A non-commutative version of Nikishin’s theorem

Let ττ be a tracial normal state on a von Neumann algebra, L1(τ)L1(τ) be the space of integrable self-adjoint operators, and SS be the space of self-adjoint measurable operators. We prove that every positive linear operator from an ordered Banach space to SS can be factorized through L1(τ)L1(τ).