C*-Convex Sets and Completely Positive Maps

Magajna, Bojan

doi:10.1007/s00020-016-2291-4

Cited by 9 publications

(8 citation statements)

References 32 publications

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“…In [15], it is generalized to the collection of unital completely positive maps. Further one can see the definition of C * -convexity being modified and studied by [26] in different settings. The notion has also been studied by Farenick et al [16] for positive operator valued measures, which is our main interest in this paper.…”

Section: Main Results On C * -Extreme Pointsmentioning

confidence: 99%

“…The notion has also been studied by Farenick et al [16] for positive operator valued measures, which is our main interest in this paper. Some general references on this topic are [25], [15], [17], [19], [38], [26], and [16]. Definition 3.1.…”

Section: Main Results On C * -Extreme Pointsmentioning

confidence: 99%

“…It should be mentioned here that there are several papers analyzing C * -convexity of UCP maps: [15] [17], [19], [38] and [26], to name a few. In [17] and [38], one can see some abstract characterizations of C * -extreme points of UCP maps.…”

Section: Introductionmentioning

confidence: 99%

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$C^*$-extreme points of positive operator valued measures and unital completely positive maps

Banerjee,

Bhat,

Kumar

2020

Preprint

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We study the quantum (C * ) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, C * -extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic C * -extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital C * -algebra with countable spectrum (in particular C n ) is C * -extreme in the set of unital completely positive maps if and only if it is a unital * -homomorphism.

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Section: Main Results On C * -Extreme Pointsmentioning

confidence: 99%

Section: Main Results On C * -Extreme Pointsmentioning

confidence: 99%

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$C^*$-extreme points of positive operator valued measures and unital completely positive maps

Banerjee,

Bhat,

Kumar

2020

Preprint

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“…As they are categorically dual to operator systems [WW99, Proposition 3.5], they introduce convex-geometric ideas and tools to the study operator systems. A Hahn-Banach separation theorem for matrix convex sets was proved by Effros-Winkler [EW97], the matricial Krein-Milman theorem is due to Webster-Winkler [WW99], and further fundamental results were developed recently by a plethora of authors: representations of convex sets by linear matrix inequalities [HV07,HM12,FNT17], further results on free spectrahedra including the convex Positivstellensatz [HKM12], inclusion problems and dilation theory [HKM13a, DDOSS17, HL+], minimal and maximal matrix convex sets [PSS18] and matrix convex hull approximation [HKM16], (absolute) extreme points of matrix convex sets and free spectrahedra [EHKM18,EH19] and matrix exposed points of free spectrahedra [Kri19], the theory of C * -convexity, i.e., fixed-dimension matrix convexity, via operator systems and the correspondence between C * -extreme points and pure states [FM97,Far04,Mag16], noncommutative Choquet theory [DK+] and the connection between nonunital operator systems and noncommutative (nc) convex sets [KKM+], the correspondence between compact rectangular matrix convex sets and operator spaces [FHL18], etc.…”

Section: Introductionmentioning

confidence: 99%

Facial structure of matrix convex sets

Klep,

Štrekelj

2021

Preprint

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This article introduces and investigates the notions of exposed points and (exposed) faces in the matrix convex setting. Matrix exposed points in finite dimensions were first defined by Kriel in 2019. Here this notion is extended to matrix convex sets in infinite-dimensional vector spaces. Then a connection between matrix exposed points and matrix extreme points is established: a matrix extreme point is ordinary exposed if and only if it is matrix exposed. This leads to a Krein-Milman type result for matrix exposed points that is due to Straszewicz-Klee in classical convexity: a compact matrix convex set is the closed matrix convex hull of its matrix exposed points.Several notions of a fixed-level as well as a multicomponent matrix face and matrix exposed face are introduced to extend the concepts of a matrix extreme point and a matrix exposed point, respectively. Their properties resemble those of (exposed) faces in the classical sense, e.g., it is shown that the C * -extreme (matrix extreme) points of a matrix face (matrix multiface) of a matrix convex set K are matrix extreme in K. As in the case of extreme points, any fixed-level matrix face is ordinary exposed if and only if it is a matrix exposed face. From this it follows that every fixed-level matrix face of a free spectrahedron is matrix exposed. On the other hand, matrix multifaces give rise to the noncommutative counterpart of the classical theory connecting (archimedean) faces of compact convex sets and (archimedean) order ideals of the corresponding function systems.

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“…The reader is referred to e.g. [11], [12], [14], [23] for more information; for a recent operator-valued separation theorem, see [19].…”

Section: Introductionmentioning

confidence: 99%

Order extreme points and solid convex hulls

Oikhberg¹,

Tursi²

2019

Preprint

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We consider the "order" analogues of some classical notions of Banach space geometry: extreme points and convex hulls. A Hahn-Banach type separation result is obtained, which allows us to establish an "order" Krein-Milman Theorem. We show that the unit ball of any infinite dimensional reflexive space contains uncountably many order extreme points, and investigate the set of positive norm-attaining functionals. Finally,we introduce the "solid" version of the Krein-Milman Property, and show it is equivalent to the Radon-Nikodým Property.

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C*-Convex Sets and Completely Positive Maps

Cited by 9 publications

References 32 publications

$C^*$-extreme points of positive operator valued measures and unital completely positive maps

$C^*$-extreme points of positive operator valued measures and unital completely positive maps

Facial structure of matrix convex sets

Order extreme points and solid convex hulls

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