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Let   $q = e^{i \theta } \in \mathbb{T}$   (where   $\theta \in \mathbb{R}$  ), 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   $c = c_{\theta }$   such that   $u,v$   can be dilated to a pair of operators   $c U, c V$  , where   $U$   and   $V$   are commuting unitaries. We show that   $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$  where   $u_{\theta }, v_{\theta }$   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   $q$  -commuting unitaries to scalar multiples of   $q^{\prime}$  -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”   $h_{\theta } = u_{\theta }+u_{\theta }^*+v_{\theta }+v_{\theta }^*$  , we recover the fact that the norm   $\|h_{\theta }\|$   is a Lipschitz continuous function of   $\theta $  , as well as the result that the spectrum   $\sigma (h_{\theta })$   is a   $\frac{1}{2}$  -Hölder continuous function in   $\theta $   with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial   $p(u_{\theta },v_{\theta })$  , which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

Dilations of q -Commuting Unitaries