Dilations of unitary tuples

We study the space of all (Formula presented.) -tuples of unitaries (Formula presented.) using dilation theory and matrix ranges. Given two such (Formula presented.) -tuples (Formula presented.) and (Formula presented.) generating, respectively, C*-algebras (Formula presented.) and (Formula presented.), we seek the minimal dilation constant (Formula presented.) such that (Formula presented.), by which we mean that there exist faithful (Formula presented.) -representations (Formula presented.) and (Formula presented.), with (Formula presented.), such that for all (Formula presented.), (Formula presented.) is equal to the compression (Formula presented.) of (Formula presented.) to (Formula presented.). This gives rise to a metric (Formula presented.) on the set of equivalence classes of (Formula presented.) -isomorphic tuples of unitaries. We compare this metric to the metric (Formula presented.) determined by (Formula presented.) and we show the inequality (Formula presented.) where (Formula presented.) is optimal. When restricting attention to unitary tuples whose matrix range contains a (Formula presented.) -neighborhood of the origin, then (Formula presented.), 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 (Formula presented.) matrix (Formula presented.), we let (Formula presented.) be the universal unitary tuple (Formula presented.) satisfying (Formula presented.), then we find that (Formula presented.). Combined with the above equivalence of metrics, this allows to recover the result of Haagerup–Rørdam (in the (Formula presented.) case) and Gao (in the (Formula presented.) case) that there exists a map (Formula presented.) such that (Formula presented.) and (Formula presented.) Of special interest are: the universal (Formula presented.) -tuple of noncommuting unitaries (Formula presented.), the (Formula presented.) -tuple of free Haar unitaries (Formula presented.), and the universal (Formula presented.) -tuple of commuting unitaries (Formula presented.). We find upper and lower bounds on the dilation constants among these three tuples, and in particular we obtain rather tight (and surprising) bounds (Formula presented.) From this, we recover Passer's upper bound for the universal unitaries (Formula presented.). In the case (Formula presented.) we obtain the new lower bound (Formula presented.), which improves on the previously known lower bound (Formula presented.).