Vector measure Maurey-Rosenthal-type factorizations and ℓ-sums of L1-spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T:X⟶L1(m), where L1(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L2(m) by means of a multiplication operator given by a function of L2(|m|), where |m| is the variation of m, extending in this way the Maurey-Rosenthal Theorem. We use this result to obtain information about the structure of the space L1(m) when m is a sequential vector measure. In this case the space L1(m) is an ℓ-sum of L1-spaces.