Malte Gerhold, Satish Pandey, Baruch Solel and I have recently posted a new paper on the arxiv. Check it out here. Here is the abstract:
We study the space of all -tuples of unitaries using dilation theory and matrix ranges. Given two -tuples and generating C*-algebras and , we seek the minimal dilation constant such that , by which we mean that is a compression of some -isomorphic copy of . This gives rise to a metric
on the set of equivalence classes of -isomorphic tuples of unitaries. We also consider the metric
and we show the inequality
Let be the universal unitary tuple satisfying , where is a real antisymmetric matrix. We find that . From this we recover the result of Haagerup-Rordam and Gao that there exists a map such that and
Of special interest are: the universal -tuple of noncommuting unitaries , the -tuple of free Haar unitaries , and the universal -tuple of commuting unitaries . We obtain the bounds
From this, we recover Passer’s upper bound for the universal unitaries . In the case we obtain the new lower bound improving on the previously known lower bound .