A Radon-Nikodym theorem for completely n-positive linear maps on pro-C*-algebras and its applications

0.17 MB·English

by  Maria Joita

A RADON-NIKODYM THEOREM FOR COMPLETELY
n-POSITIVE LINEAR MAPS ON PRO-C∗-ALGEBRAS AND ITS
7
APPLICATIONS
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2
MARIAJOIT¸A

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a
J

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Abstract. Theorderrelationonthesetofcompletelyn-positivelinearmaps

] fromapro-C∗-algebraAtoL(H),theC∗-algebraofboundedlinearoperators
A
onaHilbertspaceH,ischaracterizedintermsoftherepresentationassociated
O
witheachcompletelyn-positivelinearmap. Also,thepureelementsintheset
.
h
of all completely n-positive linear maps from A to L(H) and the extreme
t
a points inthe set of unital completely n-positivelinear maps fromA to L(H)
m
are characterized in terms of the representation induced by each completely
[
n-positivelinearmap.
1
v
3 1. Introduction and preliminaries
6
2
Stinespring [12] showed that any completely positive linear map from a C∗-
1
0 algebra A to L(H), the C∗-algebra of bounded linear operators on a Hilbert space
7
0 H,inducesarepresentationofAonanotherHilbertspacethatgeneralizestheGNS
/
h construction for positive linear functionals on C∗-algebras. In [1], Arveson proved
t
a
m a Radon-Nikodym type theorem which gives a description of the order relation on
v: the set CP∞(A,L(H)) of all completely positive linear maps from a C∗-algebra A
i toL(H)intermsoftheStinespringrepresentationassociatedwitheachcompletely
X
r positivelinearmap,andusingthistheorem,heestablishedcharacterizationsofpure
a
elementsinthesetofcompletelypositivelinearmapsfromAtoL(H)andextreme
elements in the set of all unital completely positive linear maps from A to L(H)
interms of the Stinespring representationassociatedwith eachcompletely positive
linear map.
2000Mathematical SubjectClassification:46L05
KeyWords: completely n-positivelinear maps, purecompletely n-positivelinear maps; extreme
completelyn-positivelinearmaps;pro-C∗-algebras.
ThisresearchwaspartiallysupportedbygrantCNCSIS(RomanianNationalCouncilforResearch
inHighEducation)-code A1065/2006.
1
Kaplan [10] introduced the notion of multi-positive linear functional on a C∗-
algebra A (that is, an n×n matrix of linear functionals on A which verifies the
positivity condition) and showed that any multi-positive linear functional on A
inducesarepresentationofAonaHilbertspace. Also,heprovedaRadon-Nikodym
type theorem for multi-positive linear functionals on a C∗-algebra A. In [13, 14]
Suenconsideredthe n×n matrices of continuouslinear maps froma C∗-algebraA
to L(H) and showed that a unital completely n-positive linear map from a unital
C∗-algebraA to L(H) induces a representationofA on a Hilbert space in terms of
the Stinespring construction.
A pro-C∗-algebra A is a complete Hausdorff topological ∗-algebraover C whose
topology is determined by its continuous C∗-seminorms in the sense that the net
{ai}i∈I converges to 0 in A if and only if the net {p(ai)}i∈I convergesto 0 for any
continuousC∗-seminormponA; equivalently,Aishomeomorphically∗-isomorphic
to an inverse limit of C∗-algebras. So, pro-C∗-algebras are generalizations of C∗-
algebras. Insteadofbeinggivenbyasinglenorm,thetopologyonapro-C∗-algebra
is defined by a directed family of C∗-seminorms. Besides an intrinsic interest in
pro-C∗-algebras as topological algebras comes from the fact that they provide an
important tool in investigation of certain aspects of C∗-algebras ( like multipliers
of the Pedersen ideal [2, 11]; tangent algebra of a C∗-algebra [11]; quantum field
theory [3]). In the literature, pro-C∗-algebras have been given different name such
as b∗-algebras ( C. Apostol ), LMC∗-algebras ( G. Lassner, K. Schmu¨dgen) or
locally C∗-algebras [4, 6, 7, 8, 9].
A representation of a pro-C∗-algebra A on a Hilbert space H is a continuous ∗
-morphism from A to L(H). In [4] it is showed that a continuous positive linear
functional on a pro-C∗-algebra A induces a representation of A on a Hilbert space
in terms of the GNS construction. Bhatt and Karia [2] extended the Stinespring
constructionforcompletelypositivelinearmapsfromapro-C∗-algebraAtoL(H).
In [9], we extend the KSGNS (Kasparov, Stinespring, Gel’fend, Naimark, Segal)
construction for strict completely n-positive linear maps from a pro-C∗-algebra A
to another pro-C∗-algebra B.
In this paper we generalize various earlier results by Arveson and Kaplan. The
paper is organized as follows. In Section 2, we establish a relationship between
2
the comparability of nondegenerate representations of A and the matricial order
structure of B(A,L(H)), the vector space of continuous linear maps from A to
L(H)(Proposition2.6). This is a generalizationofProposition2.2,[10]. Section 3
is devoted to a Radon-Nikodym type theorem for completely multi-positive linear
maps from A to L(H). As a consequence of this theorem, we obtain a criterion of
pureness for elements in CPn(A,L(H)) in terms of the representation associated
∞
with each completely n-positive linear map (Corollary 3.6 ). In Section 4 we prove
a sufficient criterion for a completely n-positive linear map from A to L(H) to be
pure in terms of its components (Lemma 4.1) and using this result we determine
a certain class of extreme points in the set of all unital completely positive linear
mapsfromAtoL(Hn),whereHn denotesthedirectsumofncopiesoftheHilbert
space H (Corollary 4.3). Finally, we give a characterization of the extreme points
inthe setofallunitalcompletelyn-positivelinearmapsfromAtoL(H)(Theorem
4.4) that extends the Arveson characterization of the extreme points in the set of
all unital completely positive linear maps from a C∗-algebra A to L(H).
2. Representations associated with completely n-positive linear maps
Let A be a pro-C∗-algebra and let H be a Hilbert space. An n × n matrix
n
ρ of continuous linear maps from A to L(H) can be regarded as a linear
ij i,j=1
m(cid:2) ap(cid:3)ρ from Mn(A) to Mn(L(H)) defined by
ρ [a ]n = ρ (a ) n .
ij i,j=1 ij ij i,j=1
(cid:16) (cid:17) (cid:2) (cid:3) n
Itisnotdifficulttocheckthatρiscontinuous. Wesaythat ρ isann-positive
ij i,j=1
(respectively completely n-positive) linear mapfrom Ato L(cid:2)(H(cid:3))if the linear mapρ
fromM (A)toM (L(H))ispositive(respectivelycompletelypositive). Thesetof
n n
all completely positive linear maps from A to L(H) is denoted by CP∞(A,L(H))
and the set of all completely n-positive linear maps from A to L(H) is denoted by
CPn (A,L(H)). If ρ and θ are two elements in CPn (A,L(H)), we say that θ ≤ρ
∞ ∞
if ρ−θ is an element in CPn (A,L(H)).
∞
Remark 2.1. In the same manner as in the prof of Theorem 1.4 in [5], we can
show that the map S from CP∞n (A,L(H)) to CP∞(A,Mn(L(H))) defined by
n n
S ρ (a)= ρ (a)
ij i,j=1 ij i,j=1
(cid:16)(cid:2) (cid:3) (cid:17) 3 (cid:2) (cid:3)
is an affine order isomorphism.
The following theorem is a particular case of Theorem 3.4 in [9].
Theorem 1. Let A be a pro-C∗-algebra, let H be a Hilbert space and let ρ =
n
ρ be a completely n-positive linear map from A to L(H). Then there is a
ij i,j=1
(cid:2)repr(cid:3)esentation Φρ of A on a Hilbert space Hρ and there are n elements Vρ,1,...,Vρ,n
in L(H,H ) such that
ρ
(1) ρ (a)=V∗ Φ (a)V for all a∈A and for all i,j ∈{1,...,n};
ij ρ,i ρ ρ,j
(2) {Φ (a)V ξ;a∈A,ξ ∈H,1≤i≤n} spans a dense subspace of H .
ρ ρ,i ρ
The n+2 tuple (Φ ,H ,V ,...,V ) constructed in Theorem 2.2 is said to be
ρ ρ ρ,1 ρ,n
theStinespringrepresentationofAassociatedwiththecompletelyn-positivelinear
map ρ.
Remark 2.2. The representation associated with a completely n-positive linear
map is unique up to unitary equivalence [9, Theorem 3.4].
Remark 2.3. Let (Φ ,H ,V ,...,V ) be the Stinespring representation associ-
ρ ρ ρ,1 ρ,n
n
ated with a completely n-positive linear map ρ= ρ . For each i∈{1,...,n},
ij i,j=1
wedenote by Hi theHilbert subspace of Hρ genera(cid:2)ted(cid:3)by {Φρ(a)Vρ,iξ;a∈A,ξ ∈H}
and by P the projection in L(H ) whose range is H . Then P ∈Φ (A)′, the com-
i ρ i i ρ
mutant of Φ (A) in L(H ), and a7→Φ (a)P is a representation of A on H which
ρ ρ ρ i i
is unitarily equivalent with the Stinespring representation associated with ρ for all
ii
i∈{1,...,n}. Since Φ is a nondegenerate representation of A,
ρ
limΦ (e )ξ =ξ
ρ λ
λ
for some approximate unit {eλ}λ∈Λ for A and for all ξ ∈ H [7, Proposition 4.2],
and then
P V ξ =limP Φ (e )V ξ =V ξ
i ρ,i i ρ λ ρ,i ρ,i
λ
for all ξ ∈H and for all i∈{1,...,n}. Therefore P V =V for all i∈{1,...,n}.
i ρ,i ρ,i
n
Remark 2.4. If ρ= ρ is a diagonal completely n-positive linear map from
ij i,j=1
A to L(H) (that is, ρ (cid:2)=(cid:3)0 if i6=j), then the Stinespring representation associated
ij
with ρ is unitarily equivalent with the direct sum of the Stinespring representations
associated with ρ , i∈{1,...,n}.
ii
4
Two completely n-positive linear maps ρ and θ from A to L(H) are called dis-
joint (respectively unitarily equivalent ) if the representations of A induced by ρ
respectively θ are disjoint (respectively unitarily equivalent).
ThefollowingpropositionisananalogueofProposition2.2in[10]forcompletely
n-positive linear maps.
Proposition2.5. Letρ andρ betwocompletely positive linearmaps fromAto
11 22
L(H). Then ρ and ρ are disjoint if and only if there are nononzero continuous
11 22
2
linear maps ρ and ρ such that ρ = ρ is a completely 2-positive linear
12 21 ij i,j=1
map from A to L(H). (cid:2) (cid:3)
2
Proof. Suppose that ρ and ρ are disjoint and ρ = ρ is a completely
11 22 ij i,j=1
2-positive linear map from A to L(H). Let (Φρ,Hρ,Vρ,1(cid:2),Vρ(cid:3),2) be the Stinespring
representationassociatedwithρ.Since ρ andρ aredisjoint,the centralcarriers
11 22
of projections P and P (see Remark 2.4) are orthogonal and so P P =0. Then
1 2 1 2
ρ (a)=V∗ Φ (a)V =V∗ P Φ (a)P V =V∗ Φ (a)P P V =0
12 ρ,1 ρ ρ,2 ρ,1 1 ρ 2 ρ,2 ρ,1 ρ 1 2 ρ,2
for all a ∈ A, and since ρ (a) = (ρ (a∗))∗ for all a ∈ A, we conclude that ρ =
21 12 12
ρ =0.
21
Conversely,supposethattherearenononzerocontinuouslinearmapsρ andρ
12 21
2
suchthat ρ= ρ is a completely 2-positive linear map from A to L(H). Let
ij i,j=1
Φρii,Hρii,Vρi(cid:2)i b(cid:3)e the Stinespring representation associated with ρii, i ∈ {1,2}.
(cid:0)The directsum(cid:1)Φ=Φρ11⊕Φρ22 ofthe representationsΦρ11 andΦρ22 is a represen-
tationofAontheHilbertspaceH =H ⊕H .Weconsiderthelinearoperators
0 ρ11 ρ22
V ,V ∈L(H,H ) defined by V (ξ)=V ξ⊕0 respectivelyV (ξ)=0⊕V ξ. For
1 2 0 1 ρ11 2 ρ22
eachi∈{1,2},the orthogonalprojectionof H on H is denoted by E . It is not
0 ρ i
ii
difficult to check that the range of E is generated by {Φ(a)E V ξ;a∈A,ξ ∈H},
i i i
i∈{1,2}. Moreover,ρ (a)=V∗Φ(a)V , i∈{1,2}.
ii i i
Supposethatρ andρ arenotdisjoint. Thentherearetwononzeroprojections
11 22
′
F and F in Φ(A) majorized by E respectively E , and a partial isometry V in
1 2 1 2
Φ(A)′ such that V∗V = F and VV∗ = F . It is not difficult to check that the
1 2
range of F is generatedby {Φ(a)F V ξ;a∈A,ξ ∈H}, i∈{1,2}.We consider the
i i i
5
linear map θ from A to M (L(H)) defined by
2
V∗F Φ(a)V V∗Φ(a)V∗V
θ(a)= 1 1 1 1 2 .
 V∗VΦ(a)V V∗F Φ(a)V 
2 1 2 2 2
 
It is not difficult to check that θ is continuous and
θ(a∗b)=(M(a)W)∗M(b)W,
where
Φ(a) Φ(a) F V 0
1 1
M(a)= and W = ,
 0 0   0 V∗V 
2
   
for all a and b in A. Then
∗
m m m
T∗θ(a∗a )T = M(a )WT M(a )WT ≥0
k k l l k k l l
! !
k.l=1 k=1 l=1
X X X
for all T ,...,T ∈ M (L(H)) and for all a ,...,a ∈ A. This implies that θ ∈
1 m 2 1 m
CP∞(A,M2(L(H))) and so S−1(θ) ∈ CP∞2 (A,L(H)) ( see Remark 2.1). Let ρ =
ρ 2 , where ρ = S−1(θ) and ρ = S−1(θ) . Then
ij i,j=1 12 12 21 21
(cid:2) (cid:3) (cid:0) (cid:1) (cid:0) (cid:1)
V∗(E −F )Φ(a)V 0
(S(ρ)−θ)(a)= 1 1 1 1
 0 V∗(E −F )Φ(a)V 
2 2 2 2
 
foralla∈A.ThereforeS(ρ)−θ∈CP∞(A,M2(L(H))). Fromthis factandtaking
intoaccountthatθ ∈CP∞(A,M2(L(H))),weconcludethatS(ρ)∈CP∞(A,M2(L (H)))
and then by Remark 2.1, ρ ∈ CP2 (A,L(H)). If ρ = 0, then V∗Φ(a)V∗V = 0
∞ 12 1 2
for all a∈A. This implies that
hV∗Φ(a)V ξ,Φ(a)F V ηi=hV∗Φ(a∗a)V∗V ξ,ηi=0
2 1 1 1 2
foralla∈Aandforallξ,η ∈H. SincetherangeofF isgeneratedby{Φ(a)F V ξ;
1 1 1
a∈A,ξ ∈H} and for all ξ ∈H and a∈A, V∗Φ(a)V ξ is an element in the range
2
of F , the preceding relation yields that V∗Φ(a)V ξ = 0 for all a ∈ A and for all
1 2
ξ ∈H. ThenΦ(a)F V ξ =VV∗Φ(a)V ξ =0foralla∈Aandforallξ ∈H andso
2 2 2
′
F =0. Thisisacontradiction,sinceF isanonzeroprojectioninΦ(A). Therefore
2 2
2
ρ 6=0.Thus we have found a completely 2-positive linear map ρ= ρ such
12 ij ij=1
thatρ 6=0,acontradiction. Thereforeρ andρ aredisjointandth(cid:2)ep(cid:3)roposition
12 11 22
is proved. (cid:3)
6
3. The Radon-Nikodym theorem for completely n-positive linear
maps
Let A be a pro-C∗-algebra and let H be a Hilbert space.
Lemma 3.1. Let ρ= ρ n be an element in CPn(A,L(H)). If T is a positive
ij ij=1 ∞
′
element in Φρ(A), th(cid:2)e co(cid:3)mmutant of Φρ(A) in L(Hρ), then the map ρT from
M (A) to M (L(H)) defined by
n n
ρ [a ]n = V∗ TΦ (a )V n
T ij i,j=1 ρ,i ρ ij ρ,j i,j=1
(cid:16) (cid:17) (cid:2) (cid:3)
is a completely n-positive linear map from A to L(H).
Proof. Itisnotdifficulttocheckthatρ isamatrixofcontinuouslinearmapsfrom
T
AtoL(H),the(i,j)-entryofthe matrixρ beingthecontinuouslinearmap(ρ )
T T ij
from A to L(H) defined by
(ρ ) (a)=V∗ TΦ (a)V .
T ij ρ,i ρ ρ,j
Also it is not difficult to check that
∗
S(ρT)(a∗b)= MT21 (a)V MT21 (b)V ,
(cid:16) (cid:17) (cid:16) (cid:17)
1
for all a,b∈A, where T2 is the square root of T,
1 1
T2Φρ(a) ... T2Φρ(a)
V ... 0
 0 ... 0  ρ,1
M 1 (a)= and V = . ... . .
T2  0. ...... 0.   0 ... Vρ,n 
   
 
Then
m m ∗
Tl∗S(ρT)(a∗lak)Tk = Tl∗ MT12 (al)V MT12 (ak)V Tk
kX,l=1 kX,l=1 (cid:16) (cid:17) (cid:16) (cid:17)
∗
m m
= M 1 (al)VTl M 1 (ak)VTk ≥0
T2 ! T2 !
l=1 k=1
X X
for all T ,...,T ∈M (L(H)) and for all a ,...,a ∈A. This shows that S(ρ )∈
1 m n 1 m T
CP∞(A,Mn(L(H))) and by Remark 2.1, ρT ∈CP∞n (A,L(H)). (cid:3)
Remark 3.2. Let ρ= ρ n ∈CPn(A,L(H)).
ij ij=1 ∞
(1) If I is the ide(cid:2)ntit(cid:3)y map on H , then ρ =ρ.
Hρ ρ IHρ
7
′
(2) If T and T are two positive elements in Φ (A) , then ρ =ρ +ρ .
1 2 ρ T1+T2 T1 T2
′
(3) If T is a positive element in Φ (A) and α is a positive number, then
ρ
ρ =αρ .
αT T
Lemma 3.3. Let ρ= ρ n be an element in CPn(A,L(H)) and let T and T
ij ij=1 ∞ 1 2
′
be two positive element(cid:2)s in(cid:3)Φρ(A). Then T1 ≤T2 if and only if ρT1 ≤ρT2.
Proof. First, we suppose that T ≤T . Then ρ ∈CPn (A,L(H)), and since
1 2 T2−T1 ∞
ρ −ρ [a ]n = V∗ (T −T )Φ (a )V n
T2 T1 ij i,j=1 ρ,i 2 1 ρ ij ρ,j i,j=1
(cid:0) (cid:1)(cid:16) (cid:17) = (cid:2)ρ [a ]n (cid:3)
T2−T1 ij i,j=1
(cid:16) (cid:17)
for all [a ]n ∈M (A), we conclude that ρ ≤ρ .
ij i,j=1 n T1 T2
m n
Conversely, suppose that ρ ≤ρ . Let Φ (a )V ξ ∈H . Then
T1 T2 ρ ki ρ,i ki ρ
k=1i=1
P P
m n m n
(T −T ) Φ (a )V ξ , Φ (a )V ξ
2 1 ρ ki ρ,i ki ρ ki ρ,i ki
* ! +
k=1i=1 k=1i=1
XX XX
m n
= V∗ (T −T )Φ (a∗ a )V ξ ,ξ
ρ,i 2 1 ρ ki lj ρ,j lj ki
k,l=1i,j=1
X X (cid:10) (cid:11)
m n
n n
= S ρ −S ρ (a∗ a ) ξ , ξ ≥0,
T2 T1 ki lj ljp p=1 kip p=1
kX,l=1iX,j=1(cid:28)(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) (cid:16) (cid:17) (cid:16) (cid:17) (cid:29)
e e
ξ if p=i and 1≤k ≤m
ki
where ξ = . From this fact and taking into ac-
kip 
0 if p6=i and 1≤k ≤m

count tehat {Φρ(a)Vρ,iξ;a ∈ A,ξ ∈ H, 1 ≤ i ≤ n} spans a dense subspace of Hρ,

we conclude that T −T ≥0 and the lemma is proved. (cid:3)
2 1
Lemma 2. Let ρ= ρ n and θ =[θ ]n be two elements in CPn(A,L(H)).
ij ij=1 ij ij=1 ∞
If θ ≤ρ, then there i(cid:2)s an(cid:3)element W ∈L(Hρ,Hθ) such that
(1) kWk≤1;
(2) WV =V for all i∈{1,...,n};
ρ,i θ,i
(3) WΦ (a)=Φ (a)W for all a∈A.
ρ θ
8
Proof. Since
m n m n
Φ (a )V ξ , Φ (a )V ξ
θ ki θ,i ki θ ki θ,i ki
* +
k=1i=1 k=1i=1
XX XX
m n
= V∗ Φ a∗a V ξ ,ξ
θ,j θ lj ki θ,i ki lj
k,l=1i,j=1
X X (cid:10) (cid:0) (cid:1) (cid:11)
m n
= θ a∗a ξ ,ξ
ji lj ki ki lj
k,l=1i,j=1
X X (cid:10) (cid:0) (cid:1) (cid:11)
m n
n n
= S(θ) a∗a ξ , ξ
lj ki kip ljp
p=1 p=1
kX,l=1iX,j=1(cid:28) (cid:0) (cid:1)(cid:16) (cid:17) (cid:16) (cid:17) (cid:29)
m n e e
n n
≤ S(ρ) a∗a ξ , ξ
lj ki kip ljp
p=1 p=1
kX,l=1iX,j=1(cid:28) (cid:0) (cid:1)(cid:16) (cid:17) (cid:16) (cid:17) (cid:29)
m n me n e
= Φ (a )V ξ , Φ (a )V ξ
ρ ki ρ,i ki ρ ki θ,i ki
* +
k=1i=1 k=1i=1
XX XX
for all a ∈ A and ξ ∈ H, 1 ≤ i ≤ n, 1 ≤ k ≤ m, and since {Φ (a)V ξ;a ∈
ki ki ρ ρ,i
A,ξ ∈ H, 1 ≤ i ≤ n} spans a dense subspace of H , there is a unique bounded
ρ
linear operator W from H to H such that
ρ θ
m n m n
W Φ (a )V ξ = Φ (a )V ξ .
ρ ki ρ,i ki θ ki θ,i ki
!
k=1i=1 k=1i=1
XX XX
Moreover, kWk ≤ 1. Let {eλ}λ∈Λ be an approximate unit for A, ξ ∈ H and
i∈{1,...,n}. Then
WV ξ =limWΦ (e )V ξ =limΦ (e )V ξ =V ξ.
ρ,i ρ λ ρ,i θ λ θ,i θ,i
λ λ
Therefore WV = V for all i ∈ {1,...,n}. It is not difficult to check that
ρ,i θ,i
WΦ (a)=Φ (a)W for all a∈A. (cid:3)
ρ θ
Let ρ= ρ n ∈CPn(A,L(H)). We consider the following sets:
ij ij=1 ∞
(cid:2) (cid:3)
[0,ρ]={θ∈CPn(A,L(H));θ ≤ρ}
∞
and
′
[0,I] ={T ∈Φ (A) ;0≤T ≤I }.
ρ ρ Hρ
The following theorem can be regarded as a Radon-Nikodym type theorem for
completely multi-positive linear maps.
9
Theorem 3.4. The map T 7→ ρ from [0,I] to [0,ρ] is an affine order isomor-
T ρ
phism.
Proof. According to Lemmas 3.1 and 3.3 and Remark 3.2, it remains to show that
the map T 7→ρ from [0,I] to [0,ρ] is bijective.
T ρ
LetT ∈[0,I]ρsuchthatρT =0. ThenVρ∗,iTΦρ(a∗a)Vρ,i =0andsoT12Φρ(a)Vρ,i =
0foralla∈Aandforalli∈{1,...,n}.Fromthis factandtakingintoaccountthat
{Φ (a)V ξ;a ∈ A,ξ ∈ H, 1 ≤i ≤ n} spans a dense subspace of H , we conclude
ρ ρ,i ρ
that T =0 and so the map T 7→ρ from [0,I] to [0,ρ] is injective.
T ρ
Let θ ∈ [0,ρ]. If W is the bounded linear operator from H to H constructed
ρ θ
in Lemma 3.4, then W∗W ∈[0,I] . Let T =W∗W. From
ρ
θ [a ]n = V∗ Φ (a )V n = (WV )∗Φ (a )WV n
ij i,j=1 θ,i θ ij θ,j i,j=1 ρ,i θ ij ρ,j i,j=1
(cid:16) (cid:17) = (cid:2)V∗ W∗WΦ (a(cid:3))V n(cid:2) =ρ [a ]n (cid:3)
ρ,i ρ ij ρ,j i,j=1 T ij i,j=1
(cid:16) (cid:17)
for all [a ]n ∈M (A(cid:2)), we deduce that θ =(cid:3)ρ and the theorem is proved. (cid:3)
ij i,j=1 n T
We say that a completely n-positive linear map ρ from a pro-C∗-algebra A to
L(H)ispure ifforeveryθ ∈CPn(A,L(H))withθ ≤ρthereisα∈[0,1]suchthat
∞
ρ=αθ.
A representation Φ of a pro-C∗-algebra A on a Hilbert space H is irreducible if
′
andonlyif the commutantΦ(A) ofΦ(A) inL(H)consistsofthe scalarmultiplies
of the identity map on H [4].
Corollary 3.5. Let ρ∈CPn(A,L(H)). Then the following statements are equiv-
∞
alent:
(1) ρ is pure;
(2) The representation Φ of A associated with ρ is irreducible.
ρ
Remark3.6. IfρisacontinuouspositivefunctionalonA,thenρ∈CP1 (A,L(C)).
∞
Moreover, Φ is the GNS representation of A associated with ρ, and Corollary 3.6
ρ
states that ρ is pure if and only if the representation Φ is irreducible. This is a
ρ
particular case of the known result which states that a continuous positive linear
functional ρ on a lmc∗-algebra A with bounded approximate unit is pure if and only
if the GNS representation of A associated with ρ is topological irreducible ( see, for
example, Corollary 3.7 of [4]).
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