Dilation Theory: A Guided Tour

Shalit, Orr

doi:10.1007/978-3-030-51945-2_28

Cited by 12 publications

(1 citation statement)

References 119 publications

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“…Let us write

C_{d}

for the smallest constant

c

which is the solution to the above problem. Problem 1.1 and similar problems have come up in the setting of relaxation of spectrahedral inclusion problems [17], in interpolation problems for completely positive maps and the study of the structure of operator systems [10, 11], and fit in the general paradigm of studying operator theory through dilations [30]. More recently, such problems have turned out to be connected to quantum information theory [6, 7] as well as other aspects of mathematical physics and C*‐algebras [15].…”

Section: Introductionmentioning

confidence: 99%

Dilations of unitary tuples

Gerhold

Pandey

Shalit

et al. 2021

Journal of London Math Soc

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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.

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“…Let us write