Dilations of unitary tuples

We study the space of all d-tuples of unitaries u=(u1, ... , u(d)) 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 pi: A -> B(H) and rho: B -> B(kappa), with H subset of kappa, such that for all i, pi(u(i)) is equal to the compression P-H rho(cv(i))|H of rho(cv(i)) to H. This gives rise to a metric d(D)(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 d(HR) determined by d(HR)(u,v)=inf parallel to u '-v 'parallel to:u ',v 'is an element of B(H)d,u ' similar to u and v ' similar to v}, and we show the inequality d(HR)(u,v) <= K d(D)(u,v)(1/2) where 1/2 is optimal. When restricting attention to unitary tuples whose matrix range contains a delta-neighborhood of the origin, then d(D)(u,v) <= d delta(-1)dHR(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 x d matrix Theta = (theta(k,l)), we let u(Theta) be the universal unitary tuple (u(1), ..., u(d)) satisfying u(l)u(k) = e(i theta k,l)u(k)u(l), then we find that c(u(Theta),u(Theta ')) <= e(1/4 parallel to Theta-Theta 'parallel to). Combined with the above equivalence of metrics, this allows to recover the result of Haagerup-Rordam (in the d = 2 case) and Gao (in the d >= 2 case) that there exists a map Theta bar right arrow U(Theta)is an element of B(H)(d) such that U(Theta)similar to u(Theta) and parallel to U (Theta) - U(Theta ')parallel to <= K parallel to Theta - Theta 'parallel to(1/2). Of special interest are: the universal d-tuple of noncommuting unitaries u, the d-tuple of free Haar unitaries u(f), and the universal d-tuple of commuting unitaries u(0). We find upper and lower bounds on the dilation constants among these three tuples, and in particular we obtain rather tight (and surprising) bounds 2 root 1-1d <= c(u(f), u(0)) <= 2 root -1/2d. From this, we recover Passer's upper bound for the universal unitaries c(u,u(0)) <= 2d. In the case d=3 we obtain the new lower bound c(u,u(0)) >= 1.858, which improves on the previously known lower bound c(u,u(0)) >= root 3.