Dilation distance and the stability of ergodic commutation relations

We revisit and generalize the notion of dilation distance dD(u,v) between unitary tuples and study its relation to the natural Haagerup-R{}rdam distance dHR(u,v) = inf\\|π(u) - ρ(v)\|\, where the infimum is taken over all pairs of faithful representations π C^*(u) → B(H), ρ C^*(v) → B(H). We show that dHR(u,v)≤ 10drD(u,v)1/2, where drD(u,v) is a relaxed dilation distance, improving and extending earlier results. For an antisymmetric matrix Θ, we show via a concrete dilation construction that a tuple of unitaries u that almost commutes according to Θ (i.e., \|u_ uₖ - e^i θk, uₖ u_\| is small) can be nearly dilated to a tuple of unitaries v that commutes according to Θ (i.e., v_ vₖ - e^i θk, vₖ v_ = 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 Θ-commuting unitary tuple can be approximated (in some faithful representation) by a Θ-commuting unitary tuple. Moreover, when Θ is ergodic, a Θ-commuting tuple is shown to be { almost} gauge invariant, and it follows from the results above that these can be approximated in norm by Θ-commuting tuples. In particular, we obtain the following counterpart of Lin's theorem on almost commuting unitaries: if q ∈ T is { not} a root of unity, then for every ε >0 there exists δ > 0 such that for every pair of unitaries u₁,u₂ ∈ B(H) for which \|u₁ u₂ - qu₂ u₁\| < δ, there exists two q-commuting unitaries v₁, v₂ ∈ B(H ⊗ ²) such that \|vᵢ - uᵢ ⊗ 1\| < ε ($i=1,2$).