Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843 - 7265 (print)
Volume 2 (2007), 113 – 122
CHARACTERIZATION OF THE ORDER RELATION
ON THE SET OF COMPLETELY n-POSITIVE
LINEAR MAPS BETWEEN C ∗ -ALGEBRAS
Maria Joiţa, Tania-Luminiţa Costache and Mariana Zamfir
Abstract. In this paper we characterize the order relation on the set of all nondegenerate completely n-positive linear maps between C ∗ -algebras in terms of a self-dual Hilbert module induced
by each completely n-positive linear map.
1
Introduction and preliminaries
Completely positive linear maps are an often used tool in operator algebras theory and quantum information theory. The theorems on the structure of completely
linear maps and Radon-Nikodym type theorems for completely positive linear maps
are an extremely powerful and veritable tool for problems involving characterization
and comparison of quantum operations (that is, completely positive linear maps
between the algebras of observables (C ∗ -algebras) of the physical systems under
consideration).
Given a C ∗ -algebra A and a positive integer n, we denote by Mn (A) the C ∗ algebra of all n × n matrices over A with the algebraic operations and the topology
obtained by regarding it as a direct sum of n2 copies of A.
Let A and B be two C ∗ -algebras. A linear map ρ : A → B is completely positive
if the linear maps ρ(n) : Mn (A) → Mn (B) defined by
ρ(n) ([aij ]ni,j=1 ) = [ρ(aij )]ni,j=1
are all positive, for any positive integer n.
The set of all completely positive linear maps from A to B is denoted by
CP∞ (A, B).
An n × n matrix [ρij ]ni,j=1 of linear maps from A to B can be regarded as a linear
map ρ from Mn (A) to Mn (B) defined by
ρ([aij ]ni,j=1 ) = [ρij (aij )]ni,j=1 .
2000 Mathematics Subject Classification: 46L05; 46L08
Keywords: Hilbert module, C ∗ -algebra, completely n -positive linear map, extreme points
This research was supported by CNCSIS grant code A 1065/2006.
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114
M. Joiţa, T.-L. Costache and M. Zamfir
We say that [ρij ]ni,j=1 is a completely n-positive linear map from A to B if ρ is a
completely positive linear map from Mn (A) to Mn (B).
The set of all completely n-positive linear maps from A to B is denoted by
n (A, B).
CP∞
In [8], Suen showed that any completely n-positive linear map from a C ∗ -algebra
A to L(H), the C ∗ -algebra of all bounded linear operators on a Hilbert space H, is
of the form [V ∗ Tij Φ(·)V ]ni,j=1 , where Φ is a representation of A on a Hilbert space K,
V ∈ L(H, K) and [Tij ]ni,j=1 is a positive element in Mn (Φ(A)0 ) (Φ(A)0 denotes the
commutant of Φ(A) in L(K)). In [4] we characterized the order relation on the set of
all completely n-positive linear maps from A to L(H) in terms of the representation
associated with each completely n-positive linear map by Suen’s construction.
Hilbert C ∗ -modules are generalizations of Hilbert spaces by allowing the inner
product to take values in a C ∗ -algebra rather than in the field of complex numbers.
A Hilbert A-module is a complex vector space E which is also a right A-module,
compatible with the complex algebra structure, equipped with an A-valued inner
product h·, ·i : E × E → A which is C -and A-linear in its second variable and
satisfies the following relations:
1. hξ, ηi∗ = hη, ξi for every ξ, η ∈ E;
2. hξ, ξi ≥ 0 for every ξ ∈ E;
3. hξ, ξi = 0 if and only if ξ = 0,
and which is complete
with respect to the topology determined by the norm
p
k·k given by kξk = khξ, ξik.
Given two Hilbert C ∗ -modules E and F over a C ∗ -algebra A, the Banach space
of all bounded module morphisms from E to F is denoted by BA (E, F ). The subset
of BA (E, F ) consisting of all adjointable module morphisms from E to F (that is,
T ∈ BA (E, F ) such that there is T ∗ ∈ BA (F, E) satisfying hη, T ξi = hT ∗ η, ξi for all
ξ ∈ E and for all η ∈ F ) is denoted by LA (E, F ). We will write BA (E) for BA (E, E)
and LA (E) for LA (E, E).
In general BA (E, F ) 6= LA (E, F ). So the theory of Hilbert C ∗ -modules is different from the theory of Hilbert spaces.
Any C ∗ -algebra A is a Hilbert C ∗ -module over A with the inner product defined
by ha, bi = a∗ b and the C ∗ -algebra of all adjointable module morphisms on A is
isomorphic with the multiplier algebra M (A) of A (see, for example, [5]).
The Banach space E ] of all bounded module morphisms from E to A becomes
a right A-module with the action of A on E ] defined by (aT )(ξ) = a∗ (T ξ) for all
a ∈ A, T ∈ E ] and ξ ∈ E. We say that E is self-dual if E ] = E as right A-modules.
If E and F are self-dual, then BA (E, F ) = LA (E, F ) [7, Proposition 3.4].
Suppose that A is a W ∗ -algebra. Then the A-valued inner product on E extends
to an A-valued inner product on E ] and in this way E ] becomes a self-dual Hilbert
A-module [7, Theorem 3.2].
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Surveys in Mathematics and its Applications 2 (2007), 113 – 122
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115
Order relation on the set of completely n-positive
Let E be a Hilbert C ∗ -module over a C ∗ -algebra A. The algebraic tensor product
E⊗alg A∗∗ , where A∗∗ is the enveloping W ∗ -algebra of A, becomes a right A∗∗ -module
if we define (ξ ⊗ b) c = ξ ⊗ bc, for ξ ∈ E and b, c ∈ A∗∗ .
The map [·, ·] : (E ⊗alg A∗∗ ) × (E ⊗alg A∗∗ ) → A∗∗ defined by


n
m
n X
m
X
X
X

ξi ⊗ bi ,
ηj ⊗ cj  =
b∗i hξi , ηj i cj
i=1
j=1
i=1 j=1
is an A∗∗ -valued inner product on E ⊗alg A∗∗ and the quotient module (E ⊗alg
A∗∗ )/NE , where NE = {ζ ∈ E ⊗alg B ∗∗ ; [ζ, ζ] = 0}, becomes a pre-Hilbert A∗∗ module with the inner product defined by
* n
+
m
n X
m
X
X
X
ξi ⊗ bi + NE ,
ηj ⊗ cj + NE
=
b∗i hξi , ηj i cj .
i=1
j=1
A∗∗
i=1 j=1
The Hilbert C ∗ -module E ⊗alg A∗∗ /NE obtained by the completion of (E⊗alg A∗∗ )/NE
with respect to the norm induced by the inner product h·, ·iA∗∗ is called the extension
of E by the C ∗ -algebra A∗∗ . Moreover, E can be regarded as an A-submodule of
E ⊗alg A∗∗ /NE , since the map ξ 7→ ξ ⊗ 1 + NE from E to E ⊗alg B ∗∗ /NE is an iso
#
metric inclusion. The self-dual Hilbert A∗∗ -module E ⊗alg B ∗∗ /NE
is denoted
e and we can consider E as embedded in E
e without making distinction [6, 7, 9].
by E,
If T ∈ BA (E, F ), then T extends uniquely to a bounded module morphism
Tb from E ⊗alg A∗∗ /NE to F ⊗alg A∗∗ /NF such that
!
m
m
X
X
Tb
ξi ⊗ bi + NE =
T ξi ⊗ bi + NF
i=1
and kT k =
Tb
i=1
and, by [7, Proposition 3.6], Tb extends uniquely to a bounded
e to Fe such that kT k = Te . Moreover, TfS = TeSe for
module morphism Te from E
f∗ = Te∗ .
all T ∈ BA (E, F ) and S ∈ BA (F, E), and if T ∈ LA (E, F ), then T
A representation of a C ∗ -algebra A on a Hilbert C ∗ -module E over B is a ∗morphism Φ from A to LB (E). Moreover, any representation Φ of a C ∗ -algebra A
e of A on
on a Hilbert C ∗ -module E over a C ∗ -algebra B induces a representation Φ
]
e defined by Φ
e (a) = Φ (a) for all a ∈ A.
E
A completely n-positive linear map ρ = [ρij ]ni,j=1 from A to LB (E), where E is a
Hilbert module over a C ∗ -algebra B, is nondegenerate if for some approximate unit
{eλ }λ∈Λ for A, the nets {ρii (eλ ) ξ}λ∈Λ , i ∈ {1, ..., n} are convergent to ξ for all ξ ∈ E
[5]. Using the theory of Hilbert C ∗ -modules, in [3] we extended the construction of
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116
M. Joiţa, T.-L. Costache and M. Zamfir
Suen for unital completely n-positive linear maps between unital C ∗ -algebras. Thus,
we showed that any completely n-positive linear map ρ = [ρij ]ni,j=1 from A to LB (E)
in
h ∗
fρ (·)V
fρ
fρ T ρ Φ
, where Φρ is a representation of
is of the form [ρij (·)]ni,j=1 = V
ij
E j,i=1
h in
is
A on a Hilbert B-module Eρ , Vρ is an element in LB (E, Eρ ) and T ρ = Tijρ
j,i=1
∗
fρ (A)0 with the property that V
f f ∈ LB (E) for
fρ T ρ Φ
an element in Mn Φ
ij ρ (a)Vρ
E
all a ∈ A and for all i, j ∈ {1, 2, ..., n}. Moreover, {Φρ (a) Vρ ξ; a ∈ A, ξ ∈ E} spans
n
P
a dense submodule of Eρ ,
Tiiρ = nidEfρ and hT ρ ((ηk )nk=1 ) , (ηk )nk=1 iB ∗∗ ≥ 0 for
i=1
all η1 , ..., ηn ∈ Eρ [3, Theorem 2.2]. The quadruple (Φρ , Vρ , Eρ , T ρ ) is unique up to
unitary equivalence. From the proof of [3, Theorem 2.2], we deduce that (Φρ , Vρ , Eρ )
is the KSGNS (Kasparov, Stinespring, Gel’fand, Naimark, Segal) construction [5,
n
P
Theorem 5.6] associated with the unital completely positive linear map ρe = n1
i=1
ρii . If the completely n-positive linear map ρ = [ρij ]ni,j=1 is nondegenerate, then ρe is
nondegenerate and it is not difficult to check that the above construction associated
with a unital completely n-positive linear map is still valid for nondegenerate completely n-positive linear maps. In this paper we characterize the order relation on
the set of all nondegenerate completely n-positive linear maps between C ∗ -algebras
in terms of a self-dual Hilbert module induced by each completely n-positive linear
map.
2
The main results
e we denote by T | the restriction of the map T on
For an element T ∈ LB ∗∗ (E)
E
E.
n (A, L (E)). We denote by C(ρ) the C ∗ -subalgebra of M (L ∗∗
Let ρ ∈ CP∞
n
B
B
∗
n
f
f
f
f
f
∗∗
(Eρ )) generated by {S = [Sij ]i,j=1 ∈ Mn (LB (Eρ )); Vρ Sij Φρ (a)Vρ
∈ LB (E),
E
fρ (a) = Φ
fρ (a)Sij , ∀a ∈ A, ∀i, j ∈ {1, . . . n}}.
Sij Φ
Lemma 1. Let S = [Sij ]ni,j=1 be an element in C(ρ) such that
hS ((ξi )ni=1 ) , (ξi )ni=1 iB ∗∗ ≥ 0
for all ξ1, ..., ξn in Eρ .Then the map ρS = [(ρS )ij ]ni,j=1 from Mn (A) to Mn (LB (E))
defined by
h ∗
in
fρ Sij Φ
fρ (aij )V
fρ
ρS ([aij ]ni,j=1 ) = V
E i,j=1
is a completely n-positive linear map from A to LB (E).
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117
Order relation on the set of completely n-positive
Proof. It is not difficult to see that ρS is an n × n matrix of continuous linear maps
from A to LB (E), the (i, j)-entry of the matrix ρS is the linear map (ρS )ij from A
to LB (E) defined by
fρ (a)V
fρ
fρ ∗ Sij Φ
(ρS )ij (a) = V
E
To show that ρS is a completely n-positive linear map from A to LB (E) it is
sufficient, by [2, Theorem 1.4], to prove that Γ(ρS ) ∈ CP∞ (A, Mn (LB (E))), where
n (A, L (E)) onto CP (A, M (L (E))) defined by
Γ is the isomorphism from CP∞
∞
n
B
B
Γ([ρij ]ni,j=1 )(a) = [ρij (a)]ni,j=1
for all a ∈ A.
Let a and b be two elements in A and let ξ1, ..., ξn be n elements in E. We have
in
fρ ∗ Sij Φ
fρ (a∗ b)V
fρ
V
((ξi )ni=1 )
E i,j=1
h ∗
in
fρ Sij Φ
fρ (a∗ b)V
fρ
= V
((ξi )ni=1 )
i,j=1
h ∗
in
fρ Sij Φ
fρ (a∗ )Φ
fρ (b)V
fρ
= V
((ξi )ni=1 )
i,j=1
h ∗
in
fρ
fρ Φ
fρ (a)∗ Sij Φ
fρ (b)V
= V
((ξi )ni=1 )
Γ(ρS )(a∗ b) ((ξi )ni=1 ) =
h
i,j=1
= (Ma )∗ SMb ((ξi )ni=1 ) ,
where



Ma = 

fρ (a)V
fρ
Φ
0
fρ
f
0
Φρ (a)V
·
·
0
0
...
0
...
0
...
·
f
fρ
... Φρ (a)V





Let a1 , . . . , am be m elements in A, let T1 , . . . , Tm be m elements in Mn (LB (E)) and
let ξ1 , ..., ξn be n elements in E. We have
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118
M. Joiţa, T.-L. Costache and M. Zamfir
*
=
=
=
n
X
+
(Tl∗ Γ(ρS )(a∗l ak )Tk ) ((ξi )ni=1 ) , ((ξi )ni=1 )
k,l=1
n
X
h((Mal )∗ SMak Tk ) ((ξi )ni=1 ) , Tl ((ξi )ni=1 )i
k,l=1
n
X
h((Mal )∗ SMak Tk ) ((ξi )ni=1 ) , Tl ((ξi )ni=1 )iB ∗∗
k,l=1
n D
X
E
fk ((ξi )n ) , Tel ((ξi )n )
(Mal )∗ SMak T
i=1
i=1
k,l=1
*
=
S
n
X
!
fk
Ma k T
((ξi )ni=1 ) ,
k=1
n
X
!
Mal Tel
l=1
B ∗∗
+
((ξi )ni=1 )
≥0
B ∗∗
since S = [Sij ]ni,j=1 is an element in C(ρ) such that hS ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0 for
all η1 , ..., ηn in Eρ . This implies that Γ(ρS ) ∈ CP∞ (A, Mn (LB (E))) and the lemma
is proved.
Remark 2.
1. By [4, the proof of Theorem 2.2], hT ρ ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0
n (A, L (E)). Moreover, ρ ρ = ρ.
for all η1 , ..., ηn in Eρ , and so ρT ρ ∈ CP∞
B
T
2. If S1 and S2 are two elements in C(ρ) such that hSk ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0
for all η1 , ..., ηn in Eρ , and for all k = 1, 2, then ρS1 +S2 = ρS1 + ρS2 .
3. If S is an element in C(ρ) such that hS ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0 for all
η1 , ..., ηn in Eρ , and α is a positive number, then ραS = αρS .
4. If S1 and S2 are two elements in C(ρ) such that
0 ≤ hS1 ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≤ hS2 ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗
for all η1 , ..., ηn in Eρ , then ρS1 ≤ ρS2 .
n (A, L (E)). We denote by [0, ρ] the set of all completely
Let ρ = [ρij ]ni,j=1 ∈ CP∞
B
n-positive linear maps θ = [θij ]ni,j=1 from A to LB (E) such that θ ≤ ρ (that is,
n (A, L (E))) and by [0, T ρ ] the set of all elements S = [S ]n
ρ − θ ∈ CP∞
ij i,j=1 in C(ρ)
B
such that
h(T ρ − S) ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0
for all η1 , ..., ηn in Eρ .
The following theorem gives a characterization of the order relation on the set of
completely n-positive linear maps between C ∗ -algebras in terms of the representation
associated to each completely n-positive linear map by [3, Theorem 2.2].
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119
Order relation on the set of completely n-positive
Theorem 3. The map S → ρS from [0, T ρ ] to [0, ρ] is an affine order isomorphism.
Proof. By Lemma 1 and Remark 2, we get that the map S → ρS from [0, Tρ ] to
[0, ρ] is well-defined, affine and preserves the order relation.
We show that this map is injective. Let S = [Sij ]ni,j=1 ∈ [0, T ρ ] such that ρS = 0.
fρ (a)V
fρ = 0 for all a ∈ A and i, j ∈ {1, . . . , n}.
fρ ∗ Sij Φ
Then by definition of ρS , V
E
Let i, j ∈ {1, . . . , n}. Then we have
hSij Φρ (a)Vρ ξ, Φρ (b)Vρ ηiB ∗∗
=
D
=
D
fρ (a)V
fρ ξ, Φ
fρ (b)V
fρ η
Sij Φ
E
B ∗∗
∗
E
∗
EB
fρ Φ
fρ (b)∗ Sij Φ
fρ (a)V
fρ ξ, η
V
D
∗∗
fρ (b)∗ Φ
fρ (a)V
fρ ξ, η
fρ Sij Φ
V
B ∗∗
E
D ∗
fρ (b∗ a)V
fρ ξ, η
fρ Sij Φ
=0
V
=
∗∗
=
B
for all a, b ∈ A and for all ξ, η ∈ E. From this fact and taking into account that
{Φρ (a)Vρ ξ; a ∈ A, ξ ∈ E} spans a dense submodule of Eρ and Sij |E is a bounded
module morphism from E to E # [2, Remark 2.2], we deduce that Sij |E = 0 and
then, by [9, pp. 442-443], Sij = 0, since Sij |E is a bounded module morphism from
E to E # and since Sij |E = 0. Therefore S = 0.
We prove now that the map S → ρS is surjective. Let σ = [σkl ]nk,l=1 be an
n
P
n (A, L (E)), where ρ
element in [0, ρ]. Then ρe − σ
e ∈ CP∞
e = n1
ρjj and σ
e =
B
j=1
1
n
n
P
σjj . Let (Φρ , Eρ , Vρ
, T ρ)
and Φθ , Eθ , Vθ
,Tθ
be the constructions associated
j=1
with ρ respectively θ by [3, Theorem 2.2]. Then (Φρ , Eρ , Vρ ) and (Φθ , Eθ , Vθ ) are
the KSGNS constructions associated with ρe respectively σ
e [5, Theorem 5.6]. Since
n
ρe − σ
e ∈ CP∞ (A, LB (E)), there is a bounded module morphism W : Eρ → Eθ such
that
W (Φρ (a) Vρ ξ) = Φθ (a) Vθ ξ
for all a ∈ A and for all ξ ∈ E. Moreover, W Φρ (a) =hΦθ (a) W ifor all a ∈ A,
n
f∗T θ W
f
and W Vρ = Vθ [2, the proof of Theorem 2.6]. Let S = W
. Clearly,
ij
i,j=1
fρ (A)0 ). From
S ∈ Mn (Φ
fρ ∗ Sij Φ
fρ (a)V
fρ
V
E
=
=
fρ ∗ W
f∗T θ W
fΦ
fρ (a)V
fρ
V
ij
E
∗
fθ (a)V
fθ
fθ Tijθ Φ
V
E
h ∗
in
fθ T θ Φ
f f
for all i, j ∈ {1, ..., n} and taking into account that V
ij θ (a)Vθ
E i,j=1
∈ Mn (LB (E)),
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120
M. Joiţa, T.-L. Costache and M. Zamfir
in
h ∗
fρ (a)V
fρ
fρ Sij Φ
∈ Mn (LB (E)). Therefore, S ∈ C(ρ). Morewe deduce that V
E i,j=1
over,
hS ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0
for all η1 , ..., ηn ∈ Eρ , since
=
=
=
=
S (Φρ (ak ) Vρ ξk )nk=1 , (Φρ (ak ) Vρ ξk )nk=1 B ∗∗
h
in
n
n
f∗T θ W
fΦ
fρ (aj ) V
fρ
)
,
(Φ
(a
)
V
ξ
)
((ξ
)
W
ρ
ρ k k=1
k
k k=1
ij
i,j=1
∗∗
B
h
in
∗
fρ
f ∗ Tijθ W
fΦ
fρ (aj ) V
fρ Φ
fρ (ai )∗ W
(ξk )nk=1 , (ξk )nk=1
V
i,j=1
B ∗∗
h
i
n
fθ ∗ Φ
fθ (ai )∗ Tijθ Φ
fθ (aj ) V
fθ
V
(ξk )nk=1 , (ξk )nk=1
i,j=1
∗∗
h i
B
n
Tijθ
((Φθ (ak ) Vθ ξk )nk=1 ) , (Φθ (ak ) Vθ ξk )nk=1
≥0
i,j=1
B ∗∗
for all a1 , ..., an ∈ A and for all ξ1 , ..., ξn ∈ E, and since {Φρ (a) Vρ ξ; a ∈ A, ξ ∈ E}
spans a dense submodule of Eρ . From
(ρS )ij (a) =
fρ ∗ Sij Φ
fρ (a) V
fρ
V
E
=
fρ ∗ W
f∗T θ W
fΦ
fρ (a) V
fρ
V
ij
=
fθ ∗ Tijθ Φ
fθ (a) V
fθ
V
E
E
= σij (a)
for all i, j ∈ {1, 2, ..., n} and for all a ∈ A, we deduce that σ = ρS . To proof the
surjectivity of the map S → ρS it remained to show that
h(T ρ − S) ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≥ 0
for all η1 , ..., ηn in Eρ . We have
=
=
=
≤
=
=
S (Φρ (ak ) Vρ ξk )nk=1 , (Φρ (ak ) Vρ ξk )nk=1 B ∗∗
h
in
∗
∗
n
n
f
f
f
f
Vρ Φρ (ai ) Sij Φρ (aj ) Vρ
((ξk )k=1 ) , (ξk )k=1
i,j=1
B ∗∗
h
i
n
fρ ∗ Sij Φ
fρ (a∗ aj ) V
fρ
V
((ξk )nk=1 ) , (ξk )nk=1
i
i,j=1
B ∗∗
D E
n
n
n
σ [a∗i aj ]i,j=1 ((ξk )k=1 ) , (ξk )k=1 ∗∗
D EB
n
n
n
∗
ρ [ai aj ]i,j=1 ((ξk )k=1 ) , (ξk )k=1 ∗∗
B
h
in
∗
ρ
n
∗
n
f
f
f
f
Vρ Φρ (ai ) Tij Φρ (aj ) Vρ
((ξk )k=1 ) , (ξk )k=1
i,j=1
∗∗
h i
B
n
Tijρ
(Φρ (ak ) Vρ ξk )nk=1 , (Φρ (ak ) Vρ ξk )nk=1
i,j=1
B ∗∗
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Order relation on the set of completely n-positive
for all a1 , ..., an in A and for all ξ1 , ..., ξn ∈ E. From this fact and taking into account
that {Φρ (a) Vρ ξ; a ∈ A and ξ ∈ E} spans a dense subspace in Eρ , we deduce that
hS ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗ ≤ hT ρ ((ηi )ni=1 ) , (ηi )ni=1 iB ∗∗
for all η1 , ..., ηn in Eρ , and thus the theorem is proved.
Definition 4. Let A and B be two C ∗ -algebras and let E be a Hilbert C ∗ -module
over B. A completely n-positive linear map ρ = [ρij ]ni,j=1 from A to LB (E) is said
to be pure if for every completely n-positive linear map θ = [θij ]ni,j=1 ∈ [0, ρ], there
is a positive number α such that θ = αρ.
n (A, L (E)). Then ρ is pure
Corollary 5. Let ρ = [ρij ]ni,j=1 be an element in CP∞
B
ρ
ρ
if and only if [0, T ] = {αT ; 0 ≤ α ≤ 1}.
Let A be a unital C ∗ -algebra, let B be a C ∗ -algebra and let E be a Hilbert
n (A, L (E), T 0 ), where T 0 = [V
fρ ∗ T ρ V
fn
B-module. We denote by CP∞
B
ij ρ ]i,j=1 , the set
of all completely n-positive linear maps ρ = [ρij ]ni,j=1 from A to LB (E) such that
ρ(In ) = T 0 , where In is an n × n matrix with all the entries equal to 1A , the unity
of A.
n (A, L (E), T 0 ). If the map S = [S ]n
Proposition 6. Let ρ = [ρij ]ni,j=1 ∈ CP∞
ij i,j=1 −→
B
∗
fρ Sij V
fρ ]n
e
[V
i,j=1 from C(ρ) to Mn (LB ∗∗ (E)) is injective, then ρ is an extreme point
0
n
in the set CP∞ (A, LB (E), T ).
n (A, L (E), T 0 ) and α ∈ (0, 1) such
Proof. Let θ = [θij ]ni,j=1 , σ = [σij ]ni,j=1 ∈ CP∞
B
that αθ + (1 − α) σ = ρ. Since αθ ∈ [0, ρ], by Theorem 1, there is an element
1 ]n
ρ
S1 = [Sij
i,j=1 ∈ [0, T ] such that αθ = ρS1 . Then
fρ ∗ (S 1 − αT ρ )V
f
V
ij
ij ρ
E
=
fρ ∗ S 1 V
f
V
ij ρ
E
fρ ∗ T ρ V
f
− αV
ij ρ
E
= (ρS1 )ij (1A ) − αρij (1A )
= αθij (1A ) − αρij (1A )
= αTij0 − αTij0 = 0
fρ ∗ (S 1 − αT ρ )V
f
From this fact and taking into account that V
ij
ij ρ
LB (E), we
is an element in
E
fρ (S 1 − αT ρ )V
fρ = 0 for all i, j ∈ {1, 2, . . . n} and since
conclude that V
ij
ij
ρ
fρ ∗ Sij V
fρ ]n
= [Sij ]ni,j=1 −→ [V
i,j=1 is injective, S1 = αT . Thus we showed
∗
the map S
that θ = ρ. In the same way we prove that σ = ρ and so ρ is an extreme point in
n (A, L (E), T 0 ).
CP∞
B
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Surveys in Mathematics and its Applications 2 (2007), 113 – 122
http://www.utgjiu.ro/math/sma
122
M. Joiţa, T.-L. Costache and M. Zamfir
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Maria Joiţa
Tania- Luminita Costache
Department of Mathematics,
Faculty of Applied Sciences,
Faculty of Chemistry,
University ”Politehnica”,
University of Bucharest,
Bucharest,
Romania.
Romania.
e-mail: mjoita@fmi.unibuc.ro
e-mail: lumycos@yahoo.com
Mariana Zamfir
Department of Mathematics and Informatics
Technical University of Civil Engineering Bucharest
Romania.
e-mail: zacos@k.ro
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Surveys in Mathematics and its Applications 2 (2007), 113 – 122
http://www.utgjiu.ro/math/sma