Stability of vector measures and twisted sums of Banach spaces

A Banach space X is said to have the SVM (stability of vector measures) property if there exists a constant upsilon < infinity such that for any algebra of sets F, and any function nu : F -> X satisfying parallel to nu(A boolean OR B) - nu(A) - nu(B)parallel to <= 1 for disjoint A, B is an element of F, there is a vector measure mu : F -> X with parallel to nu(A) - mu(A)parallel to <= upsilon for all A is an element of F. If this condition is valid when restricted to set algebras F of cardinality less than some fixed cardinal number kappa, then we say that X has the kappa-SVM property. The least cardinal kappa for which X does not have the kappa-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 kappa-SVM property, kappa-injectivity and the 'three-space' problem. (C) 2013 Elsevier Inc. All rights reserved.