A Factorization Problem for Normal Completely Bounded Mappings

Given an operator space X and a von Neumann algebra A, we consider a contractive mapping q: A⊗ehX⊗ehA→NCB(X*, A) formally defined by q(∑ aj⊗xjk⊗bk)=∑ xjk⊗ajbk, from the extended Haagerup tensor product A⊗ehX⊗ehA into the space of w*-continuous completely bounded maps from X* into A. We characterize elements of the range space Im(q) by a factorization property involving decomposable operators and investigate various properties of that space. In the case when X=B∗ is the predual of a von Neumann algebra B, Im(q) is included in the space DEC(B, A) of decomposable operators from B into A. Regarding q as having values in that space, we show that q is a quotient map onto its range. Then we prove that DEC(B, A) is a normal dual operator A-bimodule and that Im(q)⊂DEC(B, A) is a strong operator A-bimodule.