Andô dilations and inequalities on noncommutative varieties

We prove that there is a universal model (S⊗Iℓ2,φ(S)), where S is the unilateral shift and φ(S) is an isometric analytic Toeplitz operator on H2(D)⊗ℓ2, such that‖[prs(T1,T2)]k‖≤‖[prs(S⊗Iℓ2,φ(S))]k‖, for any commuting contractions T1 and T2 on Hilbert spaces, any k×k matrix [prs]k of polynomials in C[z,w], and any k∈N. Analogues of this result for the bi-ball Pn1,n2 and for a class of noncommutative varieties are also considered.