  Let $1<p\not=2<\infty$ and let $S^p_n$ be the associated Schatten von Neumann
class over $n\times n$ matrices. We prove new characterizations of unital
positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an
invertible complete isometry acting on a non-commutative $L^p$-space. Then we
investigate the infinite dimensional case.
