Tuesday, May 16, 2023 17:00

Location: room A.4.1 C-19

Speaker: Damian Sobota (Kurt Gödel Research Center for Mathematical Logic)

Title: On continuous operators from Banach spaces of Lipschitz functions onto $c_0$

Abstract: During my talk I will discuss some of our recent results concerning the existence of continuous operators from the Banach spaces ${\text{Lip}}_0(M)$ of Lipschitz real-valued functions on metric spaces M onto the Banach space $c_0$ of sequences converging to $0$. I will in particular prove that there is always a continuous operator onto $c_0$ from infinite-dimensional spaces of the form ${\text{Lip}}_0(C(K))$ or ${\text{Lip}}_0({\text{Lip}}_0(M))$. (Based on an ongoing joint work with C. Bargetz and J. Kąkol).

Tags: Damian Sobota

Tuesday, March 26, 2019 17:15

Room: D1-215

Speaker: Damian Sobota (University of Viena)

Title: Josefson-Nissenzweig theorem for C(K)-spaces

Abstract. The Josefson-Nissenzweig theorem is a powerful tool in Banach space theory. Its special version for Banach spaces of continuous functions reads as follows: for a given infinite compact space K there exists a sequence $({\mu }_n)$ of normalized signed Radon measures on K such that the integrals ${\mu }_n(f)$ converge to 0 for any function f in $C(K)$. During my talk I will investigate when the sequence $({\mu }_n)$ can be chosen in such a way that every ${\mu }_n$ is just a finite linear combination of Dirac point measures (in other words, ${\mu }_n$ has finite support). This will appear to have connections with the Grothendieck property of Banach spaces and complementability of the space $c_0$. In particular, I'll present a very elementary proof that $c_0$ is always complemented in a space $C(K\times K)$.

Tags: Damian Sobota

Tuesday, October 13, 2015 17:15

Room: D1-215

Speaker: Damian Sobota

Title: The Nikodym property and cardinal invariants of the continuum

Abstract. A Boolean algebra $\mathcal{A}$ is said to have the Nikodym property if every sequence $({\mu }_n)$ of measures on $\mathcal{A}$ which is elementwise bounded (i.e. $\underset{n}{sup}|{\mu }_n(a)|<\mathrm{\infty }$ for every $a\in \mathcal{A}$) is uniformly bounded (i.e. $\underset{n}{sup}\Vert {\mu }_n\Vert <\mathrm{\infty }$). The property is closely related to the classical Banach-Steinhaus theorem for Banach spaces.

My recent study concerns the problem how (and whether at all) we can describe the structure of the class of Boolean algebras with the Nikodym property in terms of well-known objects occuring inside $\mathrm{\wp }(\omega )$ or ${\omega }^{\omega }$, e.g. countable Boolean algebras, dominating families, Lebesgue null sets etc. During my talk I will present an attempt to obtain such a description via families of antichains in countable subalgebras of $\mathrm{\wp }(\omega )$ having some special measure-theoretic properties.

Tags: Damian Sobota