Dilations of unitary tuples
Malte Gerhold, Satish K. Pandey, Orr Shalit, Baruch Solel

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 -tuples of unitaries using dilation theory and matrix ranges. Given two -tuples and generating C*-algebras and , we seek the minimal dilation constant such that , by which we mean that is a compression of some -isomorphic copy of . 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 be the universal unitary tuple satisfying , where is a real antisymmetric matrix. We find that $c(u_\Theta,…
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