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Extreme Points in Spaces of Completely Positive Maps

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Extreme Points in Spaces of Completely Positive Maps C*-EXTREME POINTS IN SPACES OF COMPLETELY POSITIVE MAPS A THESIS SUBMITTEDTO THE FACULTYOF GRADUATESTUDIES AND RESEARCH IN PARTIALFULFILLMENT OF THE REQUIREMENTS FOR THE DEGREEOF DOCTOROF PHILOSOPHY rrÿ MATHEMATICS UNIVERSITYOF REGINA BY Hongding Zhou Regina, Saskatchewan July 1998 @ Copyright 1998: Hongding Zhou uisitions and Acquisitions et "1Bib iogmpti'i Services services bibliographques The author has granted a non- L'airtemr a accordé une licence non exclusive licence aüowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distri-b~eor sel1 reproduire, prêter, distri'buer ou copies of this thesis m rnicroform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film de reproduction sur papier ou sur fomiat électronique. The author retains ownership of the L'auteur conserve la proprieté du copyright in this thesis. Neither the droit d'amqui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otheMrise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Canada Abstract Given a Hilbert space H and an operator system S, the space SH(S)of al1 unital completely positive linear maps cp : S + B(H) is a BW-compact Cu-convexset. An element 9 E SH(S)is said to be a Cg-extreme point of SH(S) if the only way in which <p can be expressed as a proper Ca-convex combination p = t;vlt + t;p2t2? of pl, 92 E SH(S),using invertible coefficients tl, t2 E B(H), is by taking 91:$2 unitarily equivalent to <p. In this thesis, general properties of the Cœ-extremepoints of SH(S)are studied, with particular emphasis placed on direct sums of completely positive maps, Arveson-Hahn-Banach type extensions, maps with irreducible ranges, and the relationship between Ca-extreme points and linear extreme points. In the case of unital C'-algebras A, a characterisation of a Cm-extremepoint of SH(A)is obtained in terms of the data provided by the Stinespring representation, and in the case of finite-dimensional Hilbert space a precise description of the st mcture of such completely positive linear maps is determined. Acknowledgement s Fint, I am much indebted to the Facuity of Graduate Studies and Research for providing funding for my studies in the Department of Mathematics and Statistics of the University of Regina. 1 wish to express my thanks to Dr. D. Farenick who aroused my interest in the theory of cornpletely positive linear maps and guided me in this research. Wit hout his encouragement, patience, advice and support, this thesis could not have been done. Funding from Dr. Farenick's NSERC research grant is also gratefuily acknowledged. 1 express my appreciation to my supervisory cornmittee, Dr. B. Gilligan, Dr. E. Koh, Dr. N. Mobed, and Dr. D. Hanson, and to my extemal examiner, Dr. M. Khoshkom, for their very constructive suggestions for this thesis. I wodd also like to thnnk Dr. M. Tsatsomeros for his advice and assistance in many ways to my research. Lastly, 1 wodd aIso Say thanks to everyone who helped me and my family during t hese t hree years . Contents Abstract Acknowledgements Table of Contents iii 1 rnTRODUCTION 1.1 Outlineof the thesis . , . 1.2 Basic definitions and notations . - . 1.3 Minimal Stinespring represent at ions . 2 Cm-EXTREMEPOINTS 12 2.1 Definit ion of C'extreme points . 12 2.2 The relation between Cm-extremepoints and extreme points . 14 3 A CHARACTERISATION OF THE Cm-EXTREMEPOINTS OF SH(A) 21 3.1 General Hilbert spaces .......................... 21 3.2 For finite-dimensional Hilbert space ................... 28 3.3 Determination of a Cm-extremepoint .................. 29 3.4 The direct sum of disjoint completely positive maps ......... 36 4 DECOMPOSITION OF Ca-EXTREME POINTS INTO DIRECT SUMS 39 4.1 Pure completely positive maps ...................... 39 4.2 The extension problem for an operator system ............. 44 4.3 Finitely reducible completely positive maps ............... 48 4.4 For finite-dimensional Hilbert space ................... 56 5 THE STRUCTURE OF A C'-EXTREME POINT 58 5.1 The matrix expression .......................... 58 5.2 The decomposition into disjoint maps .................. 62 5.3 The stmcture theorem ......................... 10 6 SOME OTHER TOPICS ON Cm-EXTREMEPOINTS 76 6.1 The problem of extreme points ..................... 76 6.2 The problem of extension from an operator systern .......... 78 7 CONCLUSIONS 82 7.1 Major results of the theory in the literature .............. 82 7.2 The major new works of this thesis . - . 84 Bibliogaphy 86 List of Symbols 88 Index 89 Chapter 1 INTRODUCTION In Ca-algebra theory, the concepts of positivity and order are at the core of the subject, and as a consequence, the states (that is, the positivity preserving Iinear functionals) on a Cg-dgebra are of considerable interest and importance. One of the most useful ways in which to analyse a state d on a Cm-algebraA is to consider the state in its Gelfand-Naimark-Segd representation, or decomposition, as a triple (n, f, K), where K is a Hilbert space, T is a representation of A as an algebra of operators acting on K, and is a unit vector that is cyclic for the operator algebra n(A);the representation is 4(a) = (*(a)&(), for a.ll a E A, where (-, -) denotes the inner product on K. In modem operator algebra theory, however, it is sometirnes not suaicient to consider a C'-algebra A on its own; thus it is now a common technique to consider A and ail matrix algebras M,(A) over A together. Of course, the Cm-algebras M,(A) are noncommutative, even if A is commutative, and so the analysis of A in tandem with the induced matrix algebras M,(A) is part of that subject known as "quantized" or "noncommut ative" functional analysis. With regards to positivity, the idea for "quantum States" seems to have originated in the 1955 paper [13] of Stinespring. In this paper he shows that an operator-vdued version of the Gelfand-Naimark-Segd representation of a state on A occurs only wit h linear maps that presenre positivity at the level of all matrix dgebras over A; he cded these linear maps completely positive. Through his two seminal papers of 1969 and 1972, Arveson (11, [2] demonstrated the importance of Stinespring's concept of complete positivity, which has since assumeci a central role in operator dgebra theory. In the second of his famous "Subalgebras of Cm-algebras"papers? Arveson notes that in passing from O: to B(H) (i.e. from compex numbers to operators on complex Hilbert spaces) with regards to positivity, it is natural and appropriate to move fiom numbers to operators with regards to convexity as weU, and over the years this observation led to the study of C'-convexity, which is, essentially, convexity with operator coefficients. The initial concerns of Cm-convexitywere in comection with matricial ranges of operators, as treated by Arveson [2] and Bunce and Salinas [3] early on. This specific case evolved into the general concept of a Ce-convex set (Loebl and Paulsen [il]),and by 1981 on appropriate notion of "extreme point" was put forward (Hoppenwasser, Moore, and Paulsen [8]),which is now cded C'extreme point. In considering the matricial ranges of hyponormal operators, Farenick and Morenz [5] showed that Cm- extreme points rather than Linear extreme points mode1 best the theory of the nu- merical range at the level of the matricial range. They then turned their attention to the Cm-extrernepoints of generalised state spaces of C*-algebras (61, and obtained a Krein-Milman-type theorem and several results concerning the structure of Cm- extreme points. This thesis completes the program in [6],and the principal theorem (Theorem 7.1.5) describes in detail the Stinespring representat ion of a C'-extrema1 generalised state. An even more general notion of noncommutative convexity. which is called ma- trix convexity, has been introduced and studied by Winkler [15] and by Webster and Winkler j14]. Although matnx convexity and C'-convexity do have apparent simi- larities, the differences between the two notions are significant, and the t heories have developed in independent directions. Nevertheless, Example 2.3 of [14] is a direct application of a result from C'-convexity (Proposition 7.1.1 ) , and hirther interaction between the two concepts should not be unexpected. 1.1 Outline ofthe thesis The contents of this work are outlined below. In Chapter 1, the context and the objects under study will be explained. In Chapter 2, C'extreme points are introduced and some of their properties are studied; in particular, we shd address the general open question as to whether every Cg-extreme point is necessarily a linear extreme point. In Chapter 3, the property of being " Cg-extreme" is shown to be equivalent to the existence of certain solutions (in the cornmutant of the Stinespring dilation) to an operator equation, and this equivalence is put to use for some explicit calculations. In Chapter 4, the concept of an irreducible completely positive Linear map is introduced, and with this concept being employed, we study direct surns of generalised states. The structure of C*-extreme points is given in Chapter 5, and the results generalise some of those that will be published in my joint paper [?] with Farenick. Chapter 6 describes certain problems which remain unresolved and Chapter 7 is a summary of the principal results of the thesis. 1.2 Basic definitions and notations Let H be a Hilbert space and B(H) be the set of d bounded Linear operaton on H,and let A be a unital C*-algebra If S is a self-adjoint subspace of A containing 1, then we cdS an operutor system.
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