedy gdy $$C_{p}(X)$$ jest wyróżniona. Ten analityczny opis pozwolił uzyskać szereg nowych wyników dotyczących $$\Delta$$ -zbiorów i $$\Delta$$-przestrzeni [2], [3], [5]. Alternatywne spojrzenie na wyróżnione przestrzenie $$C_p(X)$$ prezentowane było w [1].

Między innymi pokazano w pracy [2], każda zupełna w senie Čecha $$\Delta$$ -przestrzeń jest rozproszona (scattered) i każda rozproszona zwarta przestrzeń Eberleina jest $$\Delta$$-przestrzenią; istnieją a jednak rozproszone zwarte przestrzenie $$X$$ , które nie są a $$\Delta$$-przestrzeniami, np. $$[0,\omega_{1}]$$ . Każda metryzowalna rozproszona przestrzeń topologiczna jest $$\Delta$$-przestrzenią. Podamy zastosowania tych wyników do badania przestrzeni Banacha $$C(K)$$ oraz przestrzeni $$C_{p}(K)$$ .

[1] J. C. Ferrando, S. A. Saxon, If not distinguished, is $$C_p(X)$$ even close?, Proc. Amer. Math. Soc. 149 (2021) Volume 149, 2583–-2596.

[2] J. Kąkol, A. Leiderman, A characterization of $$X$$ for which spaces $$C_p(X)$$ are distinguished and its applications, Proc Amer. Math. Soc. 8 (2021), 86–99.

[3] J. Kąkol, A. Leiderman, Basic properties of $$X$$ for which spaces $$C_p(X)$$ are distinguished, Proc. Amer. Math. Soc. 8 (2021), 257–280.

[4] R. W. Knight, $$\Delta$$ -Sets, Trans. Amer. Math. Soc. 339 (1993), 45–-60.

[5] A. Leiderman, V. V. Tkachuk, Pseudocompact $\Delta$-spaces are often scattered, Monatshefte für Math. 196 (2021).

[6] T. C. Przymusiński, Normality and separability of Moore spaces, Set-Theoretic Topology, Acad. Press, New York, 1977, 325–-337.

[7] G. M. Reed, On normality and countable paracompactness, Fund. Math. 110 (1980), 145–-152.

## Certain properties of Demyanov difference of convex sets

Adding and subtracting subsets of a vector space plays important role in nonsmooth analysis. There is no obvious difference of sets corresponding to the Minkowski (algebraic, vector) sum of sets. Demyanov difference is strictly related to Clarke subdifferential. We are going to compare Demyanov difference with other possible differences, present its new
properties and pose a few questions.