标题： 膨胀距离和遍历对易关系的稳定性 显示英文标题

标题： Dilation distance and the stability of ergodic commutation relations

摘要： 我们重新审视并推广了单位元组之间的膨胀距离${\rm d_{D}}(u,v)$的概念，并研究其与自然 Haagerup-R{\o }rdam 距离${\rm d_{HR}}(u,v) = \inf\{\|\pi(u) - \rho(v)\|\}$的关系，其中下确界是在所有忠实表示对$\pi \colon C^*(u) \to B(\mathcal{H})$,$\rho \colon C^*(v) \to B(\mathcal{H})$上取到的。 我们证明了${\rm d_{HR}}(u,v)\leq 10\operatorname{d_{rD}}(u,v)^{1/2}$，其中${\rm d_{rD}}(u,v)$是一种放松的膨胀距离，改进并扩展了以前的结果。 对于反对称矩阵$\Theta$，我们通过一个具体的扩张构造证明，根据$\Theta$近似交换的单位阵元组$u$（即，$\|u_\ell u_k - e^{i \theta_{k,\ell}} u_k u_\ell\|$很小）可以近似扩张为根据$\Theta$交换的单位阵元组$v$（即，$v_\ell v_k - e^{i \theta_{k,\ell}} v_k v_\ell = 0$很小）。我们证明可以通过第二次应用扩张构造来“反转”该扩张，这导致了原始阵元组的一个旋转版本。 因此，一个几乎与$\Theta$交换的规范不变酉元组可以在某个忠实表示中被一个$\Theta$交换的酉元组逼近。 此外，当$\Theta$是遍历的时，一个$\Theta$交换的元组被证明是{\em 几乎}规范不变的，并且由上述结果可知，这些元组可以用$\Theta$交换的元组在范数下逼近。 In particular, we obtain the following counterpart of Lin's theorem on almost commuting unitaries: if $q \in \mathbb{T}$ is {\em 不是} a root of unity, then for every $\varepsilon >0$ there exists $\delta > 0$ such that for every pair of unitaries $u_1,u_2 \in B(\mathcal{H})$ for which $\|u_1 u_2 - qu_2 u_1\| < \delta$, there exists two $q$-commuting unitaries $v_1, v_2 \in B(\mathcal{H} \otimes \ell^2)$ such that $\|v_i - u_i \otimes 1\| < \varepsilon$ ($i=1,2$). 显示英文摘要

摘要： We revisit and generalize the notion of dilation distance ${\rm d_{D}}(u,v)$ between unitary tuples and study its relation to the natural Haagerup-R{\o}rdam distance ${\rm d_{HR}}(u,v) = \inf\{\|\pi(u) - \rho(v)\|\}$, where the infimum is taken over all pairs of faithful representations $\pi \colon C^*(u) \to B(\mathcal{H})$, $\rho \colon C^*(v) \to B(\mathcal{H})$. We show that ${\rm d_{HR}}(u,v)\leq 10\operatorname{d_{rD}}(u,v)^{1/2}$, where ${\rm d_{rD}}(u,v)$ is a relaxed dilation distance, improving and extending earlier results. For an antisymmetric matrix $\Theta$, we show via a concrete dilation construction that a tuple of unitaries $u$ that almost commutes according to $\Theta$ (i.e., $\|u_\ell u_k - e^{i \theta_{k,\ell}} u_k u_\ell\|$ is small) can be nearly dilated to a tuple of unitaries $v$ that commutes according to $\Theta$ (i.e., $v_\ell v_k - e^{i \theta_{k,\ell}} v_k v_\ell = 0$). We show that the dilation can be "reversed" by a second application of the dilation construction, which leads to a rotated version of the original tuple. Thus, a gauge invariant almost $\Theta$-commuting unitary tuple can be approximated (in some faithful representation) by a $\Theta$-commuting unitary tuple. Moreover, when $\Theta$ is ergodic, a $\Theta$-commuting tuple is shown to be {\em almost} gauge invariant, and it follows from the results above that these can be approximated in norm by $\Theta$-commuting tuples. In particular, we obtain the following counterpart of Lin's theorem on almost commuting unitaries: if $q \in \mathbb{T}$ is {\em not} a root of unity, then for every $\varepsilon >0$ there exists $\delta > 0$ such that for every pair of unitaries $u_1,u_2 \in B(\mathcal{H})$ for which $\|u_1 u_2 - qu_2 u_1\| < \delta$, there exists two $q$-commuting unitaries $v_1, v_2 \in B(\mathcal{H} \otimes \ell^2)$ such that $\|v_i - u_i \otimes 1\| < \varepsilon$ ($i=1,2$).