Factorization of Completely Bounded Bilinear Operators and Injectivity

A completely bounded bilinear operatorφ: M×M→M on a von Neumann algebra M is said to have a factorization in M if there exist completely bounded linear operatorsψj, θj: M→M such thatφ(x, y)=sum;j∈Λ ψj(x) θj(y), x, y∈M,where convergence of the sum is made precise below. The main result of the paper is that all completely bounded bilinear operatorsφ: M×M→M have factorizations in M if and only if M is injective.