An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

Ambrozie, Călin-Grigore; Gheondea, Aurelian

doi:10.1080/03081087.2014.903253

Cited by 5 publications

(23 citation statements)

References 25 publications

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“…The basic result is that there is a UCP map as in Problem 1.1 if and only if W(B) ⊆ W(A), where W(A) and W(B) denote the matrix ranges of A and B, respectively (see Theorem 5.1). This generalizes results of many authors regarding Problem 1.1 [1,2,9,17,18,21,22,24].…”

Section: Introductionsupporting

confidence: 90%

“…Finally, we show that (2) implies (1). Indeed, suppose that X ∈ S, so that by the inclusion (2) there is a normal commuting d-tuple N on some Hilbert space H with σ(N ) ⊂ S 1 so that cN dilates X.…”

Section: Dilations and Matricial Relaxation Of Inclusion Problemsmentioning

confidence: 94%

“…Definition 7.6. Let λ := {λ (m) : 1 ≤ m ≤ k} be a k-tuple of rank one real d × d matrices such that I d ∈ conv{λ (1) , . .…”

Section: D Such Thatmentioning

confidence: 99%

“…The set W max (D d ) therefore has a representation as a free spectrahedron W max (D d ) = D sa A for an appropriate d-tuple of self-adjoint matrices A. Therefore, when S = D sa B is also a free spectrahedron (determined by a tuple of matrices B), the problem of determining whether or not W max (D d ) ⊆ S = D sa B is amenable to the algorithms described in [18] (see also [1,2]). Thus we obtain a relaxation for the "matrix cube problem" of Ben-Tal and Nemirovski [7] which is of widespread interest: to rule out the containment of the cube [−1, 1] d inside a spectrahedron D sa B it suffices to rule out the containment of W max (D d ) inside D sa B .…”

Section: 3mentioning

confidence: 99%

“…. , T d ) ∈ M d n : Ti ≤ 1}, is that {x ∈ R d : |xj | ≤ 1} ⊆ 1 d T1, i.e., that 1 d T1 contains the polar dual of the cube [−1, 1] d = C (d) 1 . Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer.…”

Section: Introductionmentioning

confidence: 99%

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Dilations, Inclusions of Matrix Convex Sets, and Completely Positive Maps

Davidson

Dor-On

Shalit

et al. 2016

Int Math Res Notices

140

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A matrix convex set is a set of the form S = ∪ n≥1 Sn (where each Sn is a set of d-tuples of n×n matrices) that is invariant under UCP maps from Mn to M k and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer.Given two matrix convex sets S = ∪ n≥1 Sn, and T = ∪ n≥1 Tn, we find geometric conditions on S or on T , such that S1 ⊆ T1 implies that S ⊆ CT for some constant C.For instance, under various symmetry conditions on S, we can show that C above can be chosen to equal d, the number of variables. We also show that C = d is sharp for a specific matrix convex set W max (B d ) constructed from the unit ball B d . This led us to find an essentially unique self-dual matrix convex set D, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant C = √ d. For a certain class of polytopes, we obtain a considerable sharpening of such inclusion results involving polar duals. An illustrative example is that a sufficient condition for T to contain the free matrix cube C (d) = ∪n{(T1, . . . , T d ) ∈ M d n : Ti ≤ 1}, is that {x ∈ R d : |xj | ≤ 1} ⊆ 1 d T1, i.e., that 1 d T1 contains the polar dual of the cube [−1, 1] d = C (d) 1 . Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the "number of variables" d. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.2010 Mathematics Subject Classification. 47A13, 47B32, 12Y05, 13P10. Key words and phrases. matrix convex set, free spectrahedra, matrix range, completely positive maps.

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