Dilation properties of measurable Schur multipliers and Fourier multipliers

In the article, we find new dilatation results on non-commutative $$L^p$$ spaces. We prove that any self-adjoint, unital, positive measurable Schur multiplier some $$B(L^2(\Sigma ))$$ admits, for all $$1\leqslant p<\infty $$ , an invertible isometric dilation -space. obtain a similar result completely Fourier VN(G), when G is unimodular locally compact group. Furthermore, establish multivariable versions of these results.