Dilation distance and the stability of ergodic commutation relations

TL;DR

This paper introduces and analyzes a generalized dilation distance between unitary tuples, establishing bounds with the Haagerup-R{ }ordam distance, and applies these results to approximate almost commuting unitaries by exactly commuting ones, extending Lin's theorem.

Contribution

It generalizes the notion of dilation distance, relates it to the Haagerup-R{ }ordam distance, and applies these concepts to approximate almost commuting unitaries by commuting unitaries, especially for non-root of unity cases.

Findings

Established a bound: d_{HR}(u,v) ≤ 10 * d_{rD}(u,v)^{1/2}.

Constructed dilations showing almost commuting unitaries can be approximated by commuting ones.

Extended Lin's theorem to non-root of unity cases for almost commuting unitaries.

Abstract

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 $\|v_i - u_i \otimes 1\| < \varepsilon$ ($i=1,2$).