Twisted sums and a problem of Klee

关于 $C(ω^α)$ 的扭曲和的的多面体性

Polyhedrality for twisted sums with $C(ω^α)$

本文研究同构多面体性的三空间问题，证明了 $C(α)$ 与具有 BAP 的可分同构多面体空间的扭曲和，以及 $c_0(κ)$ 与具有性质 $(*)$ 边界的空间的扭曲和，均为同构多面体。

Twisted sums of $c_0$ and $C(K)$-spaces: A solution to the CCKY problem

We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis $c_0$ has a nontrivial twisted sum with every space of the form $Y=C(K)$, where $K$ is compact and not metrizable. This gives a consistent positive solution to a problem posed by Cabello, Castillo, Kalton and Yost.

Singular twisted sums generated by complex interpolation

We present new methods to obtain singular twisted sums $X\oplus_ΩX$ (i.e., exact sequences $0\to X\to X\oplus_ΩX \to X\to 0$ in which the quotient map is strictly singular), in which $X$ is the interpolation space arising from a complex interpolation scheme and $Ω$ is the induced centralizer. Although our methods are quite general, in our applications we are mainly concerned with the choice of $X$ as either a Hilbert space, or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces, including the only known example so f…

Type, cotype and twisted sums induced by complex interpolation

This paper deals with extensions or twisted sums of Banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension. The results are presented in the context of interpolation of families of Banach spaces, and are applied to the study of submodules of Schatten classes. We also obtain nontrivial extensions of spaces without the CAP which also fail the CAP.

Extension operators and twisted sums of $c_0$ and $C(K)$ spaces

We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^…

关于扭曲和的扭曲性的一点注记

A remark on the distortion of twisted sums

本文指出Kalton-Peck空间 $Z_p$ 容许必然的双正交系统，从而将其加入任意可扭曲空间的经典族列表中。

Group actions on twisted sums of Banach spaces

We study bounded actions of groups and semigroups $G$ on exact sequences of Banach spaces from the point of view of quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of $G$-centralizer and $G$-equivariant map. We will show that when (A) $G$ is an amenable group and (U) the target space is complemented in its bidual by a $G$-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of $G$…

Stability of vector measures and twisted sums of Banach spaces

A Banach space $X$ is said to have the $\mathsf{SVM}$ (stability of vector measures) property if there exists a constant $v<\infty$ such that for any algebra of sets $\mathcal F$, and any function $ν\colon\mathcal F\to X$ satisfying $$\|ν(A\cup B)-ν(A)-ν(B)\|\leq 1\quad{for disjoint}A,B\in\mathcal F,$$there is a vector measure $μ\colon\mathcal F\to X$ with $\|ν(A)-μ(A)\|\leq v$ for all $A\in\mathcal F$. If this condition is valid when restricted to set algebras $\mathcal F$ of cardinality less than some fixed cardinal number $κ$, then we say that $X$ has the $κ$-$\mathsf{SVM}$ property. The le…

Splitting chains, tunnels and twisted sums

We study splitting chains in $\mathscr{P}(ω)$, that is, families of subsets of $ω$ which are linearly ordered by $\subseteq^*$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{ZFC}$. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of $C(ω^*)=\ell_\infty/c_0$ and $c_0(\mathfrak c)$. Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.

Spectra of Abelian C*-Subalgebra Sums

Let $C_b(X)$ be the C*-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space $X$. Moreover, let $A_0$ be some ideal and $A_1$ be some unital C*-subalgebra of $C_b(X)$. For $A_0$ and $A_1$ having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate before. Within the whole framework, e.g., the one-point compactification of $X$ and the spectrum of th…