Dilations of unitary tuples

Abstract

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 scriptA and scriptB, 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 PscriptHρfalse(cvifalse)false|scriptH of ρ(cvi) to scriptH. This gives rise to a metric prefixdnormalDfalse(u,vfalse)=logmaxfalse{c(u,v),c(v,u)false}on the set of equivalence classes of ∗‐isomorphic tuples of unitaries. We compare this metric to the metric dHR determined by prefixdHRfalse(u,vfalse)=inf∥u′−v′∥:u′,v′∈Bfalse(scriptHfalse)d,u′∼uandv′∼v,and we show the inequality prefixdHRfalse(u,vfalse)⩽KprefixdnormalD(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 prefixdnormalDfalse(u,vfalse)⩽dδ−1prefixdHRfalse(u,vfalse), 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×d matrix Θ=(θk,ℓ), we let uΘ be the universal unitary tuple (u1,…,ud) satisfying uℓuk=eiθk,ℓukuℓ, then we find that cfalse(uΘ,unormalΘ′false)⩽e14false∥normalΘ−Θ′false∥. Combined with the above equivalence of metrics, this allows to recover the result of Haagerup–Rørdam (in the d=2 case) and Gao (in the d⩾2 case) that there exists a map Θ↦Ufalse(normalΘfalse)∈B(H)d such that Ufalse(normalΘfalse)∼unormalΘ and false∥U(Θ)−Ufalse(Θ′false)false∥⩽Kfalse∥normalΘ−normalΘ′false∥1/2. Of special interest are: the universal d‐tuple of noncommuting unitaries normalu, the d‐tuple of free Haar unitaries uf, and the universal d‐tuple of commuting unitaries u0. We find upper and lower bounds on the dilation constants among these three tuples, and in particular we obtain rather tight (and surprising) bounds 21−1d⩽cfalse(uf,u0false)⩽21−12d.From this, we recover Passer's upper bound for the universal unitaries cfalse(normalu,u0false)⩽2d. In the case d=3 we obtain the new lower bound c(u,u0)⩾1.858, which improves on the previously known lower bound cfalse(normalu,u0false)⩾3.

Dilations of q-Commuting Unitaries

“…Theorem 1.2 falls within a broader long term project of applying dilation techniques to stability questions, building on and refining our work from [6,7,9]. We now turn to describing our framework.…”

“…In [6], we considered three distance functions on the space U(d). Perhaps the most natural way to measure a distance between u, v ∈ U(d) is given by the following distance function which we call the Haagerup-Rørdam-distance:…”

“…It is worth pointing out that that d HR (u, v) = 0 if and only if u ∼ v (see [6,Proposition 2.3]; alternatively this can be shown directly be constructing a representation into an appropriate direct sum).…”

Dilations of unitary tuples

“…It is not a difficult matter to numerically find this maximum to reasonable precision, and one can then estimate an upper bound on the resulting error in the estimate for c θ . The lower bound c 6π 7 ≥ 1.858, together with the simple MATLAB code and the analysis leading to it can be found in [12].…”

Subhomogeneous Operator Systems and Classification of Operator Systems Generated by $Λ$-Commuting Unitaries

“…, d , from which irreducibility implies K = H, and H is finite dimensional. Since the space is spanned by at most N d vectors, we also have the bound on the dimension.Note that the example in[12, Proposition B.1] shows a 3-tuple of unitaries which are Λ-commuting for λ i,i+1 = e 2πi m…”