Dilations of unitary tuples

• Published in: Journal of the London Mathematical Society 2021-12, Vol.104 (5), p.2053-2081
• Main Authors: Gerhold, Malte, Pandey, Satish K., Shalit, Orr Moshe, Solel, Baruch
• Format: Article
• Language: English
• Subjects: 46L54 (primary); 47A13
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms.12491

We study the space of all d‐tuples of unitaries u=(u1,…,ud) using dilation theory and matrix ranges. Given two such d‐tuples u and v generating, respectively, C*‐algebras A and B, we seek the minimal dilation constant c=c(u,v) such that u≺cv, by which we mean that there exist faithful ∗‐representations π:A→B(H) and ρ:B→B(K), with H⊆K, such that for all i, π(ui) is equal to the compression PHρ(cvi)|H of ρ(cvi) to H. This gives rise to a metric dD(u,v)=logmax{c(u,v),c(v,u)}on the set of equivalence classes of ∗‐isomorphic tuples of unitaries. We compare this metric to the metric dHR determined by dHR(u,v)=inf∥u′−v′∥:u′,v′∈B(H)d,u′∼uandv′∼v,and we show the inequality dHR(u,v)⩽KdD(u,v)1/2where 1/2 is optimal. When restricting attention to unitary tuples whose matrix range contains a δ‐neighborhood of the origin, then dD(u,v)⩽dδ−1dHR(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. For particular classes of unitary tuples, we find explicit bounds for the dilation constant. For example, if for a real antisymmetric d×d matrix Θ=(θk,ℓ), we let uΘ be the universal unitary tuple (u1,…,ud) satisfying uℓuk=eiθk,ℓukuℓ, then we find that c(uΘ,uΘ′)⩽e14∥Θ−Θ′∥. Combined with the above equivalence of metrics, this allows to recover the result of Haagerup–Rørdam (in the d=2 case) and Gao (in the d⩾2 case) that there exists a map Θ↦U(Θ)∈B(H)d such that U(Θ)∼uΘ and ∥U(Θ)−U(Θ′)∥⩽K∥Θ−Θ′∥1/2. Of special interest are: the universal d‐tuple of noncommuting unitaries u, the d‐tuple of free Haar unitaries uf, and the universal d‐tuple of commuting unitaries u0. We find upper and lower bounds on the dilation constants among these three tuples, and in particular we obtain rather tight (and surprising) bounds 21−1d⩽c(uf,u0)⩽21−12d.From this, we recover Passer's upper bound for the universal unitaries c(u,u0)⩽2d. In the case d=3 we obtain the new lower bound c(u,u0)⩾1.858, which improves on the previously known lower bound c(u,u0)⩾3.