Stability of vector measures and twisted sums of Banach spaces

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 $\nu :\mathcal{F}\to X$ satisfying $$\Vert \nu (A\cup B)-\nu (A)-\nu (B)\Vert \le 1fordisjointA,B\in \mathcal{F},$$there is a vector measure $\mu :\mathcal{F}\to X$ with $\Vert \nu (A)-\mu (A)\Vert \le v$ 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.