Dilations of unitary tuples

We study the space of all $d$-tuples of unitaries $u=(u_1,\dots ,u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal{A}$ and $\mathcal{B}$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $\ast$-isomorphic copy of $cv$. This gives rise to a metric $$d_D(u,v)=\log \max \{c(u,v),c(v,u)\}$$ on the set of equivalence classes of $\ast$-isomorphic tuples of unitaries. We also consider the metric $$d_{HR}(u,v)=\inf \{\mathrm{\parallel }{u}^{\mathrm{\prime }}-v^{\mathrm{\prime }}\mathrm{\parallel }:u^{\mathrm{\prime }},v^{\mathrm{\prime }}\in B(H{)}^d,u^{\mathrm{\prime }}\sim u\text{ and }{v}^{\mathrm{\prime }}\sim v\},$$ and we show the inequality $$d_{HR}(u,v)\le K{d}_D(u,v{)}^{1\mathrm{/}2}\mathrm{.}$$ Let $u_{\mathrm{\Theta }}$ be the universal unitary tuple $(u_1,\dots ,u_d)$ satisfying $u_{\mathrm{\ell }}{u}_k=e^{i{\theta }_{k,\mathrm{\ell }}}{u}_k{u}_{\mathrm{\ell }}$, where $\mathrm{\Theta }=({\theta }_{k,\mathrm{\ell }})$ is a real antisymmetric matrix. We find that $c(u_\Theta, u_{\Theta’})\leq e^{\frac{1}{4}|\Theta-\Theta’|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $\Theta\mapsto U(\Theta)\in B(H)^d$suchthat$U(\Theta)\sim u_\Theta$ and $$\mathrm{\parallel }U(\mathrm{\Theta })-U({\mathrm{\Theta }}^{\mathrm{\prime }})\mathrm{\parallel }\le K\mathrm{\parallel }\mathrm{\Theta }-{\mathrm{\Theta }}^{\mathrm{\prime }}{\mathrm{\parallel }}^{1\mathrm{/}2}\mathrm{.}$$ Of special interest are: the universal $d$-tuple of noncommuting unitaries $\mathrm{u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds $$2\sqrt{1-\frac{1}{d}}\le c(u_f,u_0)\le 2\sqrt{1-\frac{1}{2d}}\mathrm{.}$$ From this, we recover Passer’s upper bound for the universal unitaries $c(\mathrm{u},u_0)\le \sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c(\mathrm{u},u_0)\ge 1.858$ improving on the previously known lower bound $c(\mathrm{u},u_0)\ge \sqrt{3}$.