Every metric space of weight $λ=λ^{\aleph_0}$ admits a condensation onto a Banach space

Note on the Banach Problem 1 of condensations of Banach spaces onto compacta

It is consistent with any possible value of the continuum $\mathfrak{c}$ that every infinite-dimensional Banach space of density $\leq \mathfrak{c}$ condenses onto the Hilbert cube. Let $μ$ be a cardinal of uncountable cofinality. It is consistent that the continuum be arbitrary large, no Banach space $X$ of density $γ$, $μ< γ< \mathfrak{c}$ condenses onto a compactum, but any Banach space of density $μ$ admit a condensation onto a compactum. In particular, for $μ=ω_1$, it is consistent that $\mathfrak{c}$ is arbitrarily large, no Banach space of density $γ$, $ω_1< γ< \mathfrak{c}$, condenses …

关于第一个巴拿赫问题：绝对 $κ$-博雷尔集到紧空间的凝聚

On the first Banach problem, concerning condensations of absolute $κ$-Borel sets onto compacta

本文研究第一个巴拿赫问题中绝对 $κ$-博雷尔集到紧空间的凝聚性质，通过力迫构造证明连续统可任意大且凝聚行为可由自然数集合 $A$ 精确控制，揭示了该问题与模型论结果的深刻联系。

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a Valdivia compact and every separable subspace of $C(K)$ is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.

Spaces of continuous functions over Dugundji compacta

We show that for every Dugundji compact $K$ of weight aleph one the Banach space $C(K)$ is 1-Plichko and the space $P(K)$ of probability measures on $K$ is Valdivia compact. Combining this result with the existence of a non-Valdivia compact group, we answer a question of Kalenda.

Thin-very tall compact scattered spaces which are hereditarily separable

We strengthen the property $Δ$ of a function $f:[ω_2]^2\rightarrow [ω_2]^{\leq ω}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fréchet sm…

Automorphisms in spaces of continuous functions on Valdivia compacta

We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(I). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(I), all isomorphism between subspaces of C(K) of size less than aleph_omega extend to automorphisms of C(K).