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 thethe-ory. 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):M
n(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).
Ann×nmatrix [ρij]ni,j=1 of linear maps fromAtoB can be regarded as a linear
mapρ 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, completelyn-positive linear map, extreme points
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 fromMn(A) to Mn(B).
The set of all completely n-positive linear maps from A to B is denoted by
CPn
∞(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 spaceH, is of the form [V∗T
ijΦ(·)V]ni,j=1, where Φ is a representation ofAon a Hilbert spaceK,
V ∈L(H, K) and [Tij]ni,j=1 is a positive element in Mn(Φ(A)′) (Φ(A)′ 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=pkhξ, ξik.
Given two Hilbert C∗
-modulesE and F over a C∗
-algebraA, the Banach space of all bounded module morphisms fromE toF is denoted byBA(E, F). The subset
of BA(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 LA(E) for LA(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, thenBA(E, F) =LA(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
LetEbe a HilbertC∗-module over aC∗-algebraA. The algebraic tensor product
module with the inner product defined by
* n
with respect to the norm induced by the inner producth·,·iA∗∗ is called the extension
of E by the C∗-algebra A∗∗. Moreover, E can be regarded as an A-submodule of
E⊗algA∗∗/NE, since the map ξ7→ξ⊗1 +NE from E toE⊗algB∗∗/NE is an
iso-metric inclusion. The self-dual Hilbert A∗∗-module E⊗
algB∗∗/NE
A representation of a C∗
-algebra A on a Hilbert C∗
Hilbert module over aC∗
-algebraB, is nondegenerate if for some approximate unit {eλ}λ∈ΛforA, the nets{ρii(eλ)ξ}λ∈Λ, i∈ {1, ..., n}are convergent toξ for allξ ∈E
Suen for unital completelyn-positive linear maps between unitalC∗-algebras. Thus,
we showed that any completelyn-positive linear mapρ= [ρij]i,jn =1 fromAtoLB(E)
is of the form [ρij(·)]ni,j=1=
h f
Vρ
∗
TijρΦfρ(·)fVρ
Ein
j,i=1, where Φρ is a representation of
A on a HilbertB-module Eρ,Vρ is an element inLB(E, Eρ) and Tρ=
h
Tijρin
j,i=1 is
an element in Mn
f
Φρ(A)′
with the property that fVρ
∗
TijρΦfρ(a)fVρ
E ∈ LB(E) for
all a∈A and for all i, j ∈ {1,2, ..., n}. Moreover, {Φρ(a)Vρξ;a∈ A, ξ∈ E} spans
a dense submodule of Eρ, n
P
i=1
Tiiρ = nidE❢
ρ and hT
ρ((η
k)nk=1),(ηk)nk=1iB∗∗ ≥ 0 for
all η1, ..., ηn ∈Eρ [3, Theorem 2.2]. The quadruple (Φρ, Vρ, Eρ, Tρ) 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,
Theorem 5.6] associated with the unital completely positive linear map ρe= n1
n
P
i=1
ρii. If the completelyn-positive linear mapρ= [ρij]ni,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 ∈LB∗∗(Ee) we denote by T|E the restriction of the mapT on
E.
Let ρ ∈ CPn
∞(A, LB(E)). We denote by C(ρ) the C
∗-subalgebra of M
n(LB∗∗
(Efρ)) generated by {S = [Sij]ni,j=1 ∈ Mn(LB∗∗(Efρ));fVρ ∗
SijΦfρ(a)fVρ
E ∈ LB(E),
SijΦfρ(a) =Φfρ(a)Sij,∀a∈A,∀i, j∈ {1, . . . n}}.
Lemma 1. Let S= [Sij]ni,j=1 be an element in C(ρ) such that
hS((ξi)ni=1),(ξi)ni=1iB∗∗ ≥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
ρS([aij]ni,j=1) =
h f
Vρ
∗
SijΦfρ(aij)Vfρ
Ein
i,j=1
Proof. It is not difficult to see thatρS is an n×nmatrix of continuous linear maps
from A toLB(E), the (i, j)-entry of the matrix ρS is the linear map (ρS)ij from A
toLB(E) defined by
(ρS)ij(a) = Vfρ
∗
SijΦfρ(a)fVρ
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
Γ is the isomorphism from CP∞n(A, LB(E)) ontoCP∞(A, Mn(LB(E))) defined by
Γ([ρij]ni,j=1)(a) = [ρij(a)]ni,j=1
for all a∈A.
Let aandb be two elements inA and letξ1,..., ξn ben elements inE. We have
Γ(ρS)(a∗b) ((ξi)ni=1) =
h f
Vρ
∗
SijΦfρ(a∗b)Vfρ
Ein
i,j=1((ξi)
n i=1)
= hfVρ
∗
SijΦfρ(a∗b)fVρ
in
i,j=1((ξi)
n i=1)
= hfVρ
∗
SijΦfρ(a∗)Φfρ(b)Vfρ
in
i,j=1((ξi)
n i=1)
= hfVρ
∗ f
Φρ(a)∗SijΦfρ(b)Vfρ
in
i,j=1((ξi)
n i=1)
= (Ma)∗SMb((ξi)ni=1),
where
Ma=
f
Φρ(a)Vfρ 0 ... 0
0 Φfρ(a)fVρ ... 0
· · ... ·
0 0 ... Φfρ(a)Vfρ
Leta1, . . . , am bemelements inA, letT1, . . . , Tmbemelements inMn(LB(E)) and
* n X
k,l=1
(T∗
lΓ(ρS)(a∗lak)Tk) ((ξi)ni=1),((ξi)ni=1)
+
=
n
X
k,l=1
h((Mal)
∗
SMakTk) ((ξi)
n
i=1), Tl((ξi)ni=1)i
=
n
X
k,l=1
h((Mal)
∗
SMakTk) ((ξi)
n
i=1), Tl((ξi)ni=1)iB∗∗
=
n
X
k,l=1
D
(Mal)
∗
SMakfTk
((ξi)ni=1),Tel((ξi)ni=1)
E
B∗∗
=
*
S
n
X
k=1
MakfTk !
((ξi)ni=1), n
X
l=1
MalTel !
((ξi)ni=1)
+
B∗∗
≥0
sinceS = [Sij]ni,j=1 is an element in C(ρ) such that hS((ηi)ni=1),(ηi)ni=1iB∗∗ ≥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], hTρ((η
i)ni=1),(ηi)ni=1iB∗∗ ≥ 0
for allη1, ..., ηn in Eρ, and soρTρ∈CP∞n(A, LB(E)). Moreover,ρTρ =ρ. 2. If S1 and S2 are two elements inC(ρ) such that hSk((ηi)ni=1),(ηi)ni=1iB∗∗ ≥0
for allη1, ..., 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=1iB∗∗ ≥ 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)ni=1),(ηi)ni=1iB∗∗ ≤ hS2((ηi)
n
i=1),(ηi)ni=1iB∗∗
for allη1, ..., ηn in Eρ, then ρS1 ≤ρS2.
Letρ= [ρij]ni,j=1∈CP∞n(A, LB(E)). We denote by [0, ρ] the set of all completely
n-positive linear maps θ = [θij]ni,j=1 from A to LB(E) such that θ ≤ ρ (that is,
ρ−θ∈CPn
∞(A, LB(E))) and by [0, T
ρ] the set of all elementsS = [S
ij]ni,j=1inC(ρ)
such that
h(Tρ−S) ((ηi)ni=1),(ηi)ni=1iB∗∗ ≥0
for all η1, ..., ηn inEρ.
The following theorem gives a characterization of the order relation on the set of completelyn-positive linear maps betweenC∗
Theorem 3. The mapS→ρS from[0, Tρ]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. LetS = [Sij]ni,j=1 ∈[0, Tρ] such thatρS = 0.
the KSGNS constructions associated with ρerespectively eσ [5, Theorem 5.6]. Since
we deduce thathfVρ
spans a dense submodule ofEρ. From
(ρS)ij(a) = fVρ
surjectivity of the map S→ρS it remained to show that
for alla1, ..., aninAand for allξ1, ..., ξn∈E. From this fact and taking into account
that{Φρ(a)Vρξ;a∈A and ξ∈E} spans a dense subspace in Eρ, we deduce that
hS((ηi)ni=1),(ηi)ni=1iB∗∗ ≤ hT
ρ
((ηi)ni=1),(ηi)ni=1iB∗∗
for all η1, ..., ηn inEρ, 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]i,jn =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.
Corollary 5. Let ρ = [ρij]ni,j=1 be an element in CP∞n(A, LB(E)). Then ρ is pure
if and only if [0, Tρ] ={αTρ; 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 CPn
∞(A, LB(E), T0), where T0 = [fVρ ∗
TijρVfρ]ni,j=1, the set
of all completely n-positive linear maps ρ = [ρij]i,jn =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= [ρij]ni,j=1 ∈CP∞n(A, LB(E), T0). If the mapS = [Sij]
n
i,j=1−→
[Vfρ
∗
SijfVρ]ni,j=1 from C(ρ) to Mn(LB∗∗(Ee)) is injective, then ρ is an extreme point
in the set CPn
∞(A, LB(E), T 0).
Proof. Let θ = [θij]ni,j=1, σ = [σij]ni,j=1 ∈ CP∞n(A, LB(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= [Sij1]ni,j=1 ∈[0, Tρ] such thatαθ=ρS1. Then
f
Vρ
∗
(Sij1 −αTijρ)fVρ
E = fVρ
∗
Sij1fVρ
E −αVfρ
∗
TijρfVρ
E
= (ρS1)ij(1A)−αρij(1A)
= αθij(1A)−αρij(1A)
= αTij0 −αTij0 = 0
From this fact and taking into account that fVρ
∗
(S1
ij −αT ρ ij)Vfρ
E is an element in
LB(E), we conclude that fVρ
∗
(Sij1 −αTijρ)Vfρ = 0 for alli, j ∈ {1,2, . . . n} and since
the map S = [Sij]ni,j=1 −→[fVρ
∗
SijfVρ]ni,j=1 is injective, S1 =αTρ. Thus we showed
that θ= ρ. In the same way we prove that σ =ρ and so ρ is an extreme point in
CPn
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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: mjoita@fmi.unibuc.ro e-mail: lumycos@yahoo.com
Mariana Zamfir
Department of Mathematics and Informatics Technical University of Civil Engineering Bucharest Romania.
