C*-EXTREME POINTS IN SPACES OF COMPLETELY POSITIVE MAPS
A THESIS
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DOCTOROF PHILOSOPHY
rrÿ MATHEMATICS
UNIVERSITYOF REGINA
BY
Hongding Zhou
Regina, Saskatchewan
July 1998
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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
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. If S is an operator system, a linear map
4 : S + B(H) is cded positive provided that 4 maps positive elements of S to positive elements of B(H). If S is an operator system and 4 : S + B(H) is a iinear map, then we define #,, : M,(S) -t M,(B(H)) by A((b,))= (4(bij)). In particular, q5 is n-positive if & is positive and we cal1 # completely positive if q5 is n-positive for all n. Similarly 4 is completely contractive if & is contractive for al1 n, and 4 is completely isometrïc if #,, is isometnc for ail n. Similar to the concept of positive
linear hinctional, we denote by CP(S,H) the set of ail completely positive maps of
S into B(H), and analogous to the concept of state, we denote by SH(S)the set
of all unital completely positive maps of S into B(H). We cdSH(S) a generalised
state space. In particular, when H has dimension one, the generalised state space is
nothing more than the state space.
Suppose that V is a locally convex topological vector space. By the closed convez - hull of a subset Y of V we meaa the closure coY of the convex hull co Y. An element
xo of a convex set X in V is an eztreme point of X if the only way in which xo can
be expressed as a convex combination xo = (1 - a)xi + ax2, with O < a < 1 and rl
22 in X, is by taking to = xl = 22. As to the existence of extreme points of a closed
convex set, the following fundamental theorem is weil-known.
Theorem 1.2.1 (Krein-Milman) If X is a non-ernpty compact convez set in a locally - convez space V, then X has an eztreme point. Moreover, X = coE, where E is the
set of ail eztreme points of X.
To consider a new kind of extreme point problem in the context of noncommutative convexity, we need to introduce a certain topology. We now describe a certain topol- ogy on the space of all operator-valued linear maps of a subspace of a Cs-algebra. Let
S be a subspace of a C'-algebra A and let H be a Hilbert space. Denote by B(S?H) the set of ail bounded linear maps of S into B(H). Define
Let Br(S,H) be endowed with the (pointwise) weak operator topology. A convex subset u of B(S,H) is open if u i7 B,(S,H) is an open subset of &(S, H), for every r > O. The convex open sets form a base for a locally convex Hausdorff topology on
B(S,H), which is cded the bounded-weak-topdogy. The importance of this topology is revealed by the following proposition [Il.
Proposition 1.2.2 Let S be a closed operator system of a Cœ-algebra A. Then
CP,(S, H) and Sw(S)are compact in the bounded-weak-topology. - By the Proposition 1.2.1, SH (S)has an extreme point. Moreover, Su(S) = CO E, where E is the set of all extreme points of Sk(S). One of the most important facts concerning completely positive maps is that they satisfi a Hahn-Banach extension theorem, which was proved by Arveson [l].
Proposition 1.2.3 Let S be an operator system of a C'-algebra A and let H be a
Hilbert space. Then for eueq cornpletely positive map 4: S -t B(H), there is a completely positive map &: A A B(H) eztending 9.
In this thesis, we shd be concerned with extensions of Cm-extremepoints. Below is a description of how the most basic results work.
6 Proposition 1.2.4 An eztreme point 4 of C P(S,H) can be extended to an extreme point O of CP(A,H).
ProoJ By Proposition 1.2.3, every extreme point 4 of CP(S,H) can be extended to
& of CP(A,H). Suppose & = (1 - a)@* + ad2, with O < a < 1 and ta*, w2 in
C P(A, H). Then their restrictions i&,hlS E CP(S,H), and
Since 4 is an extreme point of CP(S,H),
Thus li>l, & are al30 extensions of 4. Note that any extension of b is unifody bounded because S contains 1, the identity of A. The set of dI the extensions of
4 is convex and closed, hence it is compact in the bounded-weak-topology. Thus there is an extreme point of the set, denoted by a, which is also an extreme point of
CP(A,H). rn
Corollary 1.2.5 Edteme points ofSw(S)can be eztended to eztreme points of SH(A).
Definition 1.2.6 Let S be an operator system and let 4 be a completely positive map of S into B(H). q5 is cded pure if the only linear maps S> : S + B(H) for which both + and 4 - $ are completely positive axe scalar multiples of &. In the sequel, 7 we mite i1, 5 4 for the situation in which both + and 4 - + are completely positive linear maps, if there exists no confusion in the context.
It follows that if q) is pure, then 4 is an extreme point of the set. When H = (C. the converse is tme: extreme points 4 of SH(S)are also pure.
Proposition 1.2.7 A pure map 4 in SH(S)can be eztended to a pure mup d in
SH(~
Proof. Since 4 E SH(S)is pure, 4 is an extreme point of SH(S).By Corollary 1.2.5,
4 can be extended to an extreme point O of &(A). If there is a completely positive map iY : A -t S(H), for which
Vt Is E CP(S,H), and # - @ ls is completely positive . Because Q, in SH(S)is pure,
0 1, is a scalar multiple of #. That is, 9Is= At$. It is clear that any extension of XQ has the form AGI where al is an extension of 4. Thus 9 is a scalar multiple of an extension of t$ : iy = ND1. Since Q1(I) = 1, 0(1)= A. If X = O, then for each x E A,
and 9 = O(= O a). If X = 1, then a similar argument shows that the completely positive map O - is O (and = 1 O). Finally, if O < X < 1, we have
Since , 01E SH(A),and since @ is an extreme point of &(A), O = al. Thus ik = AG1 = A@. Therefore iP is pw. 1.3 Minimal Stinespring representations
It is well known that if A is a unital Cm-algebraand 9 : A + B(H) is a cornpletely
positive map, then there exists a Hilbert space K, a unital '-homornorphism T : A +
B(K),and a bounded Linear operator V : H -t K with Il~(1)ll= llVl12 such that
p(a) = V**(a)V, for every a E A.
This dilation is called the Stinespring representation of y.
If there exists another dilation W'zW, where W : H + K, then let KI be the
closed linear span of *(A)WH and let p be the projection of K onto KI;it is easily
verified that h; reduces s(A), so that p E n(A)',where a( A)' denotes the cornmutant
in B(K)of x(A),and the restriction of T to Kl defines a '-homomorphism al: A -+
B(h;). The projection p gives rise to a bounded linear operator Wl : H -, Ki such
that Wl= p W and p = W;rIWl . Now Kl = [nl(A) Wl Hl, the closure of the set of
all finite linear combinations of elements of nl(A)WIH.Since KI c A?, if Kl # A'
then we Say the tnple (nl,Wl, KI)is smder than the triple (r,V, K),and this defines
a partial order on the set of all Stinespring representations of p. So a question about
this dilation is whether or not there exists a minimal Stinespring representation of 9.
The following proposition gives a certain mswer.
Theorem 1.3.1 There ezists a minimal Stinespring representation of y.
ProoJ Let {(r,, Va,Ka)} be a diain of the set of all Stinespring representations of
9 9,where cr E 1, an index set. Since there is no need to consider the largest one, we
may assume that the largest Ka is Ki.Let p, be the projection of KI onto Ka. Let
Ko = nK, and let po be the projection of KI onto ho. It follows from page 111 in
[9j that Ko is a Hilbert space and Pa -t fi in the strong-operator topology. Since
this is a chain, accordingly rr,(a) converges to rri(a)pofor each a in nom. The limit
also defines a '-homomorphism. Denote it by xo : A ct B(&). Since the unit bal1 of
B(H) is compact in the weak-operator topology and (IVa((= 1, it follows that there
exists a bounded linear operator KJ : H ++ Ko such that Val + &, namely, for each
E H, we have
Lm Val( = b[. Cr'
Moreover, since
IlVa412 = 11~11~7
it follows that Val -t in the strong-operator topology. So we have
~(a)= lim V~R~J(~)V~~= Krro(a)Vo. Q
We have shown that each chah has a lower bound, it foUows from Zorn's lemma that
there exists a minimal Stinespring representation of y. rn
From the above it follows that when we apply a Stinespring representation of 9,
we can choose the one for which (r,V, K) enjoys the property that K = [n(A)VH], and moreover if there exists any other dilation cp = W'rW, then we dso have K = [*(A)WH]. in this sense, we Say that V'rV is a minimal StinespRng representation of y. Suppose that there exist two such minimal Stinespring representations of p:
V;rl&, VI : H -t Kiand V;rraL$, : K H K2.Then it can be verified that there exists a unitary map U: Kt + K2 satidying IIK = Va and Ur,u' = ira Therefore if there exist two such minimal Stinespring representations as
then there exists a unitary operator U E n(A)' such that W = UV.
In many ways, the Stinespring representation is used in the same fashion as the
Gelfand-Naimark-Segd representation of a state, and we study cornpletely positive maps 4 using information about the dilation n. Of interest in this thesis is the following Radon-Nikodym- type Theorem of Arveson in [l].
Proposition 1.3.2 The function h H &, : &(a) = Vœh*(a)Vis an ufine isomor- phism of the partially ordered convez set of operators {h E *(A)' : O 5 h 5 1) onto the order interval [O, q5] of completely positive linear maps u5 : A + B(H).
Also for a general description of the extreme points of SH(A).Arveson established the following result .
Proposition 1.3.3 Let ip E SH(A)have a minimal Stinespring representation p =
V%V. Then is an edreme point of SH(A)if and only if the conditions h E n(A)' and V'hV = O irnply that h = 0. Chapter 2
C*-EXTREME POINTS
2.1 Definition of C*-extreme points
Su(S)is linearly convex. Our purpose bere is to introduce another kind of con- vexity which is called Ca-cmvexity. If pl and pz E SH(S),and ti, t2 € B(H) and t;tl + tjtz = 1, then it is eaw to see that
In this sense we say that SR(S)is C'-conuex. Let p and ~ E SH(S). We say y is unitarily equivalent to $ if there exists a unitary operator u E B( H) such that
9 = uR+u.
Definition 2.1.1 Let S be an operator system. An element 9 E SH(S)is said to be a Cm-eztemepoint of SH(S)if the only way in which 9 caa be expressed as a Cg-convex combination
'P = t; In the above we understand that in the context there should be no danger of confusing 1, as it arises as either the identity of S or the identity of B(H). Proposition 2.1.2 rp is a Cœ-extremepoint of SH(S)if and only if the conditions imply that vi is unitarily equivalent to rp for ail i. Pmof- One direction is straightfonvard. It is just the case in which n = 2. For the converse, suppose that cp is a C'extreme point of SH(S)and that 'P = t;4ih where & E SH(S),ti is invertible for ail i, and C:=, tl ti = 1. Let O be the positive square root of CL2ti ti ; then o is invertible and t;t + o'o = 1. With we get By the assumption this equation implies that & is unitarily equivalent to p and by induction 4; is unitarily equivalent to 9 for al1 i. rn 2.2 The relation between C*-extreme points and extreme points Loebl and Paulsen studied the concept of Cs-convexity for matrices, [Il]. Follow- ing these ideas, Fitrenick and Morenz in [6]take the condition described in Proposition 2.1.2 as the definition of C*-extreme points in Sw(A).Directly from the definition, the est problem is to determine how the extreme points and Cm-extremepoints are related. Let C(H)be the set of ail compact operaton in B(H),and let C(H)+C be the algebra {XI + K;A E C, K E C(H)). Considering that the presence of the unitary equivdence clause does not mean that a Cm-extremepoint is autornaticdy a linear extrerne point, Farenick and Morenz proved that if the range of 4 is contained in C(H)+G, in particdar, if H bas finite dimension. then a C'sxtreme point of SH(.4) is also a linear extreme point. We give here a new and self-contained proof of this result. We need the following lemma, which is a generalization of an assertion, fkst given by Loebl in [IO],that every compact operator is an extreme point in the convex hulI of its unitary orbit. Lernma 2.2.1 Let T be a compact self-adjoint operator acting on a Hilbed space H. If t;tl, t;t2 an nonzero scalars and t;t 1 + t;t2 = 1, and if then the coeficients tl, t2 cornmute tmth T. Proof. Consider the equation We can assume by the spectral theorem that T = XjEj, where Ej are mutudy orthogonal projections from H onto Hj and C E, = 1. E XI is the eigenvaiue with largest absolu te value, t hen And without loss of generdity, we assume that XI > 0. Let z E Hl be a unit vector. Then XI = C AjIIEjtlzl12 + 1hIIEjf2rl12 - Because and because Aj < Al for j > 1, we conclude that We get and so (1 - El)tlEl = 0; The above shows that the range of tl El is within Hl. If tl El [ = O, then t;tl El ( = 0. But t;tl is a nonzero scalar, thus El = O- So tlEl is injective from Hl to Hl and since Hl is a finite dimensional space, Elti El is onto Hl. Since tit is a scalar, (1 - EL)t;t1 El = O, and so (1 - E1)t;El EltlEl =O; But EltlEl maps onto Hl and the range of Eltl(l - El) is contained in HI,and so we get This proves that and therefore we have By induction, ttEj = Ejtl, for dl j. Clearly this is also true for t2, and this completes the proof of the lemma. W Theorem 2.2.2 [6] Let H be a Hilbert space and let the range p(S) c {C(H)+Cl. If 9 is a C'-eztreme point of SH(S),then ;3 is also an eztreme point of SH(S).In purticular, when H is a finite-dimensional space, a Cm-eztremepoint of SH(S)is also an eztreme point of Su(S). Proof. If ~f= ((1 - + with O < a < 1 and 111, t,bz in SH(S). then The fact that y is a Cm-extremepoint of SH(S)implies that @; = u;9ui, i = 13. where ui is a unitary operator. Let T E p(S) be self-adjoint, and consider Since T E {C(H)+ (C), and since t ,, t2 satisfy the conditions of Lemma 2.2.1, it foliows that tl cornmutes with T, and therefore t commutes with p(S). Since fi is scalar, ul commutes with v(S),and we have The same is true for &. It follows from the definition that p is an extreme point of &(SI* rn Corollary 2.2.3 Let H be a Hilbert space and suppose that C(H)+ (C is dense in v(S)in the weak-operator topology. If q is a C'-extnme point of SH(S),then i, is also an eztrerne point of Su(S). PruoJ Using the proof of Theorem 2.2.2, it follows, from the assumption +(S) n {C(H)+ (C) is dense in p(S) in the weak-operator topoiogy, that for any operator T in q(S),there exists a sequence of T, E y(S) n {C(H)+C) such that T,, -r T in the weak-operator topology. Aence 9 is an extreme point of SH(S). rn Theorem 2.2.4 Let H be a Hilbert space and let 9 be a C*-eztreme point of SH(S). If y(S) is an algebra, then 9 is an ettreme point of SH(S). 18 Pmof. If rp = (1 - a)$l+ d2,"th O < a < 1 and $1, $2 in SH(S),we need to prove that +1 = ip. Since y is a C'-extreme point of SH(S),we have (I>~= u' 9 u, $2 = u* p v, where u. v are unitary operaton. Suppose that T = p(a) for some a E S, then we get T = (1 -cr)u*Tu+<~v~Tu, We can assume that T is self-adjoint. Since ip(S) is an algebra, dl polynomids in T Le in EA= (1 -a)u'EAu+av'Exu. Multiplying by (1 - EA)from both sides, we have EAu(l-EA)=O. In applying this reasoning to 1 - Ex, we get (1- EA)uEA=O, and hence EAU = uEx. By the spectral resolution of T, we have Tu = UT.Thus u cornmutes with ail of p(S), and so p is an extreme point of SH(S). H Chapter 3 A CHARACTERISATION OF THE C*-EXTREME POINTS OF &(A) In this chapter, we study some characteristic properties of a Cg-extreme point of SH(A)* 3.1 General Hilbert spaces Definition 3.1.1 Let -4 be a unital Cm-algebmand let 9 : A + B(H) be a unital compktely positive map Wth a minimal Stinespring representation 9 = VanV. -4 positive operator h E n(A)' is said to be y-znvertible if V'hV is invertible in B(H). Theorem 3.1.2 y is a Cm-eztnmepoint of SH(A)if and only if for each 9-invertible h E r(A)' there ezists a unitary operator U E ?r(A)' and an inuertible X E B(H) such that lrhi~= VX. Proof. Suppose that p is a C'extreme point of Sw(4)and h is pinvertible. Then let M be a sufnciently large positive scalar for which h hl = - and h2 = 1 -hl M are both pinvertible. Let tl be the unique positive solution of t;tl = VmhlV:since V'hV is invertible in B(H), it follows that tl is invertible in B(H). Define then Observe t hat and y is a C'sxtreme point of SH(A).Accordingly u'(~;)-~v*T~~VtrLu axe two minimai Stinespring representations of p, and hence they are unitarily equivalent. Therefore there exists a unitary operator Li E n( A)' such that 1 Uhf Vt;'u = V; 1 where hf is the unique positive square root of hl. Therefore we get lihi~= VX, where X = uutlais invertible in B(H). Conversely, assume that for each pinvertibb h there exists a unitary U E n(A)' and an invertible X E B(H)such that ~hfv = VX. Suppose that p is expressed as a Cœ-convexcombination of generalised states 91, 92: where t;', t;' exist in B(H), and t;tl + t& = 1. We need to prove that pl, 92 are unitarily equivalent to p. Since t;pltl 5 yY = V'*V, by Proposition 1.3.2, there exists a positive hl E n(A)' such that t; Since t;' exists in B(H) and p(1) = 23 X E B(H) such that ~lhfv = VX. Therefore Since p(1) = ~~(1)= 1 and X is invertible in B(H), it follows that Xt;' is a unitw element in B( H). Clearly the same is true for 9,. We have several immediate applications of Theorem 3.1.2, the first of which is Proposition 1.2 (3) of [6],but here we give a new proof. Corollary 3.1.3 Let 9 E SH(A) with a minimal Stinespeng npresentation p = V'RV. If the subspace VH is invariant for the cornmutant R(A)', then y is a Cm- extreme point of SH(A) . Proof. Since the subspace VH is invariant for the comrnutant a( A)', it follows that for any T E K(A)'?TVH C VH. Let p = VV' be the projection of K onto VH; so Tp = pTp. Since *(A)' is self-adjoint, T' E n(A)', thus T'p = pT'p Therefore PT = pTp = Tp. Since p = VV' commutes with *(A)', if h is a p-invertible element, then Thus hi^ is invertible. Denoting v*~+vby X, it follows that hf~= hf(VV')V = v(v'~~v)= VX. By Theorern 3.1.2, cp is a Cm-extremepoint of SH(A). Remark 3.1.4 The condition that the subspace VH be invariant for the cornmutant r(A)' is very strong. For one thing, this condition is not necessary for rp to be a C'- extreme point of Su(A). For another, with this condition, it is not hard to prove that a Ce-extreme point of SH(A)is also an extreme point. We indicate now how to do this. If h E a(A)' and V'hV = O, then we can assume that h is a self-adjoint operator. Then V'hV - V'hV = O. That is, VœhphV= O. Since p = VV' commutes with *(A)', we get V'h2V = O. But this impiies that Ihl V = O. Since n(A) is a Cm-algebra, V*- Ihl T(Y*Z) Ihl V = O, Since [r(A)Vq= K, it follows that h2 = O, thus h = O. According to Proposition 1.3.3, Q is an extreme point of SH(A).Later on, we will see this condition again in Proposition 3.4.1. Recd the definition of pure maps: a completely positive map s is a pure map if for each completely positive map 11 for which w 5 O. there exists a scdar X such that 11> = A0 = ~k#d.With the following Theorern 3.1.5. we can view a Cm-extreme point of SH(A)as an extension of the concept of a pure map to the noncommutative case. Theorem 3.1.5 Let y E SH(A).y is a Cm-eztremepoint if and only if the condition @ 5 9 irnplies S, = X'yX for some operator X. tohere ut least one such X is inrertible zoheneuer +(1) is invettible. Proof. Suppose that 9 is a Cm-extremepoint of Sw(d)with a minimal Stinespring representation y = V'TV. If + 5 y, then Proposition 1.3.2 If +(l)is invertible, then h is pinvertible. hccordingly by Theorem 3.1.2. there exists an invertible X such that tl> = X'yX. So it suflices to consider the case that o(1)is not invertible. Define Then hAis ginvertible. By Theorem 3.1.2, there exists a unitary UAE B(K)n s(A)' and a linear operator XAE B(H) such that 1 Since h f -t hi uniformly in nom and since the bounded bdls of B( K) and B( H) are weak-operator compact, we get UA+ U and XA+ X in the weak-operator topology. as X + O. Thus it follows that ~h f v = VX. It is readily seen that Lr E ;r(ll)'is uni tary, accordingly Conversely, suppose that the condition IO 9 implies that tu = X'gX. such that when $(l)is invertible, then at least one such X is invertible. We need to prove that 9 is a C'sxtreme point of SH(A).In fm, suppose that where t;', tg1 exist in B(H),and t;tl + t;tz = 1. Then t&tI 5 p. By assumption, where X is invertible, since tl is invertible and t;pl(l)tlis invertible. It follows that Xt ;' is unitary. This completes the proof. m 3.2 For finit e-dimensional Hilbert space Corollary 3.2.1 Suppose that H is a finite-dimensional Hilbert space. Then r, is a Cg-eztreme point of SH(A)if and only if for each h such that h 2 O, h E ?r(A)', there eezLsts a unitary opemtor U E n( A)' and a heur operator X E B( H) such that UhV = VX. PmoJ If h 2 O and h E z(A)', then h2 2 O, h2 E ir(A)'. If 9 is a Cm-extrernepoint of SH (A), then let h>.= h2 + XI, X > 0. Then hA is a pinvertible element. By Theorem 3.1.2. there exists a unitary oprator UAE s(A)' and a linear operator XAE B(H) such that I Since h f -+ h uniforrnly in nom and the bounded bah of B(K) and B( H) are weak-operator compact, we get UhV = VX as X -t O. For the converse, if for each positive h such that h E a( A)', there exists a unitary U E r(A)' and a linear operator X E B(H) such that UhV = VX, then hf E *(A)', and also there exists a unitary U E r(A)' and a Linear operator .Y E B(H) such that ~hfV = VX. Thus we need to prove that if h is pinvertible, then X is invertibie. In fact, for a pinvertible element h, since V'V = 1, we get Since V'hV is invertible in B( H) and H is a finite dimensional Hilbert space. X is invertible. By Theorem 3.1.2, 9 is a C'extreme point of SH(A). 3.3 Determination of a C*-extreme point Let us consider some examples of completely positive linear maps and see how the results of this chapter can be used to check whether a completely positive Lnear map is a C'-extreme point and/or an extreme point. Example 3.3.1 p : M2 -+ M2 defineci by This is clearly a unital cornpletely positive linear map. But since it follows that it is not an extreme point nor a C'extreme point. In the following, we show how to prove this by Corollary 3.2.1. A minimal Stinespring representation of p cm be given by or in a short fom where Let For any h E a(A)', U E a(A)' we have where hij,Uij are scdars. Suppose that X E M2is given by If for each positive operator h E x(A)', there exists a unitary Lr E n(4)' and a linear operator X E B( H)such that LJhV = VX,then thus we get where hl and h2 are row vecton of h, and Ul and U2 are row vectors of CI. Note that this expression is correct because h is a positive definite matrix. Since hl, h2 can vary arbitrarily, we can take hl = h2 # O in particular; however, since C' is unitary. the above conditions imply that hl must be O. Thus there is no solution U. By Corollary 3.2.1. 9 is not a Cm-exttemepoint of &(Ma). Note that 9 is not an extreme point of S&I (lb).In fact, if h E *(A)',then since and since TVi =v;&=O, it follows that to get V'hV = O we only need Therefore by Proposition 1.3.3, ;p is not an extreme point of & (Mz). Exarnple 3.3.2 The "mixing" state q5 E &4(M3) given by is not a C'extreme point of &4 ( M3). To prove this, let n : 1W3 + B(C6) be given by a(x) = x x and let V :C4 + C6 be the isor.net ry Then 4 = V'rV is a minimal Stinespring representation of 6. Observe that the cornmutant K(A)' is '-isomorphic to M2;concretely, let For any h E *(A>',U E rr(A)' we have X E B(C4) is given by Suppose that for each h such that h 2 O, h E s(A)',there exists a unituy U E *(A)' and a linear operator X E B(H) such that UhV = VX. Then let we have however, and Thus we get where hl and hz are row vectors of h, and & and Clz are row vectors of U. Since hl, h2 can vaq arbitrarily, we can take hl = h2 # O in particdu; however, since Li is unitary, the above conditions imply that hl must be O. Therefore there is no solution U. By Corollary 3.2.1, 4 is not a Cm-extremepoint of &(&)- But cp is an extreme point of &4M3). In fact, if h E *(A)', then since and since and it follows that to get V'hV = O we must have each hij= 0 . Therefore by Proposition 1.3.3?c$ is an extreme point of sr(M3). Example 3.3.3 Let A = M2 $ Ml and 4: A + Ma be given by 1 , where z= This is a C*-extreme point of &p (A).In fact, we can write and where K = C3. It is easy to see that [AvH] = K, and hence v'xv is a minimal Stinespring representation of #. Observe that because any h E A' has the form Since UV* cornmutes with A', it follows from Corollary 3.1.3 that 9 is a Cm-extreme point of &p (A). More careful observation shows t hat From this decomposition it follows that 4 is a direct sum of disjoint pure maps, which are considered in the upcoming section. 3.4 The direct sum of disjoint completely positive maps In this section, we show that the direct sum of disjoint pure completely positive maps always produces Cm-extremepoints. 36 Let {s)be a set of representations of Ce-algebra A on Hilbert spaces {Ki).By the expression {ni) are disjoint we mean that any two representations xi and ?r, have no equivalent subrepresentations. We Say a set of unit al cornpletely positive maps #i : A A B( Hi) is a set of disjoint maps if the set of representations {ri}is disjoint, where #i = usœ ri^, is a minimal Stinespring representation of Qifor each i. Let us have a look at the relation between disjointness and the diagonality of the coxnmutant [ni $ %](A)'. If ?F* and rj are not disjoint, then there exist projections p E ri(A)' and q E q(A)' such that pn;p = uœvjqu, where u is unitary. Since pip = prri and qrjq = Tjq and u is unitary, it follows that uprr, = r,qu. Thus there exists a nondiagond 2 x 2 operator matrix h E [ri@ r,,] ( 4)'. On the other hand, if there exists a nondiagond h E (9 8 rj](A)'?then h;,,*ihl.~= h;,&l.2rj - It can be verified that h;p2hl,2 rj(A)'. Accordingly these two representations of the Cm-algebraA have equivalent subrepresentations. The above shows that ni and nj are disjoint if and only if any h E [q @ q](A)' is diagonal 2 x 2 operator matriz . When {ri)are d irreducible, group together aiI quivalent representations of the set 37 {ri),after absorbing the unitary factor to vi, any ni and ir, aie either &the same* or disjoint. Then h E [C$q](A)' has a very simple matrix form for which any entry in the cross position (1, j) is a scalar if ri = r;,, or always O if xi and rr, are disjoint. Proposition 3.4.1 If a set of unital pure maps di : A -r B(Hi)are disjoint. then the direct surn of a11 4 is a Cm-eztremepoint of &(A). where H = C 3Hi. Proof. Since each & has a minimal Stinespnng representation v,txivi. where ui : Hi -t Ki,the direct sum, 4 = $4; has a Stinespring representation V'rl;. where ~r= C @?ri, V = C @vi, K = aKj. Since {xi) is disjoint. ri)' = S~i(rl)'and [r(A)VH]= K. Therefore VarV is a minimal Stinespring representation of o. Since it is a direct S-, p = VV' = C @vivf. AS is irreducible, ri(-4)'consists of scdars in $(Ki). Therefore p coinmutes with a(A)', and thus d is a Co-extreme point of SH(A)by Corollary 3.1.3. I Chapter 4 DECOMPOSITION OF C*-EXTREME POINTS INTO DIRECT SUMS In this chapter, we are concerned with the decompositioc of a C'-extreme point of SH(S)into a direct sum of lower rank Cm-extremepoints. In certain cases. these lower rank C*-extreme points are pure completely positive maps. Hence we begin with those Cm-extremepoints whose range is irreducible. 4.1 Pure completely positive maps The range of a completely positive map is a family ïZ of operators in B(H). The family R of bomded Iinear operators on H is said to be zmdueib[e if 'R bas only trivial (closed) invariant subspaces. If 72 is a self-adjoint family of bounded linear operaton acting on H, then 7Z acts irreducibly on H if and only if W, the cornmutant of the farnily, consists of scalars, or equivalently, 72'' = B(H). If the family R is not irreducible, then there exists a nontrivial projection in the commutant of R. Thus when we say a completely positive map Suppse that y is a unital completely positive rnap of A into B( H). Let p = V**V be a minimal Stinespring representation of y, where V : H -t K is a Iinear map. Then it is known that rp is a pure completely pmitive map if and only if r(A)is irreducible. Let q denote the quotient map of B(H) onto the Calkin dgebra B(H)/C(H). Let C*(S) denote the smdest CR-dgebra that contains S. Suppose t hat S is an irreducible subspace of B(H) such that C*(S)contains C(H). Arveson's boundary theorem, Theorem 2.1.1 of [2], asserts that the identity map restricted to S has a unique completely positive extension to B(H) if and only if q is not completely isometric on S + S'. This rather difficdt theorem is given a simple proof in [1]. For ou use let us state here the Remark 2 of the Boundary Theorem, pz88 of [2], explicit ly. Lemma 4.1.1 If + : B(H) + B(H) is a completely positiue map of norm 1 whose set + of fied points is irreducible and is such that the quotient map q : B(H) + B(H)/C(H) is not completely isometric on 3,then 3 is a C'-algebra . Theorem 4.1.2 Let H be a Hilbert space and let 9 be a CR-eztremepoint of SH(A). 40 If 9 is irreducible and the quotient map q : B(H) + B(H)/C(H) is not completely isometric on the range of 9,then p is pure. Pmof. Let p = V'rV be a minimal Stinespring representation of 9. If n(A) is ineducible, then cp is pure. So in the following we assume that r(A)is not ineducible, and we will show that this assumption leads to a contradiction. Because *(A) is reducible, there exists a proper projection p E n(A)' such that *(a) = pr(a)p + (I - p)a(a)(I- p) for ail a E A. Denote the range of p by KI ând the range of 1 - p by K2.Then Kt and K2 are mutudy orthogonal closed subspaces of K. Clearly r(A)IKi is a Cm-subdgebraof B(Kl),and r(A)IK2is a C*-subalgebra of B(K2).n(A) has invariant subspaces h;, K2.Set n~ = plK, r2 = (1 - ~)~l~~ Vl=pV:H*Kl, h=(I-p)V: Hct Kz. Since is a Cm-extremepoint of SH(A),it follows from Theorem 3.1.5 that there exist unitary operators Ut, CI2 in r(A)', and operators XI, X2in B(H) such that Therefore we have with X;Xl + X;X2 = 1. Define $J by $J(z)= X;zXl+X;zXz. (4.1.1) Then $.J is a unit al completely positive rnap of B(H) into itself. Since the fixed points set of + contains the range of 9 and the range of p is irreducible, the set of fixed points of S> is irreducible. By assumption the quotient rnap q : B(H) -+ B(H)/C( H) is not completely isometric on the range of 9. Thus q is not completeiy isometric on the fixecl points set of $. Thus by Lemma 4.1.1, the fixed points set of ?/I is a C*-algebra and thus by the double cornmutant theorem, the strong-operator topology closure of this Cm-algebrais the hl1 B(H). This means that the equality extends to d z E B(H). Analogous to the proof of Theorern 2.2.4, we get Xi, X2are in the cornmutant of B(H). Thus Xl, X2 are scalars. If Xi is nonzero and Xz is zero, then If Xl and Xz are both nonzero, then since we get In both cases, we find that cp has a smder Stinespring representation. This con- tradicts the assumption that Hence there exists no proper projection in r(A)',and this completes the proof of the t harem. rn We now apply Theorem 4.1.2 to obtain new information regarding Proposition 1.1 of 161, which asserts that if p is a C*-extreme point of SH(A),and for some x E 4, p(x) E C(H)+C and is irreducible, then cp is an extreme point of SH(A).We prove here that this extreme point is actudy a pure point of SH(A). In fact, if for some z f A, T = ~(x)= n + A1 and is irreducible, where rc is a nonzero compact operator and A is a scalor, then the range of p is irreducible. Moreover since I E A, substituting x by x - XI, so that n = p(x - AI) is a nonzero compact operator and lies in the range of 9, yields O = Ilq(~)ll< Iltcll. This shows that q is not completely isometric on the range of 9. By Theorem 4.1.2, 9 is a pure point of SH(A). Note again that if the range of 9 contains a nonzero compact operator, then the quotient map q : B(H) + B(H)/C(H)is not completely isometric on the fixed point set of 11 defined by (4.1.1), because q annihilates the compact operators. Conse quently, if H is a finite dimensional Hilbert space and is a C*-eztreme point of SH(A)whose range is irreducible, then p is pure. 4.2 The extension problem for an operator system Let S be an operator system and A a C'-algebra that contains S. In this section, we consider the problem of extending a C'-extreme point of SH(S)to a Cg-extreme point of SH(A). Theorem 4.2.1 Let H be a Hilbert space and let S be an operator system contained in a C'-algebra A. Suppose that cp is a C'-eztieme point of SH(S).If 9 is irreducible and the quotient map q is not completely isometric on the range of 9,then 9 can be edended to a Cm-eztnmepoint of Sw(A). Proof. The set of all the extensions of 9 is a convex and closed set; hence is compact in the bounded-weak-topology. Accordingly there is an extreme point of the set. Denote the extreme point by 6. Suppose that can be expressed as a C'-convex combination with $i, 3b2 02 SH(A),where t;', ty' exist, and t;tt + t;t2 = 1. Then their restrictions satisfy 'P = 1; ~il,tl+ t; 11121St2 So we get a Cs-convex combination in Sa(S). Since cp is a C'extreme point of SH(S),there exist unitary operators ul, u2 in B(H) such that hls = u;pu1 , &Is = u;v2 - So we get 'P = (ultl)*v(ultl)+ (u2t2)"P(uztz)- By assumption, the range of 9 is irreducible and the quotient map q is not completely isometric on the range of 2 = (ultl)=(ulh)+ (u2t2)*z(u*t,) can be extended from z E y(S) to all B(H). Accordingly, ultl, u2tz are scdars. Moreover, $1 and i12 are respectively extensions of u;9ul and u;ipuz. So (U;)-'P~UT' and (u;)-l7,!9u;' are respectively extensions of 9. Since @ = t;$ltl + t;&t2, we have = (~ltl)*((~;)-'él~;~)(~ltl)+ (~2t2)*((~;)-~+2~;~)(~2tZ) - Because ultl, u2t2 are scdars and since 6 is an extreme point of the set of al1 extensions of 9, it follows that ~;-I,b~u;~= 0, u;'&u;l = e, and thus we get ?,6, = u;9u1, 11, = u;au2. By definition, @ is a C'-extreme point of SH(A). 45 Corollary 4.2.2 If Pmof. By the above Theorem 4.2.1, p can be extended to a C'-extreme point of SH(A),denoted by 8. Since the range of is irreducible and the quotient map q is not completely isometric on the range of 9,it follows that the range of O is irreducible and The quotient map q is not completely isometric on the range of cP, Accordingly by Theorem 4.1.2 O is a pure element of SH(A). rn With the following example, we can see that 4 being a Cm-extremepoint of SH(S) is a necessary condition for the concIusion of Theorem 4.2.1. Example 4.2.3 Let H be a Hilbert space and let S c C(II) be the operator system defined by S = {p + : p, q polynornids ), and where C(n)denotes the CR-algebra of continuous functions from the unit circle II to C. If T is a compact operator in B(H),then the map t/> : S + B(H) defined by is completely positive if IlTl1 $ 1. Also the map 4 : S -t B(H)defined by is completely positive if the numerical radius w (T) 5 1. It is known that if IlTl1 5 1, then w(T) 5 1. So both + and 4 are completely positive when IlTl1 5 1. Since S is uniformly dense in A = C(II), there exists a 46 unique positive extension for both $ and #. Denote them by cU and @ respectively. From the Fejer-Riesz theorem, page 11 of [12], it foI1ows that any positive element r in S can expressed as IpI2, where p is a polynomial. Accordingly if an element p + Q in S is positive, then p(0) + Q(0) in S is positive. Thus for any positive element p+ tj, Since S is dense, the above inequality implies that 9 and 8 - ik' are completely positive maps, that is, Q 5 é. Suppose that T is irreducible and compact. Then the range of p is irreducible and the quotient map q is not completely isometric on the range of 9. Combining these two facts, frorn the Corollary 4.2.2, it follows that 4 is not a Cm-extremepoint of SH(S).In fact, if 4 is a Cm-extremepoint of SH(S),then by Corollary 4.2.2, cb can be extended uniquely to a pure element of SH(A),then it is pure on S. Accordingly where X is ascalar O 5 X 9 1, andp+q~S. IfX = 1, then This is impossible. If A < 1, then This is also impossible. This shows # is not a C'extrerne point of SH(S). 47 4.3 Finit ely reducible completeky positive maps In this section, we are concerned with those completely positive maps whose range is a finite direct sum of irreducible sets. Fint we need the following result. Lemma 4.3.1 Suppose that $, 9 are positive Iinear maps of S into B(H) and T is a positive operator wch that $(a) = Tv(a)Vu E S. Then T cornmutes with 9. Proof. Since $ = TV,for any self-adjoint element z E S, we have ?m= Tp(4- By taking ' on both sides of the equation we get Because +, p are positive linear maps and T is a positive operator. then Because any element z E S has a decomposition z = z + iy, where I. y E S are self-adjoint, we get So T cornmutes with 9. The foUowing theorem is the key technique for Our purpose. Theorern 4.3.2 Let H = HI $ Hz be a direct sum of Hilbert spaces and let y be a C'-tztreme point O/ SH(S).If 9 = rple 92 toith vj : S -+ B(Hj),j = 172, and 92 Ls imducible, then Proof. Suppose that there exist $2 E Ciw, (S) such that where O;', 0;' exist in B(Hl), and a;al + o;oz = la. Let where rpz = O1 = B2, and sl = sz = hla2.Then we have a Cm-convexcombinat ion in SH(S) = t; 01 ti + t; 92t2, where al = +i $4, 02 = $2 @ û2, and tl = ul @ sl, t2 = 02 8 SZ. Clearly, t;', t;' exist in B(H), and t;tl + t;tz = 1. Since we get Yote that and therefore we get Since the range of pz is irreducible, u2.2 is a scalar. If uzs # O, then ul.1 is invertible. In kt,if there exists a nonzero vector ,f E Hl such that ul,lc = O, then since But u;,, is a nonzero scalar, and so However, since we get = O. This contradiction shows that ul.1 must be injective. Also since uu' = lH,it follows that Because u2.2 is a nonzero scalar, uiv1is injective, and therefore the range of ut-1 is dense in Hl.So ut: is densely dehed. If u,: is not bounded, then there exists a 51 sequeuce of unit vectors t,, E Hl such that ulv& + O. Since u;,~is bounded and 4.a~i.i+ ~;,2~2,1= 0 9 ~;~~2.1&+ 0 However because u;,~is a nonzero scalar, it follows that ~~,~cn+ O. Combining this fact and the fact tbat we get This contradiction shows that u,: must be bounded. So u1.l is invertible. Applying (4.3.2), we have and multiplying respectively, we get Hence we obtain Substituting back and we get (11)l = ~;~~l~l.l *ll= u;.,vl~i.l Since ulvl is invertible, by Lemma 4.3.1, we get ~1 = ~~;*~~l*~~-~~;,l~l~l.~~~;*~~l.l~-~ This proves that if uzV2# O, then ?,bl is unitady equivalent to 91. So in the remainder of the proof we cmassume ~2,~= O. But if this OCCUTS. then from (4.3.2), = u;,~~~uIJand Since we get Sirnilady we can prove that ww- = IHi So in both cases, ?,bi is unitarily equivalent to 91. This completes the proof. A major result of this chapter is the following theorem : Theorem 4.3.3 Let 9 be a C'-eztreme point of SH(S),where (o = ql @ - $ pn is a finite direct sum, for which the range of each vj : S -t $(Hi)is irreducible. Then each 9; is a Cm-eztremepoint of SH,(S). 54 Proof. As The folIowing result is a consequence of Theorem 4.3.3 adCoroiiary 4.2.2. Corollary 4.3.4 Let H be a Hilbert space and let 9 be a Cm-eztremepoint of SH(S). If p = 91 -$~p, is a finite direct surn, for which the range of each yj : S -P B(Hj) is irreducible and each qj contains a nonzen, compact operator in its range, then each vj is the restriction of a pure map o/ SH,(A). Proof. If there exists only one summand in the direct sum, namely, if the range of p is irreducible, then by Corollar; 1.2.2, 9 can be extended to a pure map of SH(A). So we assume that the range of p is not imeducible. By assumption 9 = pl @ - - p, is a finite direct sum, for which the range of each pj : S -+ B(Hj)is irreducible, and H = Hl$ - $ H,. By Theorem 4.3.3, each pj is a C'extreme point of Sw,(S). Then by Corollary 4.2.2, each pj can be extended to a pure rnap of SHI(A), which is denoted by aj. It is clear that iP = 8.- - $ b, lies in SH(A). U 4.4 For finit e-dimensional Hilbert space The above result is of particular interest in the fite dimensional case: Farenick and Morenz first proved in [6] that if H has finite dimension and if rp is a C*- eztreme point of SH(A),then They dso gave an example to show that this theorem does not hold if H has iniînite dimension. The above Corollary 4.3.4 is a generalisation of their theorem in two points. First, it points out that when H has infinite dimension. the decomposition of the given completely positive map into a finite direct sum of completely positive maps whose ranges are irreducible and contain a compact operator is an important condition. Secondly S can be an operator system. If H has finite dimension, the above Corollary 4.3.4 can be restated as: if9 is a C'-eztreme point ofSH(S),then 3 is the restriction on S of a finite direct sum of pure elements of SH,(A). Note that this assertion wiU be a straightforward result of the above Farenick-Morenz Theorern. if one can prove that a C'-extrerne point of the space of unital completely positive maps S -t B(H) can be extended to Cm-extremepoint among the unital completely positive maps A + B(H). However, unlike CoroUary 1-25and Theorem 4.2.1, generdy we do not know whether it is tme or not that Cœ-extremepoints of Sff(S)can extend to Cm-extremepoints of SH(A). Later on in Chapter 6 we wiil give a theorem that represents a step towards this god. For the record, we state and give our new proof of the Farenick-iMorenz decom- position, Thm2.l of (61. Theorem 4.4.1 If H 6asJinite dimension and $9 is a C'-edrerne point of &(A), then 9 is a direct sum of pure completely positive maps. Proof. The subspace p(A) is self-adjoint and finite-dimensional, and t herefore t here exists a minimal projection p E y(A)'. Thus if Hl = p(H), then the map 9, : -4 + B(Hl) given by yl(a) = ~(a)l,is a direct summand of 9; hence rp = pl $ zb, for some t,b E SHerrt(A).Because the projection p E ~(-4)'is minimal, the range of pl is irreducible in B(&) and su, by Theorem 4.3.2. pi is a pure completely positive map. NOW by Theorem 4.3.2, 11 is a Ca-extrernepoint, and therefore we cmrepeat for 11> the argument we have made above for p. Because H has finite dimension. this iteration will stop after finitely many steps, with the end result being that where each summand vj is a pure completely positive map. Chapter 5 THE STRUCTURE OF A C*-EXTREME POINT 5.1 The mat& expression Theorem 4.3.2 points out that a hi& rank C'-extreme point of SH(S)may be reduced to a lower rdCm-extreme point by way of a direct sum decornposition. Let us now consider how to construct a higher rank Cm-extremepoint of the space of the unital completely positive maps with a finite number of lower rank maps. Let Hi be a Hilbert space and let Il>i : A + B(&) be a pure unital completely positive map, i = 1, - ,m, with a minimal Stinespring representation Il>i = wf xiwi, where w; : Ki -t & is a linear bounded map, q : A + B(Ki) is an irreducible '-homomorphism and w;wi = Iq . Construct Hilbert spaces H and K and a '-hornomorphisrn x by Write E E H in a vector form Define : A -î B(H)through expanding ili and by set ting other H, components as Set Thus vi is a linear bounded map fiom H into Kiand Also and hence 9i(a)IH,= +i(a) Accordingly, by a direct sum of (tl>i) we mean a map rp : A + B(H) : Note that this rp is a unital completely positive map. It has a minimal Stinespring representation, which because of the orthogonality, we mite in a matrix form : and i=l It is readily seen that v,'vi are mutudy orthogonal projections in B(H), and also Ci=ik vrv; = 1. If xi(A) and Ki are unitarily equivdent to irj(A) and Kj respectively for some i, j, then following the discussion before the Proposition 3.4.1, we can absorb the unitary equivalences and let ai and nj be either "the same" or disjoint. Since alI ni are irreducible, an element h E [x@q]' has a simple matrk for which any entry in the off-diagonal position (i,j) is a scalar if q = ir,, or is O if x, and rj are disjoint. We can change the order of the direct sum so that the matrix of a is a series of smd blocks, in each block, ni = nj, and if ri and rj are located in different blocks, then they are disjoint. To illustrate this, let us assume that and the rest of the {q) are d disjoint from nt. Then we can write the matrix T as If h E r(A)',then h has a matrix fom where hij for 1 < i < r, 1 5 j 5 r are scdars; any hij in cross positions are 0, namely hij =O for 1 si sr,r < jsmandforr r. 5.2 The decomposition into disjoint maps Lemma 5.2.1 Suppose that vi : H -t Ki, is bounded Linear for eoch i, and e = v;vi + + v;v, is a projection in B(H). Then Proof. Since e and eL = 1 - e are projections in B(H), it follows that However, So vieL = 0, and thus V; = vie. Since 9 in (5.1.2) is a finite direct sum of pure unital maps, it follows that each off-diagonal element in the matrix of P = VV* is 0, namely, We mite n in the form of a matrix: The matrîx can be divided into k blocks. Ln each block, rri = rj; if rri and Rj are in different blocks, then they are disjoint. For example, RI = Q = - - = r,, , and they are disjoint with the rest of aJl the others. In this sense, we say (41, &, . ,&, ) is a group of maps of the same kind. Theorem 5.2.2 Let H be a Hilbert space and let rp be the ditect sum of a finite number of pure unital maps. Then 9 is a Cm-eztremepoint of SH(A)if and only if each direct sum of ail maps of the same kind in the set is a C'-eztreme point on their direct sum of comsponding subspaces. Proof. If Conversely, suppose that each direct sum of all maps of the same kind in the set is a C'extreme point on the direct suof their corresponding subspaces. We want to prove that equation ~ih)~= VX is solvable for 9 = V'xV and (sinvertible h E R(A)'. But the equation ~h = VX has a matrix form: Since t his is in diagonal form, we get Let El be the projection of H onto In general, let Ek be the projection of H onto where k is the number of different kinds of ri. By Lemma 5.2.1 we have Accordingly the above equation is just a system of equations: Consider the first equation of the system, which is where Xi maps Hl $ - - - $ HrI into itself, with v: being the restriction of vl on Hl $ $ Hr, , -,and v:, the restriction of u,, on Hl @ - - $ H,, . Since this part of the direct sum is a C*-extreme point, the equation above has solutions u(') and XI. Treat the other kinds of sums of maps in the same way. in each case obtaining solutions di) and Xi. Findy, let Hence the system is solvable, and by Theorem 3.1.2, rp is a Cm-extremepoint of SH(~- rn Remark 5.2.3 Cornparhg with Proposition 3.4.1, the above Theorem 5.3.2 says simply that if 9 is a finite direct sum of irredueible completely positive maps, then g is a C*-eztreme point of SH(A)if and only if each disjoint submap is a Cm-eztrerne point. 5.3 The structure theorem Now we consider a special C'-extreme point of SH(A)which is constructed by a direct sum of a set of the same kind of pure unita maps. Let zbl ,qb2 be two elements of CP(S,H). If there exists an operator rl.z such that Slz = rre2cli ~1.2,then we say 112 is a compression of $J*. Suppose that {?,b1 ,qb2, - - , ,&) is a set of the same kind of pure unital maps. Expand +i to vi and define vi as in 95.1. Now the necessary and sufficient condition that 9 be a C'extreme point of SH(A)is that for any positive matrix h of scalars, there exists a unitary matrix U of scalars such that where X maps Hl @ $ Hr to itself, and is hear bounded. Hence X has a matrix Let Theorem 5.3.1 Let H be a Hii6ert space and let y E SH(A) be a direct sum of the same kind of pure unital rnaps gi ,& , - , ,$+. Then 9 is a Ca-eztrerne point of SH(A)ij and only if {$1 ,& ,- , ,&) is a nested seqvence of compressions, narnely, (if necessary, change the order) we have Proof. Suppose that (5.3.6) holds, we want to prove that for any positive h, there exists a unitary scalar rnatrix U such that (5.3.5) is solvable. Denote the ith row vector of h by h(') and the ith row vector of LI by di).We construct U and X in the following way. All di) axe unit vecton and . . . U(l) 1 {&), &-11, .. . , uP)), Therefore U is unitary, and tij=O for i>j. Let ~i,j= O for i > j Xi j = ti,jc,j for i < j . Hence there exists a unitary U and a matrix X which satisfy (5.3.5) as required. Conversely, if 9 is a Ce-extreme point of &(A), then (5.3.5) is solvable, and we want to prove (5.3.6). If there exists no relation w, = WiTij, then tij = O for al1 h. In fa&, if for a positive h, there exists LI and X such that thus we can hd and the relation Wj = wiri, exists. We say that (2, j) is a restricted position, if tij = O for ail h. TWOelements wi and wj may have a relation Wi = WjTj,i. In this case we say that wi can be ezpressed by wj, and denote it by wj 2 wi. Clearly, That is, wi 5 wj and wj 5 wk imply that wi 5 wk. Clearly if there exists the birelation then wi = wj. Thus {wl, - ,w,) forms a partially ordered set with 5. To prove (5.3.6) is just to prove that the set is a chain. It is suaicent to prove that for given wi and wj, either w; 5 wj or tuj 5 wi- SO it is sufncent to prove that either wi 5 102 or w2 5 wl. By Theorem 4.3.2, $1 @ & is a Cs-extreme point of SHIieH,(A).So the foilowing equat ion is solvable: We assert that either w2 can be expressed by wl, or wl can be expressed by w2. Note that this means that the corresponding t 1,2 or tZP1must be nonrestricted. Suppose not. If t2,1 and tlVzare both restricted, then we have (u(~),h(')) = tzV1= O and (dl),hc2)) = tlVZ= O for any h. Here II, h 2 O are both 2 x 2 matrices, and we always cm find an h such that h(') = hc2) # O. But the above conditions irnply h(2)= O, because U is unitary. This contradiction shows that one of the tlV2or tzVlmust be nonrestricted. This completes the proof of Theorem 5.3.1. rn Corollary 5.3.2 If y is a C'-eztnme point of SH(A) and if& is a pure completely positive map A -+ B(Ho) which is unitady equivdent to one of those in the direct sum of pure completely positive maps for 9, then y @ is a C-extreme point of SH@& (4 Reconsider Example 3.3.1. H is @, Hl is C, H2is C. 9 E SH(A)is a direct sum of a set of same kind of pure unital maps $l ,tl>z. '1.I It is clear that there exists no linear map :Q: + (C such t hat either wl = wza or 2~2= u. Thus by Theorem 5.3.1, cp is not a Ce-extreme point of & (Mz). Chapter 6 SOME OTHER TOPICS ON C*-EXTREME POINTS 6.1 The problem of extreme points Faxenick and Morenz describe the marner in which extreme points and Cm-extrerne points are related in [ 6, Prop 1.1 1. However, either a compact operator condition or a finite dimension condition is imposed in their hypothesis. In contrast, we find here that whether a Cm-extremepoint of &(A) is also an extreme point SH(A)depends essentially upon the finite reducibility described in Section 4.3. More precisely we have the following theorem. Theorem 6.1.1 Let H be a Hi16ert space and [et 9 be a jînite direct mm of unztal pure maps. If p is a Cm-extremepoint of SH(A),then cp is also an eztreme point of SH(A)= Pmoj. Since V'hV has a matrix form where each h(j)is the scaiar matrix as in (5.2.4), we get V'hV = C hj,j~~~i. Since h E n(A)', hij = O if ri is disjoint with nj. If V'hV = O, then it follows from (5.2.3) that hijvi~;vjv,'= O for T*= ~j. By Theorem 5.3.1 we can assume that either vi = Vj7j.i or Vj = ~i7i.j-For the first case, noting that v;vj = IR], we have Since vivS* # O, it follows that scalar hii must be zero. Therefore if V'hV = O, then h = O. By Proposition 1.3.3, p is dso an extreme point SH(A). 6.2 The problem of extension from an operator system Although Theorem 3.1.2 has given a complete description of the Cm-extremepoints of the space of unit al completely positive maps from a C*-algebra to B( H), very litt le is knom about the Ca-extreme points of the space of unital completely positive maps from an operator system of that Cm-algebrato B(H). With Arveson's Hahn-Banach extension theorem, one may get more information from the extension of the given completely positive map. But to do so, the fist question is that if 9 is a Cm-extreme point of SH(S),then will its extension to A dso be a C'extreme point of Sw(A)?The foilowing theorem can be interpeted as an afkmative answer if the set of extensions is, loosely speaking, not too large. The general question, however, remains open. Theorem 6.2.1 Let H be a finite dimensional Hilbert space and let p be a C'-eztreme point of SH(S).Suppose that for any completely positive eztenszon Q of 9 /rom S to A, *(A) E p(S)". Then p cm be eztended to a C*-edreme point of SH(A). Proof. If 9 is irreducible, then (o(S)" = $(Il). Hence it is obvious that for any completely positive map iI : A + B(H), @(A)C p(S)". So in this case, this theorem is just the result of Theorem 4.2.1. If p is not irreducible, then because H has finite dimension, there exist a finite number of mutually orthogonal subspaces Hi of H, so that p = y1 @ - $ pi, for which vj : S -t B(Hj),is irreducible, j = 1, - - ,k. By Theorem 4.3.3, each pj is a Cm-extremepoint of Srr,(S).Then by CoroUary 4.2.2, each vj cmbe extended to a 78 pure element of SH,(A). Denote it by Oj. So $ = O - @ ai is an extension of 9 from S to all A. We daim that @ is a Cg-extreme point of SH(A). Suppose that 9 can be expressed as a C'-convex combination with 8, 8 E SH(A),where t-l, r-l exist, and where t't + r'r = 1. Then their restrictions, iIrls and els satisfy Since this is a C'-convex combination of SH(S)and since 9 is a Cm-extremepoint of SH(S)?there exist unitary operators ul, uz in B(H)such that and 9 = tau1puit+ r'uzcpuar, on S. Clearly, Q and 8 are respectively extensions of u;9ul and u;;puz. So ul9ri; and uz@u;are respectively extensions of 9. 9 = t'u;(ul Qu;)ul t + r'u;(u2Q~;)u2r. Writeult as t, uzr as r and ulQu; as iIi, u2Qu; as We have Q=t'iEt+r'@r, where 4 and 8 are extensions of 8 is unitarily equivalent to iP. By assumption, if @ is an extension of y, then @(A)C p(S)". Thus = Q1 @ *.$ Qb and each qj E SH,(A).Thesame is tme for 8 = QI $-*.B8k. Following the matrix expression, we have For each index (ji),we get ai 2 t;i9jtji. Since Qi is pure, t;iQjtjj = XjiQi. Since Xji is a sca(ar, tji is a multiple of a unitq operator (in lower rank). It is evident that if Xji # O, then Qj is unitarily equivalent to Qi- We need to prove that there exists a unitary operator u such that 9 = uœ@u.To this end, we apply induction on k. As mentioned at the beginning, it foilows from Theorem 4.2.1 that the assertion is tnie for k = 1. Also it is not hard to prove directly the assertion for k = 2. Now suppose that the assertion Is true for k - 1. In equation (6.2.1), if there is an index, say (kk),such that t hen IIk is unitarily equivalent to Qk. Now if t here exists anot her nonzero index in this column, then we get iIr is unitarily equivalent to another ai- Thus G1 is unitarily equivalent to another ai. It follows from the assumption for k - 1 and CoroLIary 5.3.2 that we are done. So we assume that every other index in this colurnn is zero, and then continue to consider this row. If there exists some other nonzero index in this row, say (2k),such that then Qk is unitarily equivalent to Q2. Further, if there exists another nonzero index in this column, then we exchange the positions of 42 and Gkand we are done. Therefore if there exist two nonzero indeces in one row, then anyother indecs in these two columns are aU zero indeces. Note that t is an invertible matrix, there should exist only one nonzero index in eadi row. Finally, if there exists at most one nonzero index in each row and each column in (6.2.1), then since the matrix (tij)is invertible. there does exist one nonzero index in each row and each column in (6.2.1), and accordingly 9 = al @ - - @ Ok is unitarily equivalent to a rearrangement of !If1$ -8at. Clearly, any remangement is unitarily equident to q18 @ Q1. Accordingly 9 = @ - @ di is unitady equident to \il 8 ' - - 8 gk. We htve proved that the assertion is true for k. By induction the assertion is tme for any k. This completes the proof. 1 Chapter 7 CONCLUSIONS In this chapter, we summarize the most important features of the Cm-extreme points of SH(A). 7.1 Major results of the theory in the literature First of all, Fxenick aad Morenz [6] proved the foilowing set of four propositions. Proposition 7.1.1 Let H have finite dimension. If 9 is a Cm-eztrerne point of Sw(A),then y is the direct surn of pure completely positiue maps. Although it is still open as to whether Cm-extranepoints are necessarily linear extreme points, the following partial resdt is known. Proposition 7.1.2 Let 9 be a C*-eztreme point oj SH(A). if for snme t E A, p(x) E {C(H) +a) and is imducible or if (o(x) E {C(H)+C) for al1 x E A, in particular, when H has finite dimension, then 9 is an eztreme point of SH(A). 82 Proposition 7.1.3 When H hos finite dimension, the direct surn of a chain of corn- pressions of a pure completely positive map is a C'-eztreme point O/ SH(A),where H is constructed by the corresponding direct sum of spaces. Proposition 7.1.4 If H has finite dimension, then the set of C'-conüez combina- tions of Cm-eztremepoints of &(A) is dense (in the B W-topology) in SH(A). This proposition above is the analogue, for Co-convexity in generaIised state spaces, of the usual Krein-Milman theorem, Proposition 1.2.1. However, in the Cs- combination, the coefficients may not be invertible. This difference from the usual Krein-Milman theorem is essential. More recently, Farenick and Zhou [7] proved the following Theorem 7.1.5 rp is a Cm-eztnmepoint of SH(A)if and only if for each pinvertible h there emits u unitary LI E rr(A)' and an inuertible X E B(H) such that trhf v = VX. As the application of the theorem above, they get the foiIowing structure theorem. Theorem 7.1.6 Let H have finite dimension. A unital completely positive linear rnap y : A + B(H) is a Ce-eztnme point of S'*(A) if and only if ip is the direct surn of pure completely positive maps for which each disjoint pup of pure completely positive maps fom a chain of compressions. Theorern 7.1.5 and 7.1.6 are subsumed by the results contained in this thesis. 83 7.2 The majornewworks ofthis thesis This thesis begins with a new equivalent definition of a Cm-extremepoint of SH(S) that is in accordance with the definition of linear extreme points. Theorem 4.1.2 establishes a certain relation between the irreducible Cm-extreme points of &(A) and the pure compietely positive maps, and from Theorem 4.1.2. we obtain the following proposition, which represents a key step in the analysis of the structure of the Cm-extremepoints of SH(A). Proposition 7.2.1 Let H have finite dimension and let p be a Cm-eztrernepoint of SH(A).If the range of y is irreducible, then 9 is a pure element of SH(A). We remark that Theorem 4.1.2 is a more precise result than that of Proposition 7.1.2. Proposition 7.1.1 (of Farenick and Morenz) is very important for the study of Cœ- extreme points here and in the papers [6], [7]. However, their proof of the theorem is difficult, as it relies on several complicated manipulations involving matrices. One substantial point in this thesis is that we have been able to give a completely new proof of Proposition 7.1.1 (see Theorem 4.4.1) based on conceptual arguments rather t han on arguments arising from matrix manipulations. Theorem 4.3.2 and Theorem 4.3.3 axe the key results that form the new ptoof. In two ways, moreover, these theorems generalise Proposition 7.1.1 where Fâfenick and Morenz considered Cm-algebrasand finite-dimensiond Hilbert spaces. Theorem 4.3.2 and Theorern 4.3.3 allow the use of operator systems and ded with the case of Hilbert spaces of infinite dimension. 84 Theorem 5.2.2 and Theorem 5.3.1 concern the structure of C'extreme points in SH(A). From these two theorems, when H has finite dimension, we have the two following results. Proposition 7.2.2 Proposition 7.2.3 A direct sum of pure completely positive maps of the same kind is a Ca-eztreme point of SH(A) if and only if these pure completeiy positive maps fonn a chain of compressions. Theorem 5.2.2 and Theorem 5.3.1 combine to give a generalized version of The- orem 7.1.6, which can be viewed as culmination of the work that was initiated by Farenick and Morenz in (61. This thesis iliscusses the relation between Cs-extreme points and linear extreme points. Some new results are obtained such as Theorem 2.2.4 and Theorem 6.1.1. Findy, some effort was made to explore the conditions that a Cm-extremepoint of SH(S)can be extended to a C*-extremepoint of SR(A); new redts are Theorem 42.1 and Theorem 6.2.1. From Theorem 4.2.1 we have the following extension t heorem. Corollary 7.2.4 Let H have finite dimension and let y be a C'-eztreme point of Sw(S).If the range of 9 is imducible, then y can be eztended us a Ca-eztreme point of SH(~ Bibliography [II W.B. Aweson, Subalgebros of Cs-afgebras, Acta Math., 123 (1969), 141-224. [2] W.B. Ameson. Subolgebras of Cm-algebrasII, Acta Math., 128 (1972). 271-308. [3] J. Bunce and N. Sahas, Completely posit2ue maps on C'-algebrcrs and the left matricial spectra of an opemtor, Duke Math. J., 43 (1976), 747-177. [4] K.R. Davidson, A proof of the boundary theorem, Proc. Amer Math. Soc., 82 (1981), 48-50. [5] D.R. Farenick and P.B. Morenz, C'-eztnme points of some compad C'-eonvez sets, Proc. Amer. Math. Soc., 118 (1993), 768775. [6] D.R. Farenick and P.B. Morenz, C'-eztreme points in the generalised state spaces of a Cu-algebm, Trans. Amer. Math. Soc., 349 (1997), 1735-1748. [7] D.R. Farenick and H. Zhou, The structure of C'-eztreme points in spaces of completely positive lznear maps on Cm-algebras, Proc. Amer. Math. Soc., 126 (1998), no. 5, 1467-1477. 86 [SI A. Hopenwasser, R.L. Moore, and V.I. Padsen, Cu-eztreme points, Trans. Amer. Math. Soc., 266 (1981), 291-307. [9] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Alge- bras, Vol 1, Academic Press, Inc., 1983. [IO] R.I. Loebl, A remark on unitary orbzts, Bull. Inst. Math. Acad. Sinica, 7 (1979), 401-407. [Il] R.I. Loebl and V.I. Paulsen, Some remarks on Cs-conven'ty, Linear Algebra Appl., 35 (1981), 63-78. [12] V.I. Paulsen, Completely Bounded Maps And Dilati~t~,Longman Sci. & Tech., n Pitman Research Notes in Math. Series 146, 1986. [13] W.F. Stinespring, Positive functàons on Ca-algebras, Proc. Amer. Math. Soc.. 6 (1955), 211-216. [14] C. Webster and S. Winkler, The Krein-Milman theonrn in operator contiezity, Trans. Amer. Math. Soc., to appear. [El S. Winkler, Matnt convezity, P hD thesis, University of California, Los Angeles, 1996. LIST OF SYMBOLS set of all bounded linear opearators on Hilbert space Ht4 quotient space of B(H) over C(H), 40 set of dl bounded linear maps of S into B(H), 5 ad compact operaton in B( H). 14,40 algebra of scalars and compact operators on B( H), 14.43 the smdest Cm-dgebrathat contains S, 40 set of al1 cornpietely positive maps of S into B(H),5 closure of the convex hdof Y?5 commutant of R,commutant of Rr,9,37,39,40,?8 set of all unital completely positive maps of S into B(H),5 closure of the set of fiaite iinear combinations of vectors of V?9 direct sum, 32,35,38 restriction on S of +, 44 unital *-homomorphisim of A into a B(K),9 INDEX bounded-weak-topology, 6 C'convex, Cm-extremepoint, 12 completely isometric, cornpletely positive, 4,s compression, 70 disjoint, 37 ext reme point, 5 generalised state, 5 imeducible, 39,40 minimal Stinespring representation, 9 operator system, 4 pure completely positive map, 7 restricted position, 73 group of unital completely positive maps of the same kind, 64 pinvertible, 21 IMAGE EVALUATION TEST TARGET (QA-3)