Unital positive Schur multipliers on Sₙᵖ with a completely isometric dilation

Let 1<p≠ 2<∞ and let Sᵖₙ be the associated Schatten von Neumann class over n× n matrices. We prove new characterizations of unital positive Schur multipliers Sᵖₙ→ Sᵖₙ which can be dilated into an invertible complete isometry acting on a non-commutative Lᵖ-space. Then we investigate the infinite dimensional case.