On factorization of separating maps on noncommutative L^p-spaces

For any semifinite von Neumann algebra ${\mathcal M}$ and $1\leq p<\infty$, we introduce a natutal $S^1$-valued noncommutative $L^p$-space $L^p({\mathcal M};S^1)$. We say that bounded map $T\colon L^p({\mathcal M})\to N})$ is $S^1$-bounded (resp. $S^1$-contractive) if $T\otimes I_{S^1}$ extends to contractive) $T\overline{\otimes} from $ M};S^1)$ into N};S^1)$. show completely positive $S^1$-bounded, with $\Vert T\overline{\otimes} I_{S^1}\Vert =\Vert T\Vert$. use the above as tool investigate separating maps which admit direct Yeadon type factorization, is, for there exist $w^*$-continuous $*$-homomorphism $J\colon{\mathcal M}\to{\mathcal N}$, partial isometry $w\in{\mathcal N}$ operator $B$ affiliated such $w^*w=J(1)=s(B)$, commutes range of $J$, $T(x)=wBJ(x)$ $x\in {\mathcal M}\cap M})$. Given N})$, $T$ $S^1$-contractive only it admits factorization. further $p\not=2$, holds true contractive.