Dilations of unitary tuples

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.