Dilations of unitary tuples

We study the space of all d-tuples unitaries u = ( 1 , … d ) using dilation theory and matrix ranges. Given two such v generating, respectively, C*-algebras A B, we seek minimal constant c that ≺ by which mean there exist faithful ∗-representations π : → B H ρ K with ⊆ for i, i is equal to compression P | H. This gives rise a metric D log max { } on set equivalence classes ∗-isomorphic tuples unitaries. compare this HR determined inf ∥ ′ − ∈ ∼ show inequality ⩽ / 2 where optimal. When restricting attention unitary whose range contains δ-neighborhood origin, then δ so these metrics are equivalent some neighborhood origin. Moreover, Hausdorff distance between ranges tuples. For particular tuples, find explicit bounds constant. example, if real antisymmetric × Θ θ k ℓ let be universal tuple satisfying e 4 . Combined above metrics, allows recover result Haagerup–Rørdam (in case) Gao ⩾ exists map ↦ U Of special interest are: d-tuple noncommuting u, free Haar f commuting 0 upper lower constants among three in obtain rather tight (and surprising) From this, Passer's bound In case 3 new 1.858 improves previously known