Stability of vector measures and twisted sums of Banach spaces

TL;DR

This paper investigates the stability of vector measures in Banach spaces, characterizing properties using twisted sums and quasi-linear maps, and explores their connections to injectivity and the three-space problem.

Contribution

It introduces the $ ext{SVM}$ and $ ext{SVM}$ character concepts, applying twisted sum techniques to classify classical Banach spaces' stability properties.

Findings

Characterized $ ext{SVM}$ properties using twisted sums.

Determined $ ext{SVM}$ characters for many classical Banach spaces.

Explored links between $ ext{SVM}$ properties, $ ext{SVM}$-injectivity, and the three-space problem.

Abstract

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 $\nu\colon\mathcal F\to X$ satisfying $$\|\nu(A\cup B)-\nu(A)-\nu(B)\|\leq 1\quad{for disjoint}A,B\in\mathcal F,$$there is a vector measure $\mu\colon\mathcal F\to X$ with $\|\nu(A)-\mu(A)\|\leq v$ 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 $\kappa$, then we say that $X$ has the $\kappa$-$\mathsf{SVM}$ property. The least cardinal $\kappa$ for which $X$ does not have the $\kappa$-$\mathsf{SVM}$ property (if it exists) is called the $\mathsf{SVM}$ character of $X$. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine $\mathsf{SVM}$ characters for many classical Banach spaces. We also discuss connections between the $\kappa$-$\mathsf{SVM}$ property, $\kappa$-injectivity and the `three-space' problem.