1Introductionandpreliminaries MariaJoit¸a,Tania-Luminit¸aCostacheandMarianaZamfir C -ALGEBRAS n -POSITIVELINEARMAPSBETWEEN CHARACTERIZATIONOFTHEORDERRELATIONONTHESETOFCOMPLETELY SurveysinMathematicsanditsApplications

ISSN1842-6298 (electronic), 1843 - 7265 (print) Volume2(2007), 113 – 122

CHARACTERIZATION OF THE ORDER RELATION ON THE SET OF COMPLETELY n-POSITIVE

LINEAR MAPS BETWEEN C

-ALGEBRAS

Maria Joit¸a, Tania-Luminit¸a Costache and Mariana Zamfir

Abstract. In this paper we characterize the order relation on the set of all nondegenerate com- pletelyn-positive linear maps betweenC^∗-algebras in terms of a self-dual Hilbert module induced by each completelyn-positive linear map.

1 Introduction and preliminaries

Completely positive linear maps are an often used tool in operator algebras the- ory 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 M_n(A) the C^∗- algebra of alln×nmatrices overA with the algebraic operations and the topology obtained by regarding it as a direct sum ofn² copies of A.

LetA and B be twoC^∗-algebras. A linear map ρ:A→B is completely positive if the linear mapsρ⁽ⁿ⁾:Mn(A)→Mn(B) defined by

ρ⁽ⁿ⁾([a_ij]ⁿ_i,j=1) = [ρ(a_ij)]ⁿ_i,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).

Ann×nmatrix [ρij]ⁿ_i,j=1 of linear maps fromAtoB can be regarded as a linear mapρ from M_n(A) to M_n(B) defined by

ρ([aij]ⁿ_i,j=1) = [ρij(aij)]ⁿ_i,j=1.

2000 Mathematics Subject Classification: 46L05; 46L08

Keywords: Hilbert module,C^∗-algebra, completelyn-positive linear map, extreme points This research was supported by CNCSIS grant code A 1065/2006.

We say that [ρ_ij]ⁿ_i,j=1 is a completely n-positive linear map from A to B if ρ is a completely positive linear map fromM_n(A) to M_n(B).

The set of all completely n-positive linear maps from A to B is denoted by CP_∞ⁿ(A, B).

In [8], Suen showed that any completelyn-positive linear map from aC^∗-algebra AtoL(H), theC^∗-algebra of all bounded linear operators on a Hilbert space H, is of the form [V^∗T_ijΦ(·)V]ⁿ_i,j=1, where Φ is a representation ofAon a Hilbert spaceK, V ∈L(H, K) and [T_ij]ⁿ_i,j=1 is a positive element in M_n(Φ(A)⁰) (Φ(A)⁰ denotes the commutant of Φ(A) inL(K)). In [4] we characterized the order relation on the set of all completelyn-positive linear maps fromAtoL(H) in terms of the representation associated with each completelyn-positive linear map by Suen’s construction.

Hilbert C^∗-modules are generalizations of Hilbert spaces by allowing the inner product to take values in aC^∗-algebra rather than in the field of complex numbers.

AHilbert A-moduleis a complex vector spaceE which is also a rightA-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, ξifor 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 k·kgiven by kξk=p

khξ, ξik.

Given two Hilbert C^∗-modulesE and F over a C^∗-algebraA, the Banach space of all bounded module morphisms fromE toF is denoted byB_A(E, F). The subset of B_A(E, F) consisting of all adjointable module morphisms from E toF (that is, T ∈BA(E, F) such that there is T^∗∈BA(F, E) satisfying hη, T ξi=hT^∗η, ξi for all ξ∈Eand for allη∈F) is denoted byLA(E, F). We will writeBA(E) forBA(E, E) and L_A(E) for L_A(E, E).

In general BA(E, F)6=LA(E, F). So the theory of HilbertC^∗-modules is differ- ent from the theory of Hilbert spaces.

AnyC^∗-algebraAis a HilbertC^∗-module overAwith 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) ofA (see, for example, [5]).

The Banach space E^] of all bounded module morphisms from E toA 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 thatE is self-dual ifE^]=E as rightA-modules.

If E and F are self-dual, thenB_A(E, F) =L_A(E, F) [7, Proposition 3.4].

Suppose thatAis aW^∗-algebra. Then theA-valued inner product onE extends to anA-valued inner product onE^] and in this way E^] becomes a self-dual Hilbert A-module [7, Theorem 3.2].

LetEbe a HilbertC^∗-module over aC^∗-algebraA. The algebraic tensor product E⊗_algA^∗∗,whereA^∗∗ is the envelopingW^∗-algebra ofA, becomes a rightA^∗∗-module if we define (ξ⊗b)c=ξ⊗bc,for ξ∈E and b, c∈A^∗∗.

The map [·,·] : (E⊗_algA^∗∗)×(E⊗_algA^∗∗)→A^∗∗ defined by





n

X

i=1

ξi⊗bi,

m

X

j=1

ηj⊗cj



=

n

X

i=1 m

X

j=1

b^∗_i hξ_i, ηjicj

is an A^∗∗-valued inner product on E ⊗_alg A^∗∗ and the quotient module (E ⊗_alg A^∗∗)/N_E, where N_E = {ζ ∈ E ⊗_algB^∗∗; [ζ, ζ] = 0}, becomes a pre-Hilbert A^∗∗- module with the inner product defined by

* _n X

i=1

ξ_i⊗b_i+N_E,

m

X

j=1

η_j⊗c_j +N_E +

A^∗∗

=

n

X

i=1 m

X

j=1

b^∗_i hξ_i, η_jic_j.

The HilbertC^∗-moduleE⊗_algA^∗∗/N_Eobtained by the completion of (E⊗_algA^∗∗)/N_E with respect to the norm induced by the inner producth·,·i_A∗∗ is called the extension of E by the C^∗-algebra A^∗∗. Moreover, E can be regarded as an A-submodule of E⊗_algA^∗∗/N_E, since the map ξ7→ξ⊗1 +N_E from E toE⊗_algB^∗∗/N_E is an iso- metric inclusion. The self-dual Hilbert A^∗∗-module

E⊗_algB^∗∗/N_E#

is denoted byE, and we can considere E as embedded inEe without making distinction [6,7,9].

If T ∈ B_A(E, F), then T extends uniquely to a bounded module morphism Tbfrom E⊗_algA^∗∗/NE toF ⊗_algA^∗∗/NF such that

Tb

m

X

i=1

ξ_i⊗b_i+N_E

!

=

m

X

i=1

T ξ_i⊗b_i+N_F

and kTk = Tb

and, by [7, Proposition 3.6], Tb extends uniquely to a bounded module morphismTe from Ee to Fe such that kTk =

Te

. Moreover, T Sf = TeSe for all T ∈BA(E, F) andS ∈BA(F, E), and if T ∈LA(E, F), then Tf^∗=Te^∗.

A representation of a C^∗-algebra A on a Hilbert C^∗-module E over B is a ∗- morphism Φ from A toL_B(E). Moreover, any representation Φ of a C^∗-algebra A on a Hilbert C^∗-moduleE over aC^∗-algebra B induces a representationΦ ofe A on Ee defined byΦ (a) =e Φ (a) for all] a∈A.

A completelyn-positive linear mapρ= [ρ_ij]ⁿ_i,j=1 fromAtoL_B(E), whereEis a Hilbert module over aC^∗-algebraB, is nondegenerate if for some approximate unit {e_λ}_λ∈ΛforA, the nets{ρ_ii(e_λ)ξ}_λ∈Λ, i∈ {1, ..., n}are convergent toξ for allξ ∈E [5]. Using the theory of HilbertC^∗-modules, in [3] we extended the construction of

Suen for unital completelyn-positive linear maps between unitalC^∗-algebras. Thus, we showed that any completelyn-positive linear mapρ= [ρ_ij]ⁿ_i,j=1 fromAtoL_B(E) is of the form [ρ_ij(·)]ⁿ_i,j=1=h

fV_ρ^∗T_ij^ρΦf_ρ(·)fV_ρ E

in

j,i=1, where Φ_ρ is a representation of A on a HilbertB-module E_ρ,V_ρ is an element inL_B(E, E_ρ) and T^ρ=h

T_ij^ρin j,i=1 is an element in M_n

Φf_ρ(A)⁰

with the property that fV_ρ^∗T_ij^ρΦf_ρ(a)fV_ρ E

∈ L_B(E) for all a∈A and for all i, j ∈ {1,2, ..., n}. Moreover, {Φ_ρ(a)V_ρξ;a∈ A, ξ∈ E} spans a dense submodule of E_ρ,

Pn i=1

T_ii^ρ = nid_Ef

ρ and hT^ρ((η_k)ⁿ_k=1),(η_k)ⁿ_k=1i_B∗∗ ≥ 0 for all η₁, ..., η_n ∈E_ρ [3, Theorem 2.2]. The quadruple (Φ_ρ, V_ρ, E_ρ, T^ρ) is unique up to unitary equivalence. From the proof of [3, Theorem 2.2], we deduce that (Φ_ρ, V_ρ, E_ρ) is the KSGNS (Kasparov, Stinespring, Gel’fand, Naimark, Segal) construction [5, Theorem 5.6] associated with the unital completely positive linear map ρe= _n¹

n

P

i=1

ρ_ii. If the completelyn-positive linear mapρ= [ρ_ij]ⁿ_i,j=1 is nondegenerate, then ρeis 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 com- pletely n-positive linear maps. In this paper we characterize the order relation on the set of all nondegenerate completely n-positive linear maps betweenC^∗-algebras in terms of a self-dual Hilbert module induced by each completelyn-positive linear map.

2 The main results

For an elementT ∈L_B^∗∗(E) we denote bye T|_E the restriction of the mapT on E.

Let ρ ∈ CP_∞ⁿ(A, L_B(E)). We denote by C(ρ) the C^∗-subalgebra of Mn(L_B^∗∗

(Ef_ρ)) generated by {S = [S_ij]ⁿ_i,j=1 ∈ M_n(L_B^∗∗(Ef_ρ));fV_ρ^∗S_ijΦf_ρ(a)fV_ρ E

∈ L_B(E), SijΦfρ(a) =Φfρ(a)Sij,∀a∈A,∀i, j∈ {1, . . . n}}.

Lemma 1. Let S= [Sij]ⁿ_i,j=1 be an element in C(ρ) such that hS((ξi)ⁿ_i=1),(ξi)ⁿ_i=1i_B∗∗ ≥0

for all ξ1,..., ξn in Eρ.Then the map ρS = [(ρS)ij]ⁿ_i,j=1 from Mn(A) to Mn(LB(E)) defined by

ρ_S([a_ij]ⁿ_i,j=1) =h

fV_ρ^∗S_ijΦf_ρ(a_ij)fV_ρ E

in i,j=1

is a completely n-positive linear map from A toL_B(E).

Proof. It is not difficult to see thatρ_S is an n×nmatrix of continuous linear maps from A toL_B(E), the (i, j)-entry of the matrix ρ_S is the linear map (ρ_S)_ij from A toLB(E) defined by

(ρS)ij(a) = Vfρ

∗SijΦfρ(a)fVρ

E

To show that ρ_S is a completely n-positive linear map from A to L_B(E) it is sufficient, by [2, Theorem 1.4], to prove that Γ(ρ_S) ∈CP∞(A, M_n(L_B(E))), where Γ is the isomorphism from CP_∞ⁿ(A, LB(E)) ontoCP∞(A, Mn(LB(E))) defined by

Γ([ρij]ⁿ_i,j=1)(a) = [ρij(a)]ⁿ_i,j=1

for all a∈A.

Let aandb be two elements inA and letξ_1,..., ξ_n ben elements inE. We have

Γ(ρ_S)(a^∗b) ((ξ_i)ⁿ_i=1) = h

fV_ρ^∗S_ijΦf_ρ(a^∗b)fV_ρ E

in i,j=1

((ξ_i)ⁿ_i=1)

= h

fV_ρ^∗S_ijΦf_ρ(a^∗b)fV_ρin

i,j=1((ξ_i)ⁿ_i=1)

= h

fVρ

∗SijΦfρ(a^∗)Φfρ(b)Vfρ

in

i,j=1((ξi)ⁿ_i=1)

= h

fVρ

∗

Φfρ(a)^∗SijΦfρ(b)Vfρ

in

i,j=1((ξi)ⁿ_i=1)

= (M_a)^∗SM_b((ξ_i)ⁿ_i=1),

where

M_a=











Φfρ(a)Vfρ 0 ... 0 0 Φf_ρ(a)fV_ρ ... 0

· · ... ·

0 0 ... Φf_ρ(a)fV_ρ











Leta1, . . . , am bemelements inA, letT1, . . . , Tmbemelements inMn(LB(E)) and letξ₁, ..., ξ_n ben elements inE. We have

* _n X

k,l=1

(T_l^∗Γ(ρS)(a^∗_la_k)T_k) ((ξi)ⁿ_i=1),((ξi)ⁿ_i=1) +

=

n

X

k,l=1

h((M_al)^∗SM_akT_k) ((ξ_i)ⁿ_i=1), T_l((ξ_i)ⁿ_i=1)i

=

n

X

k,l=1

h((M_al)^∗SM_akT_k) ((ξ_i)ⁿ_i=1), T_l((ξ_i)ⁿ_i=1)i_B∗∗

=

n

X

k,l=1

D

(M_al)^∗SM_akfT_k

((ξ_i)ⁿ_i=1),Te_l((ξ_i)ⁿ_i=1)E

B^∗∗

=

* S

n

X

k=1

M_akfT_k

!

((ξ_i)ⁿ_i=1),

n

X

l=1

M_alTe_l

!

((ξ_i)ⁿ_i=1) +

B^∗∗

≥0

sinceS = [S_ij]ⁿ_i,j=1 is an element in C(ρ) such that hS((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≥0 for all η1, ..., ηn inEρ. 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((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≥ 0 for allη1, ..., ηn in Eρ, and soρT^ρ∈CP_∞ⁿ(A, LB(E)). Moreover,ρT^ρ =ρ.

2. If S1 and S2 are two elements inC(ρ) such that hS_k((ηi)ⁿ_i=1),(ηi)ⁿ_i=1i_B∗∗ ≥0 for allη₁, ..., η_n in E_ρ, and for all k= 1,2, then ρ_S1+S2 =ρ_S1 +ρ_S2.

3. If S is an element in C(ρ) such that hS((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≥ 0 for all η1, ..., ηn in Eρ, and α is a positive number, thenραS =αρS.

4. If S1 and S2 are two elements inC(ρ) such that

0≤ hS₁((ηi)ⁿ_i=1),(ηi)ⁿ_i=1i_B∗∗ ≤ hS₂((ηi)ⁿ_i=1),(ηi)ⁿ_i=1i_B∗∗

for allη1, ..., ηn in Eρ, then ρS1 ≤ρS2.

Letρ= [ρij]ⁿ_i,j=1∈CP_∞ⁿ(A, LB(E)). We denote by [0, ρ] the set of all completely n-positive linear maps θ = [θ_ij]ⁿ_i,j=1 from A to L_B(E) such that θ ≤ ρ (that is, ρ−θ∈CP_∞ⁿ(A, L_B(E))) and by [0, T^ρ] the set of all elementsS = [S_ij]ⁿ_i,j=1inC(ρ) such that

h(T^ρ−S) ((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≥0 for all η₁, ..., η_n inE_ρ.

The following theorem gives a characterization of the order relation on the set of completelyn-positive linear maps betweenC^∗-algebras in terms of the representation associated to each completelyn-positive linear map by [3, Theorem 2.2].

Theorem 3. The mapS→ρ_S 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 → ρ_S from [0, T_ρ] to [0, ρ] is well-defined, affine and preserves the order relation.

We show that this map is injective. LetS = [S_ij]ⁿ_i,j=1 ∈[0, T^ρ] such thatρ_S = 0.

Then by definition of ρ_S, fV_ρ^∗S_ijΦf_ρ(a)fV_ρ

E = 0 for all a∈A and i, j∈ {1, . . . , n}.

Let i, j∈ {1, . . . , n}. Then we have hS_ijΦρ(a)Vρξ,Φρ(b)Vρηi_B∗∗ =

D

SijΦfρ(a)fVρξ,Φfρ(b)fVρη E

B^∗∗

= D

fVρ

∗

Φfρ(b)^∗SijΦfρ(a)fVρξ, η E

B^∗∗

= D

fV_ρ^∗S_ijΦf_ρ(b)^∗Φf_ρ(a)fV_ρξ, ηE

B^∗∗

= D

fV_ρ^∗S_ijΦf_ρ(b^∗a)fV_ρξ, ηE

B^∗∗ = 0

for all a, b ∈ A and for all ξ, η ∈ E. From this fact and taking into account that {Φ_ρ(a)V_ρξ;a ∈ A, ξ ∈ E} spans a dense submodule of E_ρ and S_ij|_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 toE^# and since S_ij|_E = 0. ThereforeS = 0.

We prove now that the map S → ρ_S is surjective. Let σ = [σ_kl]ⁿ_k,l=1 be an element in [0, ρ]. Then ρe−eσ ∈ CP_∞ⁿ(A, L_B(E)), where ρe = ¹_n

n

P

j=1

ρ_jj and eσ =

1 n

n

P

j=1

σjj. Let (Φρ, Eρ, Vρ, T^ρ) and Φθ, Eθ, Vθ, T^θ

be the constructions associated with ρ respectively θ by [3, Theorem 2.2]. Then (Φ_ρ, E_ρ, V_ρ) and (Φ_θ, E_θ, V_θ) are the KSGNS constructions associated with ρerespectively eσ [5, Theorem 5.6]. Since ρe−σe∈CP_∞ⁿ (A, LB(E)), there is a bounded module morphismW :Eρ→Eθ such that

W(Φ_ρ(a)V_ρξ) = Φ_θ(a)V_θξ

for all a ∈ A and for all ξ ∈ E. Moreover, W Φρ(a) = Φθ(a)W for all a ∈ A, and W V_ρ = V_θ [2, the proof of Theorem 2.6]. Let S = h

Wf^∗T_ij^θWfin

i,j=1. Clearly, S∈Mn(Φfρ(A)⁰). From

Vf_ρ^∗S_ijΦf_ρ(a)fV_ρ

E = fV_ρ^∗Wf^∗T_ij^θWfΦf_ρ(a)fV_ρ E

= fV_θ^∗T_ij^θΦf_θ(a)fV_θ E

for alli, j∈ {1, ..., n}and taking into account that h

fV_θ^∗T_ij^θΦf_θ(a)fV_θ E

in i,j=1

∈Mn(LB(E)),

we deduce that h

fVρ

∗SijΦfρ(a)fVρ

E

in i,j=1

∈Mn(L_B(E)). Therefore, S∈C(ρ). More- over,

hS((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≥0 for all η₁, ..., η_n∈E_ρ,since

S (Φρ(a_k)Vρξ_k)ⁿ_k=1

,(Φρ(a_k)Vρξ_k)ⁿ_k=1

B^∗∗

= h

Wf^∗T_ij^θWfΦf_ρ(a_j)Vf_ρin

i,j=1((ξ_k)ⁿ_k=1),(Φ_ρ(a_k)V_ρξ_k)ⁿ_k=1

B^∗∗

= h

fV_ρ^∗Φf_ρ(a_i)^∗Wf^∗T_ij^θWfΦf_ρ(a_j)Vf_ρin

i,j=1(ξ_k)ⁿ_k=1,(ξ_k)ⁿ_k=1

B^∗∗

= h

fV_θ^∗Φf_θ(a_i)^∗T_ij^θΦf_θ(a_j)fV_θin

i,j=1(ξ_k)ⁿ_k=1,(ξ_k)ⁿ_k=1

B^∗∗

= h

T_ij^θin

i,j=1((Φ_θ(a_k)V_θξ_k)ⁿ_k=1),(Φ_θ(a_k)V_θξ_k)ⁿ_k=1

B^∗∗

≥0

for all a1, ..., an ∈ A and for all ξ1, ..., ξn ∈E, and since {Φ_ρ(a)Vρξ;a∈ A, ξ∈ E}

spans a dense submodule ofEρ. From

(ρ_S)_ij(a) = fV_ρ^∗S_ijΦf_ρ(a)fV_ρ E

= fV_ρ^∗Wf^∗T_ij^θWfΦf_ρ(a)fV_ρ E

= fVθ

∗T_ij^θΦfθ(a)fVθ

E =σij(a)

for all i, j ∈ {1,2, ..., n} and for all a ∈ A, we deduce that σ = ρ_S. To proof the surjectivity of the map S→ρ_S it remained to show that

h(T^ρ−S) ((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≥0 for all η1, ..., ηn inEρ. We have

S (Φρ(ak)Vρξk)ⁿ_k=1

,(Φρ(ak)Vρξk)ⁿ_k=1

B^∗∗

= h

fVρ

∗

Φfρ(ai)^∗SijΦfρ(aj)fVρ

in

i,j=1((ξk)ⁿ_k=1),(ξk)ⁿ_k=1

B^∗∗

= h

fVρ

∗SijΦfρ(a^∗_iaj)fVρ

in

i,j=1((ξk)ⁿ_k=1),(ξk)ⁿ_k=1

B^∗∗

= D

σ

[a^∗_iaj]ⁿ_i,j=1

((ξ_k)ⁿ_k=1),(ξ_k)ⁿ_k=1 E

B^∗∗

≤ D ρ

[a^∗_iaj]ⁿ_i,j=1

((ξ_k)ⁿ_k=1),(ξ_k)ⁿ_k=1 E

B^∗∗

= h

fV_ρ^∗Φf_ρ(a_i)^∗T_ij^ρΦf_ρ(a_j)fV_ρin

i,j=1((ξ_k)ⁿ_k=1),(ξ_k)ⁿ_k=1

B^∗∗

= h

T_ij^ρin

i,j=1 (Φ_ρ(a_k)V_ρξ_k)ⁿ_k=1

,(Φ_ρ(a_k)V_ρξ_k)ⁿ_k=1

B^∗∗

for alla₁, ..., a_ninAand for allξ₁, ..., ξ_n∈E. From this fact and taking into account that{Φ_ρ(a)V_ρξ;a∈A and ξ∈E} spans a dense subspace in E_ρ, we deduce that

hS((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗ ≤ hT^ρ((η_i)ⁿ_i=1),(η_i)ⁿ_i=1i_B∗∗

for all η₁, ..., η_n inE_ρ, 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]ⁿ_i,j=1 from A to LB(E) is said to be pure if for every completely n-positive linear map θ = [θij]ⁿ_i,j=1 ∈ [0, ρ], there is a positive number α such that θ=αρ.

Corollary 5. Let ρ = [ρij]ⁿ_i,j=1 be an element in CP_∞ⁿ(A, LB(E)). Then ρ is pure if and only if [0, T^ρ] ={αT^ρ; 0≤α ≤1}.

Let A be a unital C^∗-algebra, let B be a C^∗-algebra and let E be a Hilbert B-module. We denote by CP_∞ⁿ(A, L_B(E), T⁰), where T⁰ = [fV_ρ^∗T_ij^ρVf_ρ]ⁿ_i,j=1, the set of all completely n-positive linear maps ρ = [ρij]ⁿ_i,j=1 from A to LB(E) such that ρ(In) =T⁰, where In is an n×nmatrix with all the entries equal to 1A, the unity ofA.

Proposition 6. Letρ= [ρ_ij]ⁿ_i,j=1 ∈CP_∞ⁿ(A, L_B(E), T⁰). If the mapS = [S_ij]ⁿ_i,j=1−→

[Vfρ

∗SijfVρ]ⁿ_i,j=1 from C(ρ) to Mn(LB^∗∗(E))e is injective, then ρ is an extreme point in the set CP_∞ⁿ(A, L_B(E), T⁰).

Proof. Let θ = [θ_ij]ⁿ_i,j=1, σ = [σ_ij]ⁿ_i,j=1 ∈ CP_∞ⁿ(A, L_B(E), T⁰) and α ∈ (0,1) such that αθ + (1−α)σ = ρ. Since αθ ∈ [0, ρ], by Theorem 1, there is an element S1= [S_ij¹]ⁿ_i,j=1 ∈[0, T^ρ] such thatαθ=ρ_S1. Then

Vfρ

∗(S_ij¹ −αT_ij^ρ)fVρ

E = fVρ

∗S_ij¹fVρ

E −αVfρ

∗T_ij^ρfVρ

E

= (ρ_S1)_ij(1_A)−αρ_ij(1_A)

= αθ_ij(1_A)−αρ_ij(1_A)

= αT_ij⁰ −αT_ij⁰ = 0 From this fact and taking into account that fV_ρ^∗(S_ij¹ −αT_ij^ρ)fV_ρ

E is an element in L_B(E), we conclude that fV_ρ^∗(S_ij¹ −αT_ij^ρ)fV_ρ = 0 for alli, j ∈ {1,2, . . . n} and since the map S = [S_ij]ⁿ_i,j=1 −→[fV_ρ^∗S_ijfV_ρ]ⁿ_i,j=1 is injective, S₁ =αT^ρ. Thus we showed that θ= ρ. In the same way we prove that σ =ρ and so ρ is an extreme point in CP_∞ⁿ(A, L_B(E), T⁰).

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Maria Joit¸a Tania- Luminita Costache

Department of Mathematics, Faculty of Applied Sciences,

Faculty of Chemistry, University ”Politehnica”,

University of Bucharest, Bucharest,

Romania. Romania.

e-mail: [email protected] e-mail: [email protected]

Mariana Zamfir

Department of Mathematics and Informatics Technical University of Civil Engineering Bucharest Romania.

e-mail: [email protected]