  We obtain dilation results which simultaneously generalize Sz.-Nagy dilation
theorem for contractions, Ando's dilation theorem for commuting contractions,
Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the
unit ball of H^\infty, in the framework of noncommutative varieties in several
variables and Poisson kernels on Fock spaces. This leads to inequalities which,
in the particular case of two contractions, are sharper than Ando's inequality
and Agler-McCarthy's inequality in the setting of commuting contractive
matrices or arbitrary commuting contractions of class C_0. Our results extend
to the bi-ball and a large class of noncommutative varieties.
