Dilation distance and the stability of ergodic commutation relations

Abstract

We revisit and generalize the notion of dilation distance ${\mathrm{d}}_{\mathrm{D}}(u,v)$ between unitary tuples and study its relation to the natural Haagerup-Rørdam distance ${\mathrm{d}}_{\mathrm{HR}}(u,v)=\inf \{\Vert \pi (u)-\rho (v)\Vert \}$, where the infimum is taken over all pairs of faithful representations $\pi :C^{\ast }(u)\to B(\mathcal{H})$, $\rho :C^{\ast }(v)\to B(\mathcal{H})$. We show that ${\mathrm{d}}_{\mathrm{HR}}(u,v)\le 10{\mathrm{d}}_{\mathrm{rD}}(u,v{)}^{1/2}$, where ${\mathrm{d}}_{\mathrm{rD}}(u,v)$ 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., $\Vert u_{\ell }{u}_k-e^{i{\theta }_{k,\ell }}{u}_k{u}_{\ell }\Vert$ 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 $\epsilon >0$ there exists $\delta >0$ such that for every pair of unitaries $u_1,u_2\in B(\mathcal{H})$ for which $\Vert u_1{u}_2-q{u}_2{u}_1\Vert <\delta$, there exists two $q$-commuting unitaries $v_1,v_2\in B(\mathcal{H}\otimes {\ell }^2)$ such that $\Vert v_i-u_i\otimes 1\Vert <\epsilon$ ($i=1,2$).

17 pages