We study the space of all $d$-tuples of unitaries $u=(u_1,\dots ,u_d)$ using dilation theory and matrix ranges. Given two such $d$-tuples $u$ and $v$ generating, respectively, 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 there exist faithful $\ast$-representations $\pi :\mathcal{A}\to B(\mathcal{H})$ and $\rho :\mathcal{B}\to B(\mathcal{K})$, with $\mathcal{H}\subseteq \mathcal{K}$, such that for all $i$, $\pi (u_i)$ is equal to the compression $P_{\mathcal{H}}\rho (c{v}_i){|}_{\mathcal{H}}$ of $\rho (c{v}_i)$ to $\mathcal{H}$. This gives rise to a metric

$$d_{\mathrm{D}}(u,v)=\log \max \{c(u,v),c(v,u)\}$$

on the set of equivalence classes of $\ast$-isomorphic tuples of unitaries. We compare this metric to the metric $d_{\mathrm{HR}}$ determined by

$$d_{\mathrm{HR}}(u,v)=inf\left\{\parallel u^{\prime }-v^{\prime }\parallel :u^{\prime },v^{\prime }\in B{(\mathcal{H})}^d,u^{\prime }\sim u\mathrm{and}{v}^{\prime }\sim v\right\},$$

and we show the inequality

$$d_{\mathrm{HR}}(u,v)⩽K{d}_{\mathrm{D}}{(u,v)}^{1/2}$$

where $1/2$ is optimal. When restricting attention to unitary tuples whose matrix range contains a $\delta$-neighborhood of the origin, then $d_{\mathrm{D}}(u,v)⩽d{\delta }^{-1}{d}_{\mathrm{HR}}(u,v)$, so these metrics are equivalent on the set of tuples whose matrix range contains some neighborhood of the origin. Moreover, these two metrics are equivalent to the Hausdorff distance between the matrix ranges of the tuples.

中文翻译：

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单一元组的膨胀

我们研究所有人的空间$d$- 酉元组$你=({你}_1,\dots ,{你}_d)$使用膨胀理论和矩阵范围。给定两个这样的$d$-元组$你$和$v$分别生成 C*-代数$\mathcal{一个}$和$\mathcal{乙}$，我们寻求最小膨胀常数$C=C(你,v)$这样$你\prec Cv$, 我们的意思是存在忠实的$*$- 陈述$\pi ：\mathcal{一个}\to 乙(\mathcal{H})$和$\rho ：\mathcal{乙}\to 乙(\mathcal{ķ})$， 和$\mathcal{H}\subseteq \mathcal{ķ}$, 这样对于所有$\mathrm{一世}$,$\pi ({你}_{\mathrm{一世}})$等于压缩${磷}_{\mathcal{H}}\rho (C{v}_{\mathrm{一世}}){|}_{\mathcal{H}}$的$\rho (C{v}_{\mathrm{一世}})$到$\mathcal{H}$. 这产生了一个度量

$$d_{\mathrm{D}}(你,v)=\mathrm{日志}\mathrm{最大限度}\{C(你,v),C(v,你)\}$$

在等价类的集合上$*$- 同构酉元组。我们将此指标与指标进行比较$d_{\mathrm{人力资源}}$取决于

$$d_{\mathrm{人力资源}}(你,v)=信息\left\{\parallel {你}^{\text{'}}-v^{\text{'}}\parallel ：{你}^{\text{'}},v^{\text{'}}\in 乙{(\mathcal{H})}^d,{你}^{\text{'}}～你和v^{\text{'}}～v\right\},$$

我们展示了不等式

$$d_{\mathrm{人力资源}}(你,v)⩽ķd_{\mathrm{D}}{(你,v)}^{1/2}$$

在哪里$1/2$是最优的。当将注意力限制在矩阵范围包含$\delta$- 原点的邻域，然后$d_{\mathrm{D}}(你,v)⩽d{\delta }^{-1}{d}_{\mathrm{人力资源}}(你,v)$，因此这些度量在其矩阵范围包含原点的某个邻域的元组集上是等价的。此外，这两个度量等价于元组的矩阵范围之间的 Hausdorff 距离。