Stability of vector measures and twisted sums of Banach spaces

Abstract

A Banach space X is said to have the SVM (stability of vector measures) property if there exists a constant v<∞ such that for any algebra of sets F, and any function ν:F→X satisfying‖ν(A∪B)−ν(A)−ν(B)‖⩽1for disjoint A,B∈F, there is a vector measure μ:F→X with ‖ν(A)−μ(A)‖⩽v for all A∈F. If this condition is valid when restricted to set algebras F of cardinality less than some fixed cardinal number κ, then we say that X has the κ-SVM property. The least cardinal κ for which X does not have the κ-SVM property (if it exists) is called the SVM character of X. We apply the machinery of twisted sums and quasi-linear maps to characterise these properties and to determine SVM characters for many classical Banach spaces. We also discuss connections between the κ-SVM property, κ-injectivity and the ‘three-space’ problem.