Dilations of unitary tuples

Abstract

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 \[ d_{HR}(u,v)=\inf\left\{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u\textrm{ and } v'\sim v\right\}, \] and we show the inequality \[ d_{HR}(u,v)\leq K d_D(u,v)^{1/2}. \] Let $u_{\Theta }$ be the universal unitary tuple $(u_1,\dots ,u_d)$ satisfying $u_{\ell }{u}_k=e^{i{\theta }_{k,\ell }}{u}_k{u}_{\ell }$, where $\Theta =({\theta }_{k,\ell })$ is a real antisymmetric matrix. We find that $c(u_{\Theta },u_{{\Theta }^{\prime }})\le e^{\frac{1}{4}\Vert \Theta -{\Theta }^{\prime }\Vert }$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $\Theta \mapsto U(\Theta )\in B(H{)}^d$ such that $U(\Theta )\sim u_{\Theta }$ and \[ \|U(Θ)-U({Θ'})\|\leq K\|Θ-Θ'\|^{1/2}. \] 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 \[ 2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}. \] 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 $c(\mathrm{u},u_0)\ge \sqrt{3}$.

31 pages. A few minor corrections have been made and a new appendix (Appendix B) has been added. To appear in Journal of the London Mathematical Society

References in corpus (6)

• Minimal and maximal matrix convex sets
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Joint measurability of quantum effects and the matrix diamond
• Compatibility of quantum measurements and inclusion constants for the matrix jewel
• Shape, Scale, and Minimality of Matrix Ranges
• Continuous perturbations of noncommutative Euclidean spaces and tori

Cited by in corpus (1)

• Bounded perturbations of the Heisenberg commutation relation via dilation theory