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Dilations of Unitary Tuples by Gerhold, Malte; Pandey, Satish K.; Shalit, Orr; Solel, Baruch is a scholarly article available to read on EtoBox.

What is Dilations of Unitary Tuples about?

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: t

Author

Gerhold, Malte; Pandey, Satish K.; Shalit, Orr; Solel, Baruch

Published

2020

Language

EN