Dilations of unitary tuples

TL;DR

This paper investigates dilation constants and metrics for tuples of unitaries, establishing bounds and continuity properties, and applies these to universal and special classes of unitary tuples.

Contribution

It introduces new bounds and metrics for unitary tuples, connecting dilation theory with matrix ranges and providing continuity results for universal unitary constructions.

Findings

Derived bounds for dilation constants between different classes of unitaries.

Established a square-root type inequality relating two metrics on unitary tuples.

Provided new lower bounds for the dilation constant in the case of three unitaries.

Abstract

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ 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 $*$-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 $*$-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 $(u_1,\ldots,u_d)$ 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 $c(u_\Theta, u_{\Theta'})\leq e^{\frac{1}{4}\|\Theta-\Theta'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $\Theta\mapsto U(\Theta)\in B(H)^d$ such that $U(\Theta)\sim u_\Theta$ and \[ \|U(\Theta)-U({\Theta'})\|\leq K\|\Theta-\Theta'\|^{1/2}. \] 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 $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.