Dilations of q-Commuting Unitaries

Let q = eiθ ϵ T (where θ ϵ R), and let u, v be q-commuting unitaries, that is, u and v are unitaries such that vu = quv. In this paper, we find the optimal constant c = cθ such that u, v can be dilated to a pair of operators cU, cV, where U and V are commuting unitaries. We show that ( equation presented) where uθ , vθ are the universal q-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating qcommuting unitaries to scalar multiples of q_-commuting unitaries. The techniques that we develop allow us to give new and simple "dilation theoretic"proofs of well-known results regarding the continuity of the field of rotations algebras. In particular, for the so-called "almost Mathieu operator"hθ = uθ + u*θ + vθ + v*θ , we recover the fact that the norm hθ is a Lipschitz continuous function of θ, as well as the result that the spectrum σ(hθ ) is a 12 -Hölder continuous function in θ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint*-polynomial p(uθ , vθ ), which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.